Journal articles: 'High-order discretization methods'

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Gottlieb, Sigal, Chi-Wang Shu, and Eitan Tadmor. "Strong Stability-Preserving High-Order Time Discretization Methods." SIAM Review 43, no. 1 (January 2001): 89–112. http://dx.doi.org/10.1137/s003614450036757x.

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Takács, Bálint, and Yiannis Hadjimichael. "High order discretization methods for spatial-dependent epidemic models." Mathematics and Computers in Simulation 198 (August 2022): 211–36. http://dx.doi.org/10.1016/j.matcom.2022.02.021.

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Filbet, Francis, and Charles Prouveur. "High order time discretization for backward semi-Lagrangian methods." Journal of Computational and Applied Mathematics 303 (September 2016): 171–88. http://dx.doi.org/10.1016/j.cam.2016.01.024.

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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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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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Bose, Mahua, and Kalyani Mali. "High Order Time Series Forecasting using Fuzzy Discretization." International Journal of Fuzzy System Applications 5, no. 4 (October 2016): 147–64. http://dx.doi.org/10.4018/ijfsa.2016100107.

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In recent years, various methods for forecasting fuzzy time series have been presented in different areas, such as stock price, enrollments, weather, production etc. It is observed that in most of the cases, static length of intervals/equal length of interval has been used. Length of the interval has significant role on forecasting accuracy. The objective of this present study is to incorporate the idea of fuzzy discretization into interval creation and examine the effect of positional information of elements within a group or interval to the forecast. This idea outperforms the existing high order forecast methods using fixed interval. Experiments are carried on three datasets including Lahi production data, enrollment data and rainfall data which deal with a lot of uncertainty.

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Yi, Tae-Hyeong, and Francis X. Giraldo. "Vertical Discretization for a Nonhydrostatic Atmospheric Model Based on High-Order Spectral Elements." Monthly Weather Review 148, no. 1 (December 27, 2019): 415–36. http://dx.doi.org/10.1175/mwr-d-18-0283.1.

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Abstract This study addresses the treatment of vertical discretization for a high-order, spectral element model of a nonhydrostatic atmosphere in which the governing equations of the model are separated into horizontal and vertical components by introducing a coordinate transformation, so that one can use different orders and types of approximations in both directions. The vertical terms of the decoupled governing equations are discretized using finite elements based on either Lagrange or basis-spline polynomial functions in the sigma coordinate, while maintaining the high-order spectral elements for the discretization of the horizontal terms. This leads to the fact that the high-order model of spectral elements with a nonuniform grid, interpolated within an element, can be easily accommodated with existing physical parameterizations. Idealized tests are performed to compare the accuracy and efficiency of the vertical discretization methods, in addition to the central finite differences, with those of the standard high-order spectral element approach. Our results show, through all the test cases, that the finite element with the cubic basis-spline function is more accurate than the other vertical discretization methods at moderate computational cost. Furthermore, grid dependency studies in the tests with and without orography indicate that the convergence rate of the vertical discretization methods is lower than the expected level of discretization accuracy, especially in the Schär mountain test, which yields approximately first-order convergence.

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Darrigrand, E., L. Gatard, and K. Mer-Nkonga. "High order boundary integral methods forMaxwell's equations using Microlocal Discretization and Fast Multipole Methods." PAMM 7, no. 1 (December 2007): 1022705–6. http://dx.doi.org/10.1002/pamm.200700332.

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May, Georg, Koen Devesse, Ajay Rangarajan, and Thierry Magin. "A Hybridized Discontinuous Galerkin Solver for High-Speed Compressible Flow." Aerospace 8, no. 11 (October 28, 2021): 322. http://dx.doi.org/10.3390/aerospace8110322.

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We present a high-order consistent compressible flow solver, based on a hybridized discontinuous Galerkin (HDG) discretization, for applications covering subsonic to hypersonic flow. In the context of high-order discretization, this broad range of applications presents unique difficulty, especially at the high-Mach number end. For instance, if a high-order discretization is to efficiently resolve shock and shear layers, it is imperative to use adaptive methods. Furthermore, high-Enthalpy flow requires non-trivial physical modeling. The aim of the present paper is to present the key enabling technologies. We discuss efficient discretization methods, including anisotropic metric-based adaptation, as well as the implementation of flexible modeling using object-oriented programming and algorithmic differentiation. We present initial verification and validation test cases focusing on external aerodynamics.

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Chen, Jing-Bo. "Modeling the scalar wave equation with Nyström methods." GEOPHYSICS 71, no. 5 (September 2006): T151—T158. http://dx.doi.org/10.1190/1.2335505.

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High-accuracy numerical schemes for modeling of the scalar wave equation based on Nyström methods are developed in this paper. Space is discretized by using the pseudospectral algorithm. For the time discretization, Nyström methods are used. A fourth-order symplectic Nyström method with pseudospectral spatial discretization is presented. This scheme is compared with a commonly used second-order scheme and a fourth-order nonsymplectic Nyström method. For a typical time-step size, the second-order scheme exhibits spatial dispersion errors for long-time simulations, while both fourth-order schemes do not suffer from these errors. Numerical comparisons show that the fourth-order symplectic algorithm is more accurate than the fourth-order nonsymplectic one. The capability of the symplectic Nyström method in approximately preserving the discrete energy for long-time simulations is also demonstrated.

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Spiteri, Raymond J., and Steven J. Ruuth. "A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods." SIAM Journal on Numerical Analysis 40, no. 2 (January 2002): 469–91. http://dx.doi.org/10.1137/s0036142901389025.

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Li, Yu, Wei Shan, and Yanming Zhang. "High-Order Dissipation-Preserving Methods for Nonlinear Fractional Generalized Wave Equations." Fractal and Fractional 6, no. 5 (May 10, 2022): 264. http://dx.doi.org/10.3390/fractalfract6050264.

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In this paper, we construct and analyze a class of high-order and dissipation-preserving schemes for the nonlinear space fractional generalized wave equations by the newly introduced scalar auxiliary variable (SAV) technique. The system is discretized by a fourth-order Riesz fractional difference operator in spatial discretization and the collocation methods in the temporal direction. Not only can the present method achieve fourth-order accuracy in the spatial direction and arbitrarily high-order accuracy in the temporal direction, but it also has long-time computing stability. Then, the unconditional discrete energy dissipation law of the present numerical schemes is proved. Finally, some numerical experiments are provided to certify the efficiency and the structure-preserving properties of the proposed schemes.

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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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Li, Zhiqiang, and Yubin Yan. "Error estimates of high-order numerical methods for solving time fractional partial differential equations." Fractional Calculus and Applied Analysis 21, no. 3 (June 26, 2018): 746–74. http://dx.doi.org/10.1515/fca-2018-0039.

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Abstract Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. [21] for solving time fractional partial differential equation. We prove that this method has the convergence order O(τ3−α) for all α ∈ (0, 1) when the first and second derivatives of the solution are vanish at t = 0, where τ is the time step size and α is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. [21]. We show that this new method also has the convergence order O(τ3−α) for all α ∈ (0, 1). The proofs of the error estimates are based on the energy method developed recently by Lv and Xu [26]. We also consider the space discretization by using the finite element method. Error estimates with convergence order O(τ3−α + h2) are proved in the fully discrete case, where h is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.

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Jund, Sébastien, Stéphanie Salmon, and Eric Sonnendrücker. "High-Order Low Dissipation Conforming Finite-Element Discretization of the Maxwell Equations." Communications in Computational Physics 11, no. 3 (March 2012): 863–92. http://dx.doi.org/10.4208/cicp.100310.230511a.

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AbstractIn this paper, we study high order discretization methods for solving the Maxwell equations on hybrid triangle-quad meshes. We have developed high order finite edge element methods coupled with different high order time schemes and we compare results and efficiency for several schemes. We introduce in particular a class of simple high order low dissipation time schemes based on a modified Taylor expansion.

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Swarnakar, Jaydeep, Prasanta Sarkar, and Lairenlakpam Joyprakash Singh. "A unified direct approach for discretizing fractional-order differentiator in delta-domain." International Journal of Modeling, Simulation, and Scientific Computing 09, no. 06 (December 2018): 1850055. http://dx.doi.org/10.1142/s1793962318500551.

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Fractional-order differentiator is a principal component of the fractional-order controller. Discretization of fractional-order differentiator is essential to implement the fractional-order controller digitally. Discretization methods generally include indirect approach and direct approach to find the discrete-time approximation of fractional-order differentiator in the [Formula: see text]-domain as evident from the existing literature. In this paper, a direct approach is proposed for discretization of fractional-order differentiator in delta-domain instead of the conventional [Formula: see text]-domain as the delta operator unifies both analog system and digital system together at a high sampling frequency. The discretization of fractional-order differentiator is accomplished in two stages. In the first stage, the generating function is framed by reformulating delta operator using trapezoidal rule or Tustin approximation and in the next stage, the fractional-order differentiator has been approximated by expanding the generating function using continued fraction expansion method. The proposed method has been compared with two well-known direct discretization methods taken from the existing literature. Two examples are presented in this context to show the efficacy of the proposed discretization method using simulation results obtained from MATLAB.

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Yan, Zhenghu, Changfu Zhang, Jianli Jia, Baoji Ma, Xinguang Jiang, Dong Wang, and Tingguo Zhu. "High-order semi-discretization methods for stability analysis in milling based on precise integration." Precision Engineering 73 (January 2022): 71–92. http://dx.doi.org/10.1016/j.precisioneng.2021.08.024.

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Dong, Gang, Zhichang Guo, and Wenjuan Yao. "Numerical methods for time-fractional convection-diffusion problems with high-order accuracy." Open Mathematics 19, no. 1 (January 1, 2021): 782–802. http://dx.doi.org/10.1515/math-2021-0036.

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Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

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Lehrenfeld, Christoph, and Arnold Reusken. "L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems." Journal of Numerical Mathematics 27, no. 2 (June 26, 2019): 85–99. http://dx.doi.org/10.1515/jnma-2017-0109.

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AbstractIn the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken,IMA J. Numer. Anal.38(2018), No. 3, 1351–1387] ana priorierror analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in theH1-norm. In this paper we extend this analysis and derive optimalL2-error bounds.

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FENG, QUANDONG, JINGFANG HUANG, NINGMING NIE, ZAIJIU SHANG, and YIFA TANG. "IMPLEMENTING ARBITRARILY HIGH-ORDER SYMPLECTIC METHODS VIA KRYLOV DEFERRED CORRECTION TECHNIQUE." International Journal of Modeling, Simulation, and Scientific Computing 01, no. 02 (June 2010): 277–301. http://dx.doi.org/10.1142/s1793962310000171.

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In this paper, an efficient numerical procedure is presented to implement the Gaussian Runge–Kutta (GRK) methods (also called Gauss methods). The GRK technique first discretizes each marching step of the initial value problem using collocation formulations based on Gaussian quadrature. As is well known, it preserves the geometric structures of Hamiltonian systems. Existing analysis shows that the GRK discretization with s nodes is of order 2s, A-stable, B-stable, symplectic and symmetric, and hence "optimal" for solving initial value problems of general ordinary differential equations (ODEs). However, as the unknowns at different collocation points are coupled in the discretized system, direct solution of the resulting algebraic equations is in general inefficient. Instead, we use the Krylov deferred correction (KDC) method in which the spectral deferred correction (SDC) scheme is applied as a preconditioner to decouple the original system, and the resulting preconditioned nonlinear system is solved efficiently using Newton–Krylov schemes such as Newton–GMRES method. The KDC accelerated GRK methods have been applied to several Hamiltonian systems and preliminary numerical results are presented to show the accuracy, stability, and efficiency features of these methods for different accuracy requirements in short- and long-time simulations.

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Zhang, Ruming. "High Order Complex Contour Discretization Methods to Simulate Scattering Problems in Locally Perturbed Periodic Waveguides." SIAM Journal on Scientific Computing 44, no. 5 (September 29, 2022): B1257—B1281. http://dx.doi.org/10.1137/21m1421532.

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Kirby, Robert M., Zohar Yosibash, and George Em Karniadakis. "Towards stable coupling methods for high-order discretization of fluid–structure interaction: Algorithms and observations." Journal of Computational Physics 223, no. 2 (May 2007): 489–518. http://dx.doi.org/10.1016/j.jcp.2006.09.015.

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Ahmed, Fareed, Faheem Ahmed, and Yong Yang. "Numerical Solution of Compressible Euler Equations by High Order Nodal Discontinuous Galerkin Method." Applied Mechanics and Materials 392 (September 2013): 165–69. http://dx.doi.org/10.4028/www.scientific.net/amm.392.165.

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In this paper we present a robust, high order method for numerical solution of compressible Euler Equations of the gas dynamics. Euler equations are hyperbolic in nature. Our scheme is based on Nodal Discontinuous Galerkin Finite Element Method (NDG-FEM). This method combines mainly two key ideas which are based on the finite volume and finite element methods. In this method, we employ Discontinuous Galerkin (DG) technique for finite element space discretization by discontinuous approximations. Whereas, for temporal discretization, we used explicit Runge-Kutta (ERK) method. In order to compute fluxes at element interfaces, we have used Roe Approximate scheme. We used filter to remove spurious oscillations near the shock waves. Numerical predictions for Shock tube problem (SOD) are presented and compared with exact solution at different polynomial order and mesh sizes. Results show the suitability of DG method for modeling gas dynamics equations and effectiveness of high order approximations.

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Guerra, Jorge E., and Paul A. Ullrich. "A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models." Geoscientific Model Development 9, no. 5 (June 1, 2016): 2007–29. http://dx.doi.org/10.5194/gmd-9-2007-2016.

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Abstract. Atmospheric modeling systems require economical methods to solve the non-hydrostatic Euler equations. Two major differences between hydrostatic models and a full non-hydrostatic description lies in the vertical velocity tendency and numerical stiffness associated with sound waves. In this work we introduce a new arbitrary-order vertical discretization entitled the staggered nodal finite-element method (SNFEM). Our method uses a generalized discrete derivative that consistently combines the discontinuous Galerkin and spectral element methods on a staggered grid. Our combined method leverages the accurate wave propagation and conservation properties of spectral elements with staggered methods that eliminate stationary (2Δx) modes. Furthermore, high-order accuracy also eliminates the need for a reference state to maintain hydrostatic balance. In this work we demonstrate the use of high vertical order as a means of improving simulation quality at relatively coarse resolution. We choose a test case suite that spans the range of atmospheric flows from predominantly hydrostatic to nonlinear in the large-eddy regime. Our results show that there is a distinct benefit in using the high-order vertical coordinate at low resolutions with the same robust properties as the low-order alternative.

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de la Luz Sosa, Jose, Daniel Olvera-Trejo, Gorka Urbikain, Oscar Martinez-Romero, Alex Elías-Zúñiga, and Luis Norberto López de Lacalle. "Uncharted Stable Peninsula for Multivariable Milling Tools by High-Order Homotopy Perturbation Method." Applied Sciences 10, no. 21 (November 6, 2020): 7869. http://dx.doi.org/10.3390/app10217869.

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In this work, a new method for solving a delay differential equation (DDE) with multiple delays is presented by using second- and third-order polynomials to approximate the delayed terms using the enhanced homotopy perturbation method (EMHPM). To study the proposed method performance in terms of convergency and computational cost in comparison with the first-order EMHPM, semi-discretization and full-discretization methods, a delay differential equation that model the cutting milling operation process was used. To further assess the accuracy of the proposed method, a milling process with a multivariable cutter is examined in order to find the stability boundaries. Then, theoretical predictions are computed from the corresponding DDE finding uncharted stable zones at high axial depths of cut. Time-domain simulations based on continuous wavelet transform (CWT) scalograms, power spectral density (PSD) charts and Poincaré maps (PM) were employed to validate the stability lobes found by using the third-order EMHPM for the multivariable tool.

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Charest, Marc R. J., Clinton P. T. Groth, and Pierre Q. Gauthier. "A High-Order Central ENO Finite-Volume Scheme for Three-Dimensional Low-Speed Viscous Flows on Unstructured Mesh." Communications in Computational Physics 17, no. 3 (March 2015): 615–56. http://dx.doi.org/10.4208/cicp.091013.281114a.

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AbstractHigh-order discretization techniques offer the potential to significantly reduce the computational costs necessary to obtain accurate predictions when compared to lower-order methods. However, efficient and universally-applicable high-order discretizations remain somewhat illusive, especially for more arbitrary unstructured meshes and for incompressible/low-speed flows. A novel, high-order, central essentially non-oscillatory (CENO), cell-centered, finite-volume scheme is proposed for the solution of the conservation equations of viscous, incompressible flows on three-dimensional unstructured meshes. Similar to finite element methods, coordinate transformations are used to maintain the scheme’s order of accuracy even when dealing with arbitrarily-shaped cells having non-planar faces. The proposed scheme is applied to the pseudo-compressibility formulation of the steady and unsteady Navier-Stokes equations and the resulting discretized equations are solved with a parallel implicit Newton-Krylov algorithm. For unsteady flows, a dual-time stepping approach is adopted and the resulting temporal derivatives are discretized using the family of high-order backward difference formulas (BDF). The proposed finite-volume scheme for fully unstructured mesh is demonstrated to provide both fast and accurate solutions for steady and unsteady viscous flows.

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Karouma, Abdulrahman, Truong Nguyen-Ba, Thierry Giordano, and Rémi Vaillancourt. "A new class of efficient one-step contractivity preserving high-order time discretization methods of order 5 to 14." Numerical Algorithms 79, no. 1 (November 22, 2017): 251–80. http://dx.doi.org/10.1007/s11075-017-0436-4.

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Hemker, P. W., W. Hoffman, and M. H. Van Raalte. "Discontinuous Galerkin Discretization with Embedded Boundary Conditions." Computational Methods in Applied Mathematics 3, no. 1 (2003): 135–58. http://dx.doi.org/10.2478/cmam-2003-0010.

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AbstractThe purpose of this paper is to introduce discretization methods of discontinuous Galerkin type for solving second-order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary. Starting with the Lagrange multiplier formulation for the boundary conditions, we derive variational forms for the discretization of 2-D elliptic problems with embedded Dirichlet boundary conditions. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DGdiscretization is adapted in cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection-dominated boundary-value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of cubic polynomials, the boundary condition treatment appears quite effective in handling such complex situations.

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Alonso-Mallo, Isaías, and Ana M. Portillo. "Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods." Mathematics 9, no. 10 (May 14, 2021): 1113. http://dx.doi.org/10.3390/math9101113.

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The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

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Weng, Jiong, Xiaojing Liu, Youhe Zhou, and Jizeng Wang. "A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations." Mathematics 9, no. 22 (November 19, 2021): 2957. http://dx.doi.org/10.3390/math9222957.

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A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

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Hao, S., A. H. Barnett, P. G. Martinsson, and P. Young. "High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane." Advances in Computational Mathematics 40, no. 1 (June 8, 2013): 245–72. http://dx.doi.org/10.1007/s10444-013-9306-3.

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Guo, Ruihan, Francis Filbet, and Yan Xu. "Efficient High Order Semi-implicit Time Discretization and Local Discontinuous Galerkin Methods for Highly Nonlinear PDEs." Journal of Scientific Computing 68, no. 3 (February 8, 2016): 1029–54. http://dx.doi.org/10.1007/s10915-016-0170-4.

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Gregersen, Brent A., and Darrin M. York. "High-order discretization schemes for biochemical applications of boundary element solvation and variational electrostatic projection methods." Journal of Chemical Physics 122, no. 19 (May 15, 2005): 194110. http://dx.doi.org/10.1063/1.1899146.

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Chen, Shanqin. "Krylov SSP Integrating Factor Runge–Kutta WENO Methods." Mathematics 9, no. 13 (June 24, 2021): 1483. http://dx.doi.org/10.3390/math9131483.

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Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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Armenta, R. B., and C. D. Sarris. "A General Procedure for Introducing Structured Nonorthogonal Discretization Grids Into High-Order Finite-Difference Time-Domain Methods." IEEE Transactions on Microwave Theory and Techniques 58, no. 7 (July 2010): 1818–29. http://dx.doi.org/10.1109/tmtt.2010.2049921.

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Abouali, Mohammad, and Jose E. Castillo. "Solving Poisson equation with Robin boundary condition on a curvilinear mesh using high order mimetic discretization methods." Mathematics and Computers in Simulation 139 (September 2017): 23–36. http://dx.doi.org/10.1016/j.matcom.2014.10.004.

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Jia, Jun, Judith C. Hill, Katherine J. Evans, George I. Fann, and Mark A. Taylor. "A Spectral Deferred Correction Method Applied to the Shallow Water Equations on a Sphere." Monthly Weather Review 141, no. 10 (September 25, 2013): 3435–49. http://dx.doi.org/10.1175/mwr-d-12-00048.1.

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Abstract Although significant gains have been made in achieving high-order spatial accuracy in global climate modeling, less attention has been given to the impact imposed by low-order temporal discretizations. For long-time simulations, the error accumulation can be significant, indicating a need for higher-order temporal accuracy. A spectral deferred correction (SDC) method is demonstrated of even order, with second- to eighth-order accuracy and A-stability for the temporal discretization of the shallow water equations within the spectral-element High-Order Methods Modeling Environment (HOMME). Because this method is stable and of high order, larger time-step sizes can be taken while still yielding accurate long-time simulations. The spectral deferred correction method has been tested on a suite of popular benchmark problems for the shallow water equations, and when compared to the explicit leapfrog, five-stage Runge–Kutta, and fully implicit (FI) second-order backward differentiation formula (BDF2) time-integration methods, it achieves higher accuracy for the same or larger time-step sizes. One of the benchmark problems, the linear advection of a Gaussian bell height anomaly, is extended to run for longer time periods to mimic climate-length simulations, and the leapfrog integration method exhibited visible degradation for climate length simulations whereas the second-order and higher methods did not. When integrated with higher-order SDC methods, a suite of shallow water test problems is able to replicate the test with better accuracy.

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Matringe, Sebastien F., Ruben Juanes, and Hamdi A. Tchelepi. "Tracing Streamlines on Unstructured Grids From Finite Volume Discretizations." SPE Journal 13, no. 04 (December 1, 2008): 423–31. http://dx.doi.org/10.2118/103295-pa.

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Summary The accuracy of streamline reservoir simulations depends strongly on the quality of the velocity field and the accuracy of the streamline tracing method. For problems described on complex grids (e.g., corner-point geometry or fully unstructured grids) with full-tensor permeabilities, advanced discretization methods, such as the family of multipoint flux approximation (MPFA) schemes, are necessary to obtain an accurate representation of the fluxes across control volume faces. These fluxes are then interpolated to define the velocity field within each control volume, which is then used to trace the streamlines. Existing methods for the interpolation of the velocity field and integration of the streamlines do not preserve the accuracy of the fluxes computed by MPFA discretizations. Here we propose a method for the reconstruction of the velocity field with high-order accuracy from the fluxes provided by MPFA discretization schemes. This reconstruction relies on a correspondence between the MPFA fluxes and the degrees of freedom of a mixed finite-element method (MFEM) based on the first-order Brezzi-Douglas-Marini space. This link between the finite-volume and finite-element methods allows the use of flux reconstruction and streamline tracing techniques developed previously by the authors for mixed finite elements. After a detailed description of our streamline tracing method, we study its accuracy and efficiency using challenging test cases. Introduction The next-generation reservoir simulators will be unstructured. Several research groups throughout the industry are now developing a new breed of reservoir simulators to replace the current industry standards. One of the main advances offered by these next generation simulators is their ability to support unstructured or, at least, strongly distorted grids populated with full-tensor permeabilities. The constant evolution of reservoir modeling techniques provides an increasingly realistic description of the geological features of petroleum reservoirs. To discretize the complex geometries of geocellular models, unstructured grids seem to be a natural choice. Their inherent flexibility permits the precise description of faults, flow barriers, trapping structures, etc. Obtaining a similar accuracy with more traditional structured grids, if at all possible, would require an overwhelming number of gridblocks. However, the added flexibility of unstructured grids comes with a cost. To accurately resolve the full-tensor permeabilities or the grid distortion, a two-point flux approximation (TPFA) approach, such as that of classical finite difference (FD) methods is not sufficient. The size of the discretization stencil needs to be increased to include more pressure points in the computation of the fluxes through control volume edges. To this end, multipoint flux approximation (MPFA) methods have been developed and applied quite successfully (Aavatsmark et al. 1996; Verma and Aziz 1997; Edwards and Rogers 1998; Aavatsmark et al. 1998b; Aavatsmark et al. 1998c; Aavatsmark et al. 1998a; Edwards 2002; Lee et al. 2002a; Lee et al. 2002b). In this paper, we interpret finite volume discretizations as MFEM for which streamline tracing methods have already been developed (Matringe et al. 2006; Matringe et al. 2007b; Juanes and Matringe In Press). This approach provides a natural way of reconstructing velocity fields from TPFA or MPFA fluxes. For finite difference or TPFA discretizations, the proposed interpretation provides mathematical justification for Pollock's method (Pollock 1988) and some of its extensions to distorted grids (Cordes and Kinzelbach 1992; Prévost et al. 2002; Hægland et al. 2007; Jimenez et al. 2007). For MPFA, our approach provides a high-order streamline tracing algorithm that takes full advantage of the flux information from the MPFA discretization.

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Lunet, Thibaut, Christine Lac, Franck Auguste, Florian Visentin, Valéry Masson, and Juan Escobar. "Combination of WENO and Explicit Runge–Kutta Methods for Wind Transport in the Meso-NH Model." Monthly Weather Review 145, no. 9 (September 2017): 3817–38. http://dx.doi.org/10.1175/mwr-d-16-0343.1.

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This paper investigates the use of the weighted essentially nonoscillatory (WENO) space discretization methods of third and fifth order for momentum transport in the Meso-NH meteorological model, and their association with explicit Runge–Kutta (ERK) methods, with the specific purpose of finding an optimal combination in terms of wall-clock time to solution. A linear stability analysis using von Neumann theory is first conducted that considers six different ERK time integration methods. A new graphical representation of linear stability is proposed, which allows a first discrimination between the ERK methods. The theoretical analysis is then completed by tests on numerical problems of increasing complexity (linear advection of high wind gradient, orographic waves, density current, large eddy simulation of fog, and windstorm simulation), using a fourth-order-centered scheme as a reference basis. The five-stage third-order and fourth-order ERK combinations appear as the time integration methods of choice for coupling with WENO schemes in terms of stability. An explicit time-splitting method added to the ERK temporal scheme for WENO improves the stability properties slightly more. When the spatial discretizations are compared, WENO schemes present the main advantage of maintaining stable, nonoscillatory transitions with sharp discontinuities, but WENO third order is excessively damping, while WENO fifth order provides better accuracy. Finally, WENO fifth order combined with the ERK method makes the whole physics of the model 3 times faster compared to the classical fourth-order centered scheme associated with the leapfrog temporal scheme.

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Zhang, Yang, Kenan Liu, Wuyun Zhao, Wei Zhang, and Fei Dai. "Stability Analysis for Milling Process with Variable Pitch and Variable Helix Tools by High-Order Full-Discretization Methods." Mathematical Problems in Engineering 2020 (July 26, 2020): 1–14. http://dx.doi.org/10.1155/2020/4517969.

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Chatter is one of the significant limitations in the milling process, which may cause poor surface quality, reduced productivity, and accelerated tool wear. Variable pitch and variable helix tools can be used to suppress regenerative chatter. This study extends the high-order full-discretization methods (FDMs) to predict the stability of milling with variable pitch and variable helix tools. The time-periodic delay-differential equation (DDE) with multiple delays is used to model the milling process using variable pitch and variable helix tools. Then, the DDE with multiple delays is reexpressed by the state-space equation. Meanwhile, the spindle rotational period is divided into many small-time intervals, and the state space equation is integrated on the small-time interval. Then, the high-order interpolation polynomials are used to approximate the state term, and the weights related to the time delay are employed to approximate the time-delay term. The second-order, third-order, and fourth-order extended FDMs (2nd EFDM, 3rd EFDM, and 4th EFDM) are compared with the benchmark in terms of the rate of convergence. It is found that the 2nd EFDM, 3rd EFDM, and 4th EFDM converge faster than the benchmark method. The difference between the curves obtained by different EFDMs and the reference curve is very small. There is no need to extend hypersecond FDMs to analyze the stability of milling with variable pitch and variable helix tools.

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Simone, Antonio, and Stig Hestholm. "Instabilities in applying absorbing boundary conditions to high‐order seismic modeling algorithms." GEOPHYSICS 63, no. 3 (May 1998): 1017–23. http://dx.doi.org/10.1190/1.1444379.

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The problem of artificial reflections from grid boundaries in the numerical discretization of elastic and acoustic wave equations has long plagued geophysicists. Even if modern computers have made it possible to extend the synthetics over more wavelengths (equivalent to larger propagation distances), efficient absorption methods are still needed to minimize interference from unwanted reflections from the numerical grid boundaries. In this study, we examine applicabilities and stabilities of the optimal absorbing boundary condition (OABC) of Peng and Toksöz (1994, 1995) for 2-D and 3-D acoustic and elastic wave modeling. As a basis for comparison, we use exponential damping (ED) (Cerjan et. al., 1985), in which velocities and stresses are multiplied by progressively decreasing terms when approaching the boundaries of the numerical grid.

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Yang, Jun, Wei Cai, and Xiaoping Wu. "A High-Order Time Domain Discontinuous Galerkin Method with Orthogonal Tetrahedral Basis for Electromagnetic Simulations in 3-D Heterogeneous Conductive Media." Communications in Computational Physics 21, no. 4 (March 8, 2017): 1065–89. http://dx.doi.org/10.4208/cicp.oa-2016-0089.

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AbstractWe present a high-order discontinuous Galerkin (DG) method for the time domain Maxwell's equations in three-dimensional heterogeneous media. New hierarchical orthonormal basis functions on unstructured tetrahedral meshes are used for spatial discretization while Runge-Kutta methods for time discretization. A uniaxial perfectly matched layer (UPML) is employed to terminate the computational domain. Exponential convergence with respect to the order of the basis functions is observed and large parallel speedup is obtained for a plane-wave scattering model. The rapid decay of the out-going wave in the UPML is shown in a dipole radiation simulation. Moreover, the low frequency electromagnetic fields excited by a horizontal electric dipole (HED) and a vertical magnetic dipole (VMD) over a layered conductive half-space and a high frequency ground penetrating radar (GPR) detection for an underground structure are investigated, showing the high accuracy and broadband simulation capability of the proposed method.

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Ameli, A. A., and M. J. Abedini. "Performance assessment of low-order versus high-order numerical schemes in the numerical simulation of aquifer flow." Hydrology Research 47, no. 6 (January 6, 2016): 1104–15. http://dx.doi.org/10.2166/nh.2016.148.

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Numerical methods have been widely used to simulate transient groundwater flow induced by pumping wells in geometrically and mathematically complex systems. However, flow and transport simulation using low-order numerical methods can be computationally expensive with a low rate of convergence in multi-scale problems where fine spatial discretization is required to ensure stability and desirable accuracy (for instance, close to a pumping well). Numerical approaches based on high-order test functions may better emulate the global behavior of parabolic and/or elliptic groundwater governing equations with and without the presence of pumping well(s). Here, we assess the appropriateness of high-order differential quadrature method (DQM) and radial basis function (RBF)-DQM approaches compared to low-order finite difference and finite element methods. This assessment is carried out using the exact analytical solution by Theis and observed head data as benchmarks. Numerical results show that high-order DQM and RBF-DQM are more efficient schemes compared to low-order numerical methods in the simulation of 1-D axisymmetric transient flow induced by a pumping well. Mesh-less RBF-DQM, with the ability to implement arbitrary (e.g., adaptive) node distribution, properly simulates 2-D transient flow induced by pumping wells in confined/unconfined aquifers with regular and irregular geometries, compared to the other high-order and low-order approaches presented in this paper.

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Castelo, Antonio, Alexandre M. Afonso, and Wesley De Souza Bezerra. "A Hierarchical Grid Solver for Simulation of Flows of Complex Fluids." Polymers 13, no. 18 (September 18, 2021): 3168. http://dx.doi.org/10.3390/polym13183168.

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Tree-based grids bring the advantage of using fast Cartesian discretizations, such as finite differences, and the flexibility and accuracy of local mesh refinement. The main challenge is how to adapt the discretization stencil near the interfaces between grid elements of different sizes, which is usually solved by local high-order geometrical interpolations. Most methods usually avoid this by limiting the mesh configuration (usually to graded quadtree/octree grids), reducing the number of cases to be treated locally. In this work, we employ a moving least squares meshless interpolation technique, allowing for more complex mesh configurations, still keeping the overall order of accuracy. This technique was implemented in the HiG-Flow code to simulate Newtonian, generalized Newtonian and viscoelastic fluids flows. Numerical tests and application to viscoelastic fluid flow simulations were performed to illustrate the flexibility and robustness of this new approach.

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Zaiats, V., J. Majewski, T. Marciniak, and M. Zaiats. "The combination numerical method of effective processing high frequency signals." КОМП’ЮТЕРНО-ІНТЕГРОВАНІ ТЕХНОЛОГІЇ: ОСВІТА, НАУКА, ВИРОБНИЦТВО, no. 36 (November 21, 2019): 21–28. http://dx.doi.org/10.36910/6775-2524-0560-2019-36-4.

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An iterative approach to the construction of second-order numerical methods based on the Liniger-Willaby method with minimal error of discretization is proposed. The essence of the approach is to identify corrections to the Euler's explicit and implicit method at a time when their contributions to the amendment are equivalent. Improved time and accuracy in the process of determining the characteristics of quartz oscillators of the 9th order and high-speed auto-generators18 of the order with very long transients.

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Yoshida, Takumi, Takeshi Okuzono, and Kimihiro Sakagami. "A high-order explicit time-domain FEM using 15-node tetrahedral elements for room acoustics modeling: Basic performance." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 268, no. 3 (November 30, 2023): 5251–61. http://dx.doi.org/10.3397/in_2023_0740.

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Time-domain room acoustic modeling using mass-lumped higher-order tetrahedral finite elements with an explicit time-marching scheme is highly attractive because of its excellent geometrical flexibility with unstructured meshing and applicability into massively parallel computing. However, the standard mass-lumped tetrahedral elements yield an unstable scheme, and only implicit time-marching schemes are available. This study proposes a novel wave-based room acoustics solver based on a high-order explicit time-domain FEM. The present solver uses mass-lumped 15-node tetrahedral elements for spatial discretization and a dissipation-free two-stage partitioned Runge-Kutta time integration for time discretization. The 15-node tetrahedral elements had been developed to apply mass-lumping methods to 10-node tetrahedral elements, which are the most common finite element in engineering applications, but its applicability in room acoustics problems is unknown. Higher accuracy and efficiency of the proposed method over the standard method using 10-node tetrahedral elements is presented through an eigenvalue analysis of a long duct model and acoustic simulations in a small room. Additionally, frequency-dependent sound absorbing boundary is implemented by revising an auxiliary differential equation method.

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Banholzer, Stefan, Bennet Gebken, Lena Reichle, and Stefan Volkwein. "ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization." Mathematical and Computational Applications 26, no. 2 (April 15, 2021): 32. http://dx.doi.org/10.3390/mca26020032.

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The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach.

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Goodarzi, M., M. R. Safaei, A. Karimipour, K. Hooman, M. Dahari, S. N. Kazi, and E. Sadeghinezhad. "Comparison of the Finite Volume and Lattice Boltzmann Methods for Solving Natural Convection Heat Transfer Problems inside Cavities and Enclosures." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/762184.

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Different numerical methods have been implemented to simulate internal natural convection heat transfer and also to identify the most accurate and efficient one. A laterally heated square enclosure, filled with air, was studied. A FORTRAN code based on the lattice Boltzmann method (LBM) was developed for this purpose. The finite difference method was applied to discretize the LBM equations. Furthermore, for comparison purpose, the commercially available CFD package FLUENT, which uses finite volume Method (FVM), was also used to simulate the same problem. Different discretization schemes, being the first order upwind, second order upwind, power law, and QUICK, were used with the finite volume solver where the SIMPLE and SIMPLEC algorithms linked the velocity-pressure terms. The results were also compared with existing experimental and numerical data. It was observed that the finite volume method requires less CPU usage time and yields more accurate results compared to the LBM. It has been noted that the 1st order upwind/SIMPLEC combination converges comparatively quickly with a very high accuracy especially at the boundaries. Interestingly, all variants of FVM discretization/pressure-velocity linking methods lead to almost the same number of iterations to converge but higher-order schemes ask for longer iterations.

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Tang, Zhuochao, Zhuojia Fu, HongGuang Sun, and Xiaoting Liu. "An efficient localized collocation solver for anomalous diffusion on surfaces." Fractional Calculus and Applied Analysis 24, no. 3 (June 1, 2021): 865–94. http://dx.doi.org/10.1515/fca-2021-0037.

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Abstract This paper introduces an efficient collocation solver, the generalized finite difference method (GFDM) combined with the recent-developed scale-dependent time stepping method (SD-TSM), to predict the anomalous diffusion behavior on surfaces governed by surface time-fractional diffusion equations. In the proposed solver, the GFDM is used in spatial discretization and SD-TSM is used in temporal discretization. Based on the moving least square theorem and Taylor series, the GFDM introduces the stencil selection algorithms to choose the stencil support of a certain node from the whole discretization nodes on the surface. It inherits the similar properties from the standard FDM and avoids the mesh generation, which is available particularly for high-dimensional irregular discretization nodes. The SD-TSM is a non-uniform temporal discretization method involving the idea of metric, which links the fractional derivative order with the non-uniform discretization strategy. Compared with the traditional time stepping methods, GFDM combined with SD-TSM deals well with the low accuracy in the early period. Numerical investigations are presented to demonstrate the efficiency and accuracy of the proposed GFDM in conjunction with SD-TSM for solving either single or coupled fractional diffusion equations on surfaces.

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Wang, Baokun, Shaohua Wang, Yibing Peng, Youguo Pi, and Ying Luo. "Design and High-Order Precision Numerical Implementation of Fractional-Order PI Controller for PMSM Speed System Based on FPGA." Fractal and Fractional 6, no. 4 (April 12, 2022): 218. http://dx.doi.org/10.3390/fractalfract6040218.

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In this paper, the design of a fractional-order proportional integral (FOPI) controller and integer-order (IOPI) controller are compared for the permanent magnet synchronous motor (PMSM) speed regulation system. A high-precision implementation method of a fractional-order proportional integral (FOPI) controller is proposed in this work. Three commonly used numerical implementation methods of fractional operators are investigated and compared for comprehensively evaluating the numerical implementation performance in this work. Furthermore, for the impulse response invariant implementation method, the effects of different discretization orders on the control performance of the system are compared. The high-order fractional-order controller can be implemented accurately in a control system with the field-programmable gate array (FPGA) with the capability of parallel calculation. The simulation and experimental results show that the high-precision numerical implementation method of the designed high-order FOPI controller has better performance than the ordinary precision fractional operation implementation method and traditional order integer order PI controller.

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Journal articles on the topic 'High-order discretization methods'

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