Journal articles on the topic 'Discretization'
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STRUCKMEIER, JENS, and KONRAD STEINER. "SECOND-ORDER SCHEME FOR THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH MAXWELLIAN MOLECULES." Mathematical Models and Methods in Applied Sciences 06, no. 01 (February 1996): 137–47. http://dx.doi.org/10.1142/s0218202596000080.
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In the standard approach particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction on the discretization parameter as well as on the differential cross-section in the case of the general Boltzmann equation. Recently, construction of an implicit particle scheme for the Boltzmann equation with Maxwellian molecules was shown. This paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second-order particle method when using an equiweighting of explicit and implicit discretization.
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GUIHÉNEUF, PIERRE-ANTOINE. "Dynamical properties of spatial discretizations of a generic homeomorphism." Ergodic Theory and Dynamical Systems 35, no. 5 (April 23, 2014): 1474–523. http://dx.doi.org/10.1017/etds.2013.108.
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This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works on well-chosen examples (such as Gambaudo and Tresser [Some difficulties generated by small sinks in the numerical study of dynamical systems: two examples. Phys. Lett. A 94(9) (1983), 412–414]) show that the dynamical behaviours of dynamical systems and of their discretizations can be quite different. We are interested in generic homeomorphisms of compact manifolds. So our aim is to tackle the following question: can the dynamical properties of a generic homeomorphism be detected on the spatial discretizations of this homeomorphism? We will prove that the dynamics of a single discretization of a generic conservative homeomorphism does not depend on the homeomorphism itself, but rather on the grid used for the discretization. Therefore, dynamical properties of a given generic conservative homeomorphism cannot be detected using a single discretization. Nevertheless, we will also prove that some dynamical features of a generic conservative homeomorphism (such as the set of the periods of all periodic points) can be read on a sequence of finer and finer discretizations.
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Mollet, Christian. "Stability of Petrov–Galerkin Discretizations: Application to the Space-Time Weak Formulation for Parabolic Evolution Problems." Computational Methods in Applied Mathematics 14, no. 2 (April 1, 2014): 231–55. http://dx.doi.org/10.1515/cmam-2014-0001.
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Abstract. This paper is concerned with the stability of Petrov–Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov–Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trial and test spaces for sufficiently regular operators. It turns out that one needs to choose different discretization levels for the trial and test spaces in order to obtain a positive lower bound for the discrete inf-sup condition which is independent of the discretization levels. In particular, we state the required number of extra layers in order to guarantee uniform boundedness of the discrete inf-sup constants explicitly. This general result will be applied to the space-time weak formulation of parabolic evolution problems as an important model example. In this regard, we consider suitable hierarchical families of discrete spaces. The results apply, e.g., for finite element discretizations as well as for wavelet discretizations. Due to the Riesz basis property, wavelet discretizations allow for optimal preconditioning independently of the grid spacing. Moreover, our predictions on the stability, especially in view of the dependence on the refinement levels w.r.t. the test and trial spaces, are underlined by numerical results. Furthermore, it can be observed that choosing the same discretization levels would, indeed, lead to stability problems.
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Fischer, Jens V., and Rudolf L. Stens. "On the Reversibility of Discretization." Mathematics 8, no. 4 (April 17, 2020): 619. http://dx.doi.org/10.3390/math8040619.
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“Discretization” usually denotes the operation of mapping continuous functions to infinite or finite sequences of discrete values. It may also mean to map the operation itself from one that operates on functions to one that operates on infinite or finite sequences. Advantageously, these two meanings coincide within the theory of generalized functions. Discretization moreover reduces to a simple multiplication. It is known, however, that multiplications may fail. In our previous studies, we determined conditions such that multiplications hold in the tempered distributions sense and, hence, corresponding discretizations exist. In this study, we determine, vice versa, conditions such that discretizations can be reversed, i.e., functions can be fully restored from their samples. The classical Whittaker-Kotel’nikov-Shannon (WKS) sampling theorem is just one particular case in one of four interwoven symbolic calculation rules deduced below.
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Bassi, Francesco, Lorenzo Botti, and Alessandro Colombo. "Agglomeration-based physical frame dG discretizations: An attempt to be mesh free." Mathematical Models and Methods in Applied Sciences 24, no. 08 (May 4, 2014): 1495–539. http://dx.doi.org/10.1142/s0218202514400028.
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In this work we consider agglomeration-based physical frame discontinuous Galerkin (dG) discretization as an effective way to increase the flexibility of high-order finite element methods. The mesh free concept is pursued in the following (broad) sense: the computational domain is still discretized using a mesh but the computational grid should not be a constraint for the finite element discretization. In particular the discrete space choice, its convergence properties, and even the complexity of solving the global system of equations resulting from the dG discretization should not be influenced by the grid choice. Physical frame dG discretization allows to obtain mesh-independent h-convergence rates. Thanks to mesh agglomeration, high-order accurate discretizations can be performed on arbitrarily coarse grids, without resorting to very high-order approximations of domain boundaries. Agglomeration-based h-multigrid techniques are the obvious choice to obtain fast and grid-independent solvers. These features (attractive for any mesh free discretization) are demonstrated in practice with numerical test cases.
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Beaude, L., R. Masson, S. Lopez, and P. Samier. "Combined face based and nodal based discretizations on hybrid meshes for non-isothermal two-phase Darcy flow problems." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 4 (July 2019): 1125–56. http://dx.doi.org/10.1051/m2an/2019014.
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In the last 20 years many discretization schemes have been developed to approximate the Darcy fluxes on polyhedral cells in heterogeneous anisotropic porous media. Among them, we can distinguished cell based approaches like the Two Point Flux Approximation (TPFA) or the Multi Point Flux Approximation (MPFA) schemes, face based approaches like the Hybrid Finite Volume (HFV) scheme belonging to the family of Hybrid Mimetic Mixed methods and nodal based discretizations like the Vertex Approximate Gradient (VAG) scheme. They all have their own drawbacks and advantages which typically depend on the type of cells and on the anisotropy of the medium. In this work, we propose a new methodology to combine the VAG and HFV discretizations on arbitrary subsets of cells or faces in order to choose the best suited scheme in different parts of the mesh. In our approach the TPFA discretization is considered as an HFV discretization for which the face unknowns can be eliminated. The coupling strategy is based on a node to face interpolation operator at the interfaces which must be chosen to ensure the consistency, the coercivity and the limit conformity properties of the combined discretization. The convergence analysis is performed in the gradient discretization framework and convergence is proved for arbitrary cell or face partitions of the mesh. For face partitions, an additional stabilisation local to the cell is required to ensure the coercivity while for cell partitions no additional stabilisation is needed. The framework preserves at the interface the discrete conservation properties of the VAG and HFV schemes with fluxes based on local to each cell transmissibility matrices. This discrete conservative form allows to naturally extend the VAG and HFV discretizations of two-phase Darcy flow models to the combined VAG–HFV schemes. The efficiency of our approach is tested for single phase and immiscible two-phase Darcy flows on 3D meshes using a combination of the HFV and VAG discretizations as well as for non-isothermal compositional liquid gas Darcy flows on a vertical 2D cross-section of the Bouillante geothermal reservoir (Guadeloupe) using a combination of the TPFA and VAG discretizations.
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Fang, Jian, Li Na Sui, and Hong Yi Jian. "Continuous Entropy Estimation with Different Unsupervised Discretization Methods." Applied Mechanics and Materials 380-384 (August 2013): 1617–20. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1617.
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In this paper, we compare and analyze the performances of nine unsupervised discretization methods, i.e., equal width, equal frequency, k-means clustering discretization, ordinal, fixed frequency, non-disjoint, proportional, weight proportional, mean value and standard deviation discretizations in the framework of continues entropy estimation based on 15 probability density distributions, i.e., Beta, Cauchy, Central Chi-Squared, Exponential, F, Gamma, Laplace, Logistic, Lognormal, Normal, Rayleigh, Student's-t, Triangular, Uniform, and Weibull distributions.
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Voulis, Igor, and Arnold Reusken. "Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints." Journal of Numerical Mathematics 27, no. 3 (September 25, 2019): 155–82. http://dx.doi.org/10.1515/jnma-2018-0013.
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Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.
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Mönkölä, Sanna. "On the Accuracy and Efficiency of Transient Spectral Element Models for Seismic Wave Problems." Advances in Mathematical Physics 2016 (2016): 1–15. http://dx.doi.org/10.1155/2016/9431583.
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This study concentrates on transient multiphysical wave problems for simulating seismic waves. The presented models cover the coupling between elastic wave equations in solid structures and acoustic wave equations in fluids. We focus especially on the accuracy and efficiency of the numerical solution based on higher-order discretizations. The spatial discretization is performed by the spectral element method. For time discretization we compare three different schemes. The efficiency of the higher-order time discretization schemes depends on several factors which we discuss by presenting numerical experiments with the fourth-order Runge-Kutta and the fourth-order Adams-Bashforth time-stepping. We generate a synthetic seismogram and demonstrate its function by a numerical simulation.
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Oyono Ngou, Polynice, and Cody Hyndman. "A Fourier Interpolation Method for Numerical Solution of FBSDEs: Global Convergence, Stability, and Higher Order Discretizations." Journal of Risk and Financial Management 15, no. 9 (August 31, 2022): 388. http://dx.doi.org/10.3390/jrfm15090388.
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The convolution method for the numerical solution of forward-backward stochastic differential equations (FBSDEs) was originally formulated using Euler time discretizations and a uniform space grid. In this paper, we utilize a tree-like spatial discretization that approximates the BSDE on the tree, so that no spatial interpolation procedure is necessary. In addition to suppressing extrapolation error, leading to a globally convergent numerical solution for the FBSDE, we provide explicit convergence rates. On this alternative grid the conditional expectations involved in the time discretization of the BSDE are computed using Fourier analysis and the fast Fourier transform (FFT) algorithm. The method is then extended to higher-order time discretizations of FBSDEs. Numerical results demonstrating convergence are presented using a commodity price model, incorporating seasonality, and forward prices.
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Maleki-Jebeli, Saeed, Mahmoud Mosavi-Mashhadi, and Mostafa Baghani. "Hybrid Isogeometric-Finite Element Discretization Applied to Stress Concentration Problems." International Journal of Applied Mechanics 10, no. 08 (September 2018): 1850081. http://dx.doi.org/10.1142/s1758825118500813.
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Isogeometric analysis (IGA) employs non-uniform rational B-splines (NURBS) or other B-spline-based variants to represent both the geometry and the field variable. Exact geometry representation and higher order global continuity (at least [Formula: see text] even on elements’ boundaries) are two favorable properties that would make IGA an appropriate discretization technique in problems with responses associated with the derivatives of the primary field variable. As a category of these problems, in this paper, 2D elastostatic problems involving stress concentration sites are analyzed with a hybrid isogeometric-finite element (IG-FE) discretization. To exploit higher order continuity of NURBS basis functions, IGA discretization is applied selectively at pre-identified locations of high displacement gradients where the stress concentration occurs. In addition, considering computational efficiency, the rest of problem domain is discretized by means of linear Lagrangian finite elements. The connection of NURBS and Lagrangian domain is carried out through employing specially devised elements [Corbett, C. J. and Sauer, R. A. [2014] “NURBS-enriched contact finite elements”, Computer Methods in Applied Mechanics and Engineering 275, 55–75]. The methodology is applied in some 2D elastostatic examples. Increasing the number of DOFs and comparing convergence of the concentrated stress value using different discretizations, it is shown that the hybrid IG-FE discretizations generally have faster and more stable convergence response compared with pure FE discretizations especially at lower DOFs.
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Geurts, Bernard J. "Analysis of errors occurring in large eddy simulation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367, no. 1899 (July 28, 2009): 2873–83. http://dx.doi.org/10.1098/rsta.2009.0001.
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We analyse the effect of second- and fourth-order accurate central finite-volume discretizations on the outcome of large eddy simulations of homogeneous, isotropic, decaying turbulence at an initial Taylor–Reynolds number Re λ =100. We determine the implicit filter that is induced by the spatial discretization and show that a higher order discretization also induces a higher order filter, i.e. a low-pass filter that keeps a wider range of flow scales virtually unchanged. The effectiveness of the implicit filtering is correlated with the optimal refinement strategy as observed in an error-landscape analysis based on Smagorinsky's subfilter model. As a point of reference, a finite-volume method that is second-order accurate for both the convective and the viscous fluxes in the Navier–Stokes equations is used. We observe that changing to a fourth-order accurate convective discretization leads to a higher value of the Smagorinsky coefficient C S required to achieve minimal total error at given resolution. Conversely, changing only the viscous flux discretization to fourth-order accuracy implies that optimal simulation results are obtained at lower values of C S . Finally, a fully fourth-order discretization yields an optimal C S that is slightly lower than the reference fully second-order method.
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Jha, Rajesh K., and Robert G. Parker. "Spatial Discretization of Axially Moving Media Vibration Problems." Journal of Vibration and Acoustics 122, no. 3 (March 1, 2000): 290–94. http://dx.doi.org/10.1115/1.1303847.
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Spatial discretization of axially moving media eigenvalue problems is examined from the perspectives of moving versus stationary system basis functions, configuration space versus state space form discretization, and subcritical versus supercritical speed convergence. The moving string eigenfunctions, which have previously been shown to give excellent discretization convergence under certain conditions, become linearly dependent and cause numerical problems as the number of terms increases. This problem does not occur in a discretization of the state space form of the eigenvalue problem, although convergence is slower, not monotonic, and not necessarily from above. Use of the moving string basis at supercritical speeds yields strikingly poor results with either the configuration or state space discretizations. The stationary system eigenfunctions provide reliable eigenvalue predictions across the range of problems examined. Because they have known exact solutions, the moving string on elastic foundation and the traveling, tensioned beam are used as illustrative examples. Many of the findings, however, apply to more complex moving media problems, including nontrivial equilibria of nonlinear models. [S0739-3717(00)02103-6]
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Tölke, Jonas, Manfred Krafczyk, Manuel Schulz, Ernst Rank, and Rodolfo Berrios. "Implicit discretization and nonuniform mesh refinement approaches for FD discretizations of LBGK Models." International Journal of Modern Physics C 09, no. 08 (December 1998): 1143–57. http://dx.doi.org/10.1142/s0129183198001059.
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After a short discussion of recent discretization techniques for the lattice-Boltzmann equations we motivate and discuss some alternative approaches using implicit, nonuniform FD discretization and mesh refinement techniques. After presenting results of a stability analysis we use an implicit approach to simulate a boundary layer test problem. The numerical results compare well to the reference solution when using strongly refined meshes. Some basic ideas for a nonuniform mesh refinement (with non-cartesian mesh topology) are introduced using the standard discretization procedure of alternating collision and propagation.
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Yavuz, Mehmet, and Ndolane Sene. "Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate." Fractal and Fractional 4, no. 3 (July 16, 2020): 35. http://dx.doi.org/10.3390/fractalfract4030035.
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In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.
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Christiansen, Snorre H., Hans Z. Munthe-Kaas, and Brynjulf Owren. "Topics in structure-preserving discretization." Acta Numerica 20 (April 28, 2011): 1–119. http://dx.doi.org/10.1017/s096249291100002x.
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In the last few decades the concepts of structure-preserving discretization, geometric integration and compatible discretizations have emerged as subfields in the numerical approximation of ordinary and partial differential equations. The article discusses certain selected topics within these areas; discretization techniques both in space and time are considered. Lie group integrators are discussed with particular focus on the application to partial differential equations, followed by a discussion of how time integrators can be designed to preserve first integrals in the differential equation using discrete gradients and discrete variational derivatives.Lie group integrators depend crucially on fast and structure-preserving algorithms for computing matrix exponentials. Preservation of domain symmetries is of particular interest in the application of Lie group integrators to PDEs. The equivariance of linear operators and Fourier transforms on non-commutative groups is used to construct fast structure-preserving algorithms for computing exponentials. The theory of Weyl groups is employed in the construction of high-order spectral element discretizations, based on multivariate Chebyshev polynomials on triangles, simplexes and simplicial complexes.The theory of mixed finite elements is developed in terms of special inverse systems of complexes of differential forms, where the inclusion of cells corresponds to pullback of forms. The theory covers, for instance, composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge–Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.
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EVANS, JOHN A., and THOMAS J. R. HUGHES. "ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS." Mathematical Models and Methods in Applied Sciences 23, no. 08 (April 22, 2013): 1421–78. http://dx.doi.org/10.1142/s0218202513500139.
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We develop divergence-conforming B-spline discretizations for the numerical solution of the steady Navier–Stokes equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. They are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of steady Navier–Stokes flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational formulations employing the new discretization scheme are automatically momentum-conservative and energy-stable. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Euler flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions subject to a smallness condition, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. These numerical experiments also suggest our discretization methodology is robust with respect to Reynolds number and more accurate than classical numerical methods for the steady Navier–Stokes equations.
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Devaud, Denis. "Petrov–Galerkin space-time hp-approximation of parabolic equations in H1/2." IMA Journal of Numerical Analysis 40, no. 4 (October 16, 2019): 2717–45. http://dx.doi.org/10.1093/imanum/drz036.
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Abstract We analyse a class of variational space-time discretizations for a broad class of initial boundary value problems for linear, parabolic evolution equations. The space-time variational formulation is based on fractional Sobolev spaces of order $1/2$ and the Riemann–Liouville derivative of order $1/2$ with respect to the temporal variable. It accommodates general, conforming space discretizations and naturally accommodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for $hp$-time semidiscretizations with an explicit expression of stable test functions given in terms of Hilbert transforms of the corresponding trial functions; inf-sup constants are independent of temporal order and the time-step sequences, allowing quasi-optimal, high-order discretizations on graded time-step sequences, and also $hp$-time discretizations. For solutions exhibiting Gevrey regularity in time and taking values in certain weighted Bochner spaces, we establish novel exponential convergence estimates in terms of $N_t$, the number of (elliptic) spatial problems to be solved. The space-time variational setting allows general space discretizations and, in particular, for spatial $hp$-FEM discretizations. We report numerical tests of the method for model problems in one space dimension with typical singular solutions in the spatial and temporal variable. $hp$-discretizations in both spatial and temporal variables are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization.
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Sidorenko, Dmitry, Sergey Danilov, Nikolay Koldunov, Patrick Scholz, and Qiang Wang. "Simple algorithms to compute meridional overturning and barotropic streamfunctions on unstructured meshes." Geoscientific Model Development 13, no. 7 (July 23, 2020): 3337–45. http://dx.doi.org/10.5194/gmd-13-3337-2020.
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Abstract. Computation of barotropic and meridional overturning streamfunctions for models formulated on unstructured meshes is commonly preceded by interpolation to a regular mesh. This operation destroys the original conservation, which can be then artificially imposed to make the computation possible. An elementary method is proposed that avoids interpolation and preserves conservation in a strict model sense. The method is described as applied to the discretization of the Finite volumE Sea ice – Ocean Model (FESOM2) on triangular meshes. It, however, is generalizable to colocated vertex-based discretization on triangular meshes and to both triangular and hexagonal C-grid discretizations.
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Patrício, Fernanda Simões, Miguel Patrício, and Higinio Ramos. "Extrapolating for attaining high precision solutions for fractional partial differential equations." Fractional Calculus and Applied Analysis 21, no. 6 (December 19, 2018): 1506–23. http://dx.doi.org/10.1515/fca-2018-0079.
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Abstract This paper aims at obtaining a high precision numerical approximation for fractional partial differential equations. This is achieved through appropriate discretizations: firstly we consider the application of shifted Legendre or Chebyshev polynomials to get a spatial discretization, followed by a temporal discretization where we use the Implicit Euler method (although any other temporal integrator could be used). Finally, the use of an extrapolation technique is considered for improving the numerical results. In this way a very accurate solution is obtained. An algorithm is presented, and numerical results are shown to demonstrate the validity of the present technique.
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Wang, Junqi, Pei Wang, and Patrick Shafto. "Efficient Discretization of Optimal Transport." Entropy 25, no. 6 (May 24, 2023): 839. http://dx.doi.org/10.3390/e25060839.
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Obtaining solutions to optimal transportation (OT) problems is typically intractable when marginal spaces are continuous. Recent research has focused on approximating continuous solutions with discretization methods based on i.i.d. sampling, and this has shown convergence as the sample size increases. However, obtaining OT solutions with large sample sizes requires intensive computation effort, which can be prohibitive in practice. In this paper, we propose an algorithm for calculating discretizations with a given number of weighted points for marginal distributions by minimizing the (entropy-regularized) Wasserstein distance and providing bounds on the performance. The results suggest that our plans are comparable to those obtained with much larger numbers of i.i.d. samples and are more efficient than existing alternatives. Moreover, we propose a local, parallelizable version of such discretizations for applications, which we demonstrate by approximating adorable images.
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Dryja, M., and M. Sarkis. "Additive Average Schwarz Methods for Discretization of Elliptic Problems with Highly Discontinuous Coefficients." Computational Methods in Applied Mathematics 10, no. 2 (2010): 164–76. http://dx.doi.org/10.2478/cmam-2010-0009.
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AbstractA second order elliptic problem with highly discontinuous coefficients has been considered. The problem is discretized by two methods: 1) continuous finite element method (FEM) and 2) composite discretization given by a continuous FEM inside the substructures and a discontinuous Galerkin method (DG) across the boundaries of these substructures. The main goal of this paper is to design and analyze parallel algorithms for the resulting discretizations. These algorithms are additive Schwarz methods (ASMs) with special coarse spaces spanned by functions that are almost piecewise constant with respect to the substructures for the first discretization and by piecewise constant functions for the second discretization. It has been established that the condition number of the preconditioned systems does not depend on the jumps of the coefficients across the substructure boundaries and outside of a thin layer along the substructure boundaries. The algorithms are very well suited for parallel computations.
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Vashi, K. M. "Quantitative Assessment of Mass Discretization in Structural Dynamic Modeling." Journal of Pressure Vessel Technology 108, no. 4 (November 1, 1986): 401–5. http://dx.doi.org/10.1115/1.3264804.
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For dynamic analysis of majority of structures, a dynamic model is developed by discretizing the distributed mass and elastic properties. For assessing the adequacy of mass discretization, several procedures are used. These procedures include a comparison of analysis with test, parametric studies using finer mass discretizations, and lump mass spacing according to a frequency-controlled span length of a simply supported beam. This paper presents a quantitative assessment of mass discretization by utilizing exact analytical solution to the discrete problem of beam and bar vibrations. The assessment examines the effect of mass discretization on the accuracy of natural frequencies, modes and participation factors. In one modeling rule, the total number of dynamic degrees of freedom is taken to be twice the number of lower frequencies to be computed with a reasonable amount of accuracy. The assessment in this paper provides support for this rule.
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Dehotin, J., and I. Braud. "Which spatial discretization for distributed hydrological models? Proposition of a methodology and illustration for medium to large-scale catchments." Hydrology and Earth System Sciences 12, no. 3 (May 23, 2008): 769–96. http://dx.doi.org/10.5194/hess-12-769-2008.
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Abstract. Distributed hydrological models are valuable tools to derive distributed estimation of water balance components or to study the impact of land-use or climate change on water resources and water quality. In these models, the choice of an appropriate spatial discretization is a crucial issue. It is obviously linked to the available data, their spatial resolution and the dominant hydrological processes. For a given catchment and a given data set, the "optimal" spatial discretization should be adapted to the modelling objectives, as the latter determine the dominant hydrological processes considered in the modelling. For small catchments, landscape heterogeneity can be represented explicitly, whereas for large catchments such fine representation is not feasible and simplification is needed. The question is thus: is it possible to design a flexible methodology to represent landscape heterogeneity efficiently, according to the problem to be solved? This methodology should allow a controlled and objective trade-off between available data, the scale of the dominant water cycle components and the modelling objectives. In this paper, we propose a general methodology for such catchment discretization. It is based on the use of nested discretizations. The first level of discretization is composed of the sub-catchments, organised by the river network topology. The sub-catchment variability can be described using a second level of discretizations, which is called hydro-landscape units. This level of discretization is only performed if it is consistent with the modelling objectives, the active hydrological processes and data availability. The hydro-landscapes take into account different geophysical factors such as topography, land-use, pedology, but also suitable hydrological discontinuities such as ditches, hedges, dams, etc. For numerical reasons these hydro-landscapes can be further subdivided into smaller elements that will constitute the modelling units (third level of discretization). The first part of the paper presents a review about catchment discretization in hydrological models from which we derived the principles of our general methodology. The second part of the paper focuses on the derivation of hydro-landscape units for medium to large scale catchments. For this sub-catchment discretization, we propose the use of principles borrowed from landscape classification. These principles are independent of the catchment size. They allow retaining suitable features required in the catchment description in order to fulfil a specific modelling objective. The method leads to unstructured and homogeneous areas within the sub-catchments, which can be used to derive modelling meshes. It avoids map smoothing by suppressing the smallest units, the role of which can be very important in hydrology, and provides a confidence map (the distance map) for the classification. The confidence map can be used for further uncertainty analysis of modelling results. The final discretization remains consistent with the resolution of input data and that of the source maps. The last part of the paper illustrates the method using available data for the upper Saône catchment in France. The interest of the method for an efficient representation of landscape heterogeneity is illustrated by a comparison with more traditional mapping approaches. Examples of possible models, which can be built on this spatial discretization, are finally given as perspectives for the work.
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Danilov, Sergey, Carolin Mehlmann, Dmitry Sidorenko, and Qiang Wang. "CD-type discretization for sea ice dynamics in FESOM version 2." Geoscientific Model Development 17, no. 6 (March 20, 2024): 2287–97. http://dx.doi.org/10.5194/gmd-17-2287-2024.
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Abstract. Two recently proposed variants of CD-type discretizations of sea ice dynamics on triangular meshes are implemented in the Finite-VolumE Sea ice–Ocean Model (FESOM version 2). The implementations use the finite element method in spherical geometry with longitude–latitude coordinates. Both are based on the edge-based sea ice velocity vectors but differ in the basis functions used to represent the velocities. The first one uses nonconforming linear (Crouzeix–Raviart) basis functions, and the second one uses continuous linear basis functions on sub-triangles obtained by splitting parent triangles into four smaller triangles. Test simulations are run to show how the performance of the new discretizations compares with the A-grid discretization using linear basis functions. Both CD discretizations are found to simulate a finer structure of linear kinematic features (LKFs). Both show some sensitivity to the representation of scalar fields (sea ice concentration and thickness). Cell-based scalars lead to a finer LKF structure for the first CD discretization, but the vertex-based scalars may be advantageous in the second case.
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LaBianca, Ane, Mette H. Mortensen, Peter Sandersen, Torben O. Sonnenborg, Karsten H. Jensen, and Jacob Kidmose. "Impact of urban geology on model simulations of shallow groundwater levels and flow paths." Hydrology and Earth System Sciences 27, no. 8 (April 19, 2023): 1645–66. http://dx.doi.org/10.5194/hess-27-1645-2023.
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Abstract. This study examines the impact of urban geology and spatial discretization on the simulation of shallow groundwater levels and flow paths at the city scale. The study uses an integrated hydrological model based on the MIKE SHE code that couples surface water and 3D groundwater simulations with a leaky sewer system. The effect of the geological configuration was analyzed by applying three geological models to an otherwise identical hydrological model. The effect of spatial discretization was examined by using two different horizontal discretizations for the hydrological models of 50 and 10 m, respectively. The impact of the geological configuration and spatial discretization was analyzed based on model calibration, simulations of high water levels, and particle tracking. The results show that a representation of the subsurface infrastructure, and near-terrain soil types, in the geological model impacts the simulation of the high water levels when the hydrological model is simulated in a 10 m discretization. This was detectable even though the difference between the geological models only occurs in 7 % of the volume of the geological models. When the hydrological model was run in a 50 m horizontal discretization, the impact of the urban geology on the high water levels was smoothed out. Results from particle tracking show that representing the subsurface infrastructure in the hydrological model changed the particles' flow paths and travel time to sinks in both the 50 and 10 m horizontal discretization of the hydrological model. It caused less recharge to deeper aquifers and increased the percentage of particles flowing to saturated-zone drains and leaky sewer pipes. In conclusion, the results indicate that even though the subsurface infrastructure and fill material only occupy a small fraction of the shallow geology, it affects the simulation of local water levels and substantially alters the flow paths. The comparison of the spatial discretization demonstrates that, to simulate this effect, the spatial discretization needs to be of a scale that represents the local variability in the shallow urban geology.
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Glasserman, Paul, and Hui Wang. "Discretization of deflated bond prices." Advances in Applied Probability 32, no. 2 (June 2000): 540–63. http://dx.doi.org/10.1239/aap/1013540178.
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This paper proposes and analyzes discrete-time approximations to a class of diffusions, with an emphasis on preserving certain important features of the continuous-time processes in the approximations. We start with multivariate diffusions having three features in particular: they are martingales, each of their components evolves within the unit interval, and the components are almost surely ordered. In the models of the term structure of interest rates that motivate our investigation, these properties have the important implications that the model is arbitrage-free and that interest rates remain positive. In practice, numerical work with such models often requires Monte Carlo simulation and thus entails replacing the original continuous-time model with a discrete-time approximation. It is desirable that the approximating processes preserve the three features of the original model just noted, though standard discretization methods do not. We introduce new discretizations based on first applying nonlinear transformations from the unit interval to the real line (in particular, the inverse normal and inverse logit), then using an Euler discretization, and finally applying a small adjustment to the drift in the Euler scheme. We verify that these methods enforce important features in the discretization with no loss in the order of convergence (weak or strong). Numerical results suggest that these methods can also yield a better approximation to the law of the continuous-time process than does a more standard discretization.
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Glasserman, Paul, and Hui Wang. "Discretization of deflated bond prices." Advances in Applied Probability 32, no. 02 (June 2000): 540–63. http://dx.doi.org/10.1017/s0001867800010077.
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This paper proposes and analyzes discrete-time approximations to a class of diffusions, with an emphasis on preserving certain important features of the continuous-time processes in the approximations. We start with multivariate diffusions having three features in particular: they are martingales, each of their components evolves within the unit interval, and the components are almost surely ordered. In the models of the term structure of interest rates that motivate our investigation, these properties have the important implications that the model is arbitrage-free and that interest rates remain positive. In practice, numerical work with such models often requires Monte Carlo simulation and thus entails replacing the original continuous-time model with a discrete-time approximation. It is desirable that the approximating processes preserve the three features of the original model just noted, though standard discretization methods do not. We introduce new discretizations based on first applying nonlinear transformations from the unit interval to the real line (in particular, the inverse normal and inverse logit), then using an Euler discretization, and finally applying a small adjustment to the drift in the Euler scheme. We verify that these methods enforce important features in the discretization with no loss in the order of convergence (weak or strong). Numerical results suggest that these methods can also yield a better approximation to the law of the continuous-time process than does a more standard discretization.
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Ronse, Christian, Loic Mazo, and Mohamed Tajine. "Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization." Mathematical Morphology - Theory and Applications 3, no. 1 (January 1, 2019): 1–28. http://dx.doi.org/10.1515/mathm-2019-0001.
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Abstract We consider Hausdorff discretization from a metric space E to a discrete subspace D, which associates to a closed subset F of E any subset S of D minimizing the Hausdorff distance between F and S; this minimum distance, called the Hausdorff radius of F and written rH(F), is bounded by the resolution of D. We call a closed set F separated if it can be partitioned into two non-empty closed subsets F1 and F2 whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of E and D (satisfied in ℝn and ℤn), we show that given a non-separated closed subset F of E, for any r > rH(F), every Hausdorff discretization of F is connected for the graph with edges linking pairs of points of D at distance at most 2r. When F is connected, this holds for r = rH(F), and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on D of the balls of radius rH(F). However, when the closed set F is separated, the Hausdorff discretizations are disconnected whenever the resolution of D is small enough. In the particular case where E = ℝn and D = ℤn with norm-based distances, we generalize our previous results for n = 2. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called coordinate-homogeneous norms, which include the Lp norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected.
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Heida, Martin, Markus Kantner, and Artur Stephan. "Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 6 (November 2021): 3017–42. http://dx.doi.org/10.1051/m2an/2021078.
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We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.
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Banjai, Lehel, and Christian Lubich. "Runge–Kutta convolution coercivity and its use for time-dependent boundary integral equations." IMA Journal of Numerical Analysis 39, no. 3 (June 7, 2018): 1134–57. http://dx.doi.org/10.1093/imanum/dry033.
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Abstract A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretizations. It is known that this coercivity property is inherited by convolution quadrature time discretization based on A-stable multistep methods, which are of order at most 2. Here we study the question as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge–Kutta methods and hence for methods of arbitrary order. As an illustration the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretization of a nonlinear boundary integral equation that originates from a nonlinear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretization.
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Sommer, Matthias, and Sebastian Reich. "Phase Space Volume Conservation under Space and Time Discretization Schemes for the Shallow-Water Equations." Monthly Weather Review 138, no. 11 (November 1, 2010): 4229–36. http://dx.doi.org/10.1175/2010mwr3323.1.
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Abstract Applying concepts of analytical mechanics to numerical discretization techniques for geophysical flows has recently been proposed. So far, mostly the role of the conservation laws for energy- and vorticity-based quantities has been discussed, but recently the conservation of phase space volume has also been addressed. This topic relates directly to questions in statistical fluid mechanics and in ensemble weather and climate forecasting. Here, phase space volume behavior of different spatial and temporal discretization schemes for the shallow-water equations on the sphere are investigated. Combinations of spatially symmetric and common temporal discretizations are compared. Furthermore, the relation between time reversibility and long-time volume averages is addressed.
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Guillot, Martin Joseph, and Steve C. McCool. "Effect of boundary condition approximation on convergence and accuracy of a finite volume discretization of the transient heat conduction equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 4 (May 5, 2015): 950–72. http://dx.doi.org/10.1108/hff-02-2014-0033.
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Purpose – The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort. Design/methodology/approach – The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions. Findings – The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method. Practical implications – The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods. Originality/value – The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods.
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Chen, Minghua, and Weihua Deng. "Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators." Communications in Computational Physics 16, no. 2 (August 2014): 516–40. http://dx.doi.org/10.4208/cicp.120713.280214a.
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AbstractHigh order discretization schemes play more important role in fractional operators than classical ones. This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones; but for fractional operators the stencils for high order schemes and low order ones are the same. Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved. Using the fractional linear multistep methods, Lubich obtains thev-th order (v <6) approximations of theα-th derivative (α >0) or integral (α <0) [Lubich, SIAM J. Math. Anal., 17, 704-719, 1986], because of the stability issue the obtained scheme can not be directly applied to the space fractional operator withαЄ (1,2) for time dependent problem. By weighting and shifting Lubich’s 2nd order discretization scheme, in [Chen & Deng, SINUM, arXiv:1304.7425] we derive a series of effective high order discretizations for space fractional derivative, called WSLD operators there. As the sequel of the previous work, we further provide new high order schemes for space fractional derivatives by weighting and shifting Lubich’s 3rd and 4th order discretizations. In particular, we prove that the obtained 4th order approximations are effective for space fractional derivatives. And the corresponding schemes are used to solve the space fractional diffusion equation with variable coefficients.
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Ullrich, Paul A., and Mark A. Taylor. "Arbitrary-Order Conservative and Consistent Remapping and a Theory of Linear Maps: Part I." Monthly Weather Review 143, no. 6 (May 28, 2015): 2419–40. http://dx.doi.org/10.1175/mwr-d-14-00343.1.
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Abstract The design of accurate, conservative, consistent, and monotone operators for remapping scalar fields between computational grids on the sphere has been a persistent issue for global modeling groups. This problem is especially pronounced when mapping between distinct discretizations (such as finite volumes or finite elements). To this end, this paper provides a novel unified mathematical framework for the development of linear remapping operators. This framework is then applied in the development of high-order conservative, consistent, and monotone linear remapping operators from a finite-element discretization to a finite-volume discretization. The resulting scheme is evaluated in the context of both idealized and operational simulations and shown to perform well for a variety of problems.
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Ansorge, R. "Iterated discretization." Numerical Functional Analysis and Optimization 17, no. 7-8 (January 1996): 691–701. http://dx.doi.org/10.1080/01630569608816719.
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Temlyakov, V. N. "Universal discretization." Journal of Complexity 47 (August 2018): 97–109. http://dx.doi.org/10.1016/j.jco.2018.02.001.
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Silver, G. L. "Operational discretization." Applied Mathematics and Computation 188, no. 1 (May 2007): 440–45. http://dx.doi.org/10.1016/j.amc.2006.10.002.
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Gogatishvili, A., and L. Pick. "Discretization and anti-discretization of rearrangement-invariant norms." Publicacions Matemàtiques 47 (July 1, 2003): 311–58. http://dx.doi.org/10.5565/publmat_47203_02.
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Dörfler, Willy, and Stefan Sauter. "A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems." Computational Methods in Applied Mathematics 13, no. 3 (July 1, 2013): 333–47. http://dx.doi.org/10.1515/cmam-2013-0008.
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Abstract. We develop a new analysis for residual-type a posteriori error estimation for a class of highly indefinite elliptic boundary value problems by considering the Helmholtz equation at high wavenumber as our model problem. We employ a classical conforming Galerkin discretization by using hp-finite elements. In [Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79 (2010), pp. 1871–1914], Melenk and Sauter introduced an hp-finite element discretization which leads to a stable and pollution-free discretization of the Helmholtz equation under a mild resolution condition which requires only degrees of freedom, where denotes the spatial dimension. In the present paper, we will introduce an a posteriori error estimator for this problem and prove its reliability and efficiency. The constants in these estimates become independent of the, possibly, high wavenumber provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the a posteriori estimates would be amplified by a factor k.
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Abedian, Rooholah. "High-Order Semi-Discrete Central-Upwind Schemes with Lax–Wendroff-Type Time Discretizations for Hamilton–Jacobi Equations." Computational Methods in Applied Mathematics 18, no. 4 (October 1, 2018): 559–80. http://dx.doi.org/10.1515/cmam-2017-0031.
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AbstractA new fifth-order, semi-discrete central-upwind scheme with a Lax–Wendroff time discretization procedure for solving Hamilton–Jacobi (HJ) equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. Unlike most of the commonly used high-order upwind schemes, the new scheme is formulated as a Godunov-type method. The new scheme is based on the flux Kurganov, Noelle and Petrova (KNP flux). The spatial discretization is based on a symmetrical weighted essentially non-oscillatory reconstruction of the derivative. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. Various numerical experiments are performed to demonstrate the accuracy and stability properties of the new method. As a result, comparing with other fifth-order schemes for HJ equations, the major advantage of the new scheme is more cost effective for certain problems while the new method exhibits smaller errors without any increase in the complexity of the computations.
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Mardal, Kent-Andre, Marie E. Rognes, and Travis B. Thompson. "Accurate discretization of poroelasticity without Darcy stability." BIT Numerical Mathematics 61, no. 3 (March 31, 2021): 941–76. http://dx.doi.org/10.1007/s10543-021-00849-0.
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AbstractIn this manuscript we focus on the question: what is the correct notion of Stokes–Biot stability? Stokes–Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes–Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes–Biot stable Euler–Galerkin discretization schemes.
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Nilsen, Halvor M., K. A. A. Lie, and Jostein R. Natvig. "Accurate Modeling of Faults by Multipoint, Mimetic, and Mixed Methods." SPE Journal 17, no. 02 (June 7, 2012): 568–79. http://dx.doi.org/10.2118/149690-pa.
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Summary The predominant way of modeling faults in industry-standard flow simulators is to introduce so-called transmissibility multipliers in the underlying two-point discretization. Although this approach provides adequate accuracy in many practical cases, two-point discretizations are only consistent for K-orthogonal grids and may introduce significant discretization errors for grids that severely depart from being K-orthogonal. Such grid-distortion errors can be avoided by lateral or vertical stair-stepping of deviated faults at the expense of errors in the geometrical fault description. In other words, modelers have the choice of either making (geometrical) errors by adapting faults to a grid that is almost K-orthogonal, or introducing discretization errors because of the lack of K-orthogonality if the grid is adapted to deviated faults. We propose a method for accurate description of faults in solvers based on a hybridized mixed or mimetic discretization, which also includes the MPFA-O method. The key idea is to represent faults as internal boundaries and calculate fault transmissibilities directly instead of using multipliers to modify grid-dependent transmissibilities. The resulting method is geology-driven and consistent for cells with planar surfaces and thereby avoids the grid errors inherent in the two-point method. We also propose a method to translate fault transmissibility multipliers into fault transmissibilities. This makes our method readily applicable to reservoir models that contain fault multipliers.
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Hansen, Eskil, and Erik Henningsson. "A Full Space-Time Convergence Order Analysis of Operator Splittings for Linear Dissipative Evolution Equations." Communications in Computational Physics 19, no. 5 (May 2016): 1302–16. http://dx.doi.org/10.4208/cicp.scpde14.22s.
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AbstractThe Douglas-Rachford and Peaceman-Rachford splitting methods are common choices for temporal discretizations of evolution equations. In this paper we combine these methods with spatial discretizations fulfilling some easily verifiable criteria. In the setting of linear dissipative evolution equations we prove optimal convergence orders, simultaneously in time and space. We apply our abstract results to dimension splitting of a 2D diffusion problem, where a finite element method is used for spatial discretization. To conclude, the convergence results are illustrated with numerical experiments.
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Shi, Zhi Cai, Yong Xiang Xia, Chao Gang Yu, and Jin Zu Zhou. "The Discretization Algorithm Based on Rough Set and its Application." Applied Mechanics and Materials 416-417 (September 2013): 1399–403. http://dx.doi.org/10.4028/www.scientific.net/amm.416-417.1399.
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The discretization is one of the most important steps for the application of Rough set theory. In this paper, we analyzed the shortcomings of the current relative works. Then we proposed a novel discretization algorithm based on information loss and gave its mathematical description. This algorithm used information loss as the measure so as to reduce the loss of the information entropy during discretizating. The algorithm was applied to different samples with the same attributes from KDDcup99 and intrusion detection systems. The experimental results show that this algorithm is sensitive to the samples only for parts of all attributes. But it dose not compromise the effect of intrusion detection and it improves the response performance of intrusion detection remarkably.
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Zlotnik, Alexander, Anna Fedchenko, and Timofey Lomonosov. "Entropy Correct Spatial Discretizations for 1D Regularized Systems of Equations for Gas Mixture Dynamics." Symmetry 14, no. 10 (October 17, 2022): 2171. http://dx.doi.org/10.3390/sym14102171.
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One-dimensional regularized systems of equations for the general (multi-velocity and multi-temperature) and one-velocity and one-temperature compressible multicomponent gas mixture dynamics are considered in the absence of chemical reactions. Two types of the regularization are taken. For the latter system, diffusion fluxes between the components of the mixture are taken into account. For both the systems, the important mixture entropy balance equations with non-negative entropy productions are valid. By generalizing a discretization constructed previously in the case of a single-component gas, we suggest new nonstandard symmetric three-point spatial discretizations for both the systems which are not only conservative in mass, momentum, and total energy but also satisfy semi-discrete counterparts of the mentioned entropy balance equations with non-negative entropy productions. Importantly, the basic discretization in the one-velocity and one-temperature case is not constructed directly but by aggregation of the discretization in the case of general mixture, and that is a new approach. In this case, the results of numerical experiments are also presented for contact problems between two different gases for initial pressure jumps up to 2500.
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Taorong Qiu, Lu Liu, Longzhen Duan, Shilin Zhou, and Tao Liu. "An Improved Discretization Algorithm for Lightning Meteorological Data Discretization." Journal of Convergence Information Technology 7, no. 23 (December 31, 2012): 518–27. http://dx.doi.org/10.4156/jcit.vol7.issue23.61.
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Steinberg, Stanly. "A Discreate Calculus with Applications of High-Order Discretizations to Boundary-Value Problems." Computational Methods in Applied Mathematics 4, no. 2 (2004): 228–61. http://dx.doi.org/10.2478/cmam-2004-0014.
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AbstractWe develop a discrete analog of the differential calculus and use this to develop arbitrarily high-order approximations to Sturm–Liouville boundary-value problems with general mixed boundary conditions. An important feature of the method is that we obtain a discrete exact analog of the energy inequality for the continuum boundary-value problem. As a consequence, the discrete and continuum problems have exactly the same solvability conditions. We call such discretizations mimetic. Numerical test confirm the accuracy of the discretization. We prove the solvability and convergence for the discrete boundary-value problem modulo the invertibility of a matrix that appears in the discretization being positive definite. Numerical experiments indicate that the spectrum of this matrix is real, greater than one, and bounded above by a number smaller than three.
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Hu, Xing-Biao, and Guo-Fu Yu. "Integrable semi-discretizations and full-discretization of the two-dimensional Leznov lattice." Journal of Difference Equations and Applications 15, no. 3 (March 2009): 233–52. http://dx.doi.org/10.1080/10236190802101695.
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Ronse, Christian, and Mohamed Tajine. "Hausdorff Discretization for Cellular Distances and Its Relation to Coverand Supercover Discretizations." Journal of Visual Communication and Image Representation 12, no. 2 (June 2001): 169–200. http://dx.doi.org/10.1006/jvci.2000.0458.
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