Journal articles: 'Semi- implicit discretizations'

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Wu, Wenyuan, Greg Reid, and Silvana Ilie. "Implicit Riquier Bases for PDAE and their semi-discretizations." Journal of Symbolic Computation 44, no. 7 (July 2009): 923–41. http://dx.doi.org/10.1016/j.jsc.2008.04.020.

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Bartels, Sören, Lars Diening, and Ricardo H. Nochetto. "Unconditional Stability of Semi-Implicit Discretizations of Singular Flows." SIAM Journal on Numerical Analysis 56, no. 3 (January 2018): 1896–914. http://dx.doi.org/10.1137/17m1159166.

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Fernández, Miguel A., and Mikel Landajuela. "Splitting schemes and unfitted-mesh methods for the coupling of an incompressible fluid with a thin-walled structure." IMA Journal of Numerical Analysis 40, no. 2 (January 30, 2019): 1407–53. http://dx.doi.org/10.1093/imanum/dry098.

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Abstract Two unfitted-mesh methods for a linear incompressible fluid/thin-walled structure interaction problem are introduced and analyzed. The spatial discretization is based on different variants of Nitsche’s method with cut elements. The degree of fluid–solid splitting (semi-implicit or explicit) is given by the order in which the space and time discretizations are performed. The a priori stability and error analysis shows that strong coupling is avoided without compromising stability and accuracy. Numerical experiments with a benchmark illustrate the accuracy of the different methods proposed.

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Bartels, Sören. "Simulation of constrained elastic curves and application to a conical sheet indentation problem." IMA Journal of Numerical Analysis 41, no. 3 (February 24, 2021): 2255–79. http://dx.doi.org/10.1093/imanum/drab008.

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Abstract We consider variational problems that model the bending behavior of curves that are constrained to belong to given hypersurfaces. Finite element discretizations of corresponding functionals are justified rigorously via $\varGamma $-convergence. The stability of semi-implicit discretizations of gradient flows is investigated, which provide a practical method to determine stationary configurations. A particular application of the considered models arises in the description of conical sheet deformations.

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Ceniceros, Hector D., Jordan E. Fisher, and Alexandre M. Roma. "Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method." Journal of Computational Physics 228, no. 19 (October 2009): 7137–58. http://dx.doi.org/10.1016/j.jcp.2009.05.031.

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Bénard, P. "Stability of Semi-Implicit and Iterative Centered-Implicit Time Discretizations for Various Equation Systems Used in NWP." Monthly Weather Review 131, no. 10 (October 2003): 2479–91. http://dx.doi.org/10.1175/1520-0493(2003)131<2479:sosaic>2.0.co;2.

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Oliver, Marcel, and Claudia Wulff. "Stability under Galerkin truncation of A-stable Runge–Kutta discretizations in time." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144, no. 3 (May 16, 2014): 603–36. http://dx.doi.org/10.1017/s0308210512002028.

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We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup, and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semi-flow by an implicit A-stable Runge–Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semi-flow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. We then estimate the Galerkin truncation error for the semi-flow of the evolution equation, its Runge–Kutta discretization and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schrodinger equation.

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Cordero, Elisabetta, and Andrew Staniforth. "A Problem with the Robert–Asselin Time Filter for Three-Time-Level Semi-Implicit Semi-Lagrangian Discretizations." Monthly Weather Review 132, no. 2 (February 2004): 600–610. http://dx.doi.org/10.1175/1520-0493(2004)132<0600:apwtrt>2.0.co;2.

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Bonaventura, Luca, and Todd Ringler. "Analysis of Discrete Shallow-Water Models on Geodesic Delaunay Grids with C-Type Staggering." Monthly Weather Review 133, no. 8 (August 1, 2005): 2351–73. http://dx.doi.org/10.1175/mwr2986.1.

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Abstract The properties of C-grid staggered spatial discretizations of the shallow-water equations on regular Delaunay triangulations on the sphere are analyzed. Mass-conserving schemes that also conserve either energy or potential enstrophy are derived, and their features are analogous to those of the C-grid staggered schemes on quadrilateral grids. Results of numerical tests carried out with explicit and semi-implicit time discretizations show that the potential-enstrophy-conserving scheme is able to reproduce correctly the main features of large-scale atmospheric motion and that power spectra for energy and potential enstrophy obtained in long model integrations display a qualitative behavior similar to that predicted by the decaying turbulence theory for the continuous system.

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Carcano, S., L. Bonaventura, T. Esposti Ongaro, and A. Neri. "A semi-implicit, second-order-accurate numerical model for multiphase underexpanded volcanic jets." Geoscientific Model Development 6, no. 6 (November 4, 2013): 1905–24. http://dx.doi.org/10.5194/gmd-6-1905-2013.

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Abstract. An improved version of the PDAC (Pyroclastic Dispersal Analysis Code, Esposti Ongaro et al., 2007) numerical model for the simulation of multiphase volcanic flows is presented and validated for the simulation of multiphase volcanic jets in supersonic regimes. The present version of PDAC includes second-order time- and space discretizations and fully multidimensional advection discretizations in order to reduce numerical diffusion and enhance the accuracy of the original model. The model is tested on the problem of jet decompression in both two and three dimensions. For homogeneous jets, numerical results are consistent with experimental results at the laboratory scale (Lewis and Carlson, 1964). For nonequilibrium gas–particle jets, we consider monodisperse and bidisperse mixtures, and we quantify nonequilibrium effects in terms of the ratio between the particle relaxation time and a characteristic jet timescale. For coarse particles and low particle load, numerical simulations well reproduce laboratory experiments and numerical simulations carried out with an Eulerian–Lagrangian model (Sommerfeld, 1993). At the volcanic scale, we consider steady-state conditions associated with the development of Vulcanian and sub-Plinian eruptions. For the finest particles produced in these regimes, we demonstrate that the solid phase is in mechanical and thermal equilibrium with the gas phase and that the jet decompression structure is well described by a pseudogas model (Ogden et al., 2008). Coarse particles, on the other hand, display significant nonequilibrium effects, which associated with their larger relaxation time. Deviations from the equilibrium regime, with maximum velocity and temperature differences on the order of 150 m s−1 and 80 K across shock waves, occur especially during the rapid acceleration phases, and are able to modify substantially the jet dynamics with respect to the homogeneous case.

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Carcano, S., L. Bonaventura, T. Esposti Ongaro, and A. Neri. "A semi-implicit, second order accurate numerical model for multiphase underexpanded volcanic jets." Geoscientific Model Development Discussions 6, no. 1 (January 22, 2013): 399–452. http://dx.doi.org/10.5194/gmdd-6-399-2013.

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Abstract. An improved version of the PDAC (Pyroclastic Dispersal Analysis Code, Esposti Ongaro et al., 2007) numerical model for the simulation of multiphase volcanic flows is presented and validated for the simulation of multiphase volcanic jets in supersonic regimes. The present version of PDAC includes second-order time and space discretizations and fully multidimensional advection discretizations, in order to reduce numerical diffusion and enhance the accuracy of the original model. The model is tested on the problem of jet decompression, in both two and three dimensions. For homogeneous jets, numerical results are consistent with experimental results at the laboratory scale (Lewis and Carlson, 1964). For non-equilibrium gas-particle jets, we consider monodisperse and bidisperse mixtures and we quantify non-equilibrium effects in terms of the ratio between the particle relaxation time and a characteristic jet time scale. For coarse particles and low particle load, numerical simulations well reproduce laboratory experiments and numerical simulations carried out with an Eulerian-Lagrangian model (Sommerfeld, 1993). At the volcanic scale, we consider steady-state conditions associated to the development of Vulcanian and sub-Plinian eruptions. For the finest particles produced in these regimes, we demonstrate that the solid phase is in mechanical and thermal equilibrium with the gas phase and that the jet decompression structure is well described by a pseudogas model (Ogden et al., 2008). Coarse particles, on the contrary, display significant non-equilibrium effects, associated to their larger relaxation time. Deviations from the equilibrium regime occur especially during the rapid acceleration phases and are able to appreciably modify the average jet dynamics, with maximum velocity and temperature differences of the order of 150 m s−1 and 80 K across shock waves.

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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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Bürger, Raimund, Stefan Diehl, and Camilo Mejías. "On time discretizations for the simulation of the batch settling–compression process in one dimension." Water Science and Technology 73, no. 5 (November 7, 2015): 1010–17. http://dx.doi.org/10.2166/wst.2015.572.

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The main purpose of the recently introduced Bürger–Diehl simulation model for secondary settling tanks was to resolve spatial discretization problems when both hindered settling and the phenomena of compression and dispersion are included. Straightforward time integration unfortunately means long computational times. The next step in the development is to introduce and investigate time-integration methods for more efficient simulations, but where other aspects such as implementation complexity and robustness are equally considered. This is done for batch settling simulations. The key findings are partly a new time-discretization method and partly its comparison with other specially tailored and standard methods. Several advantages and disadvantages for each method are given. One conclusion is that the new linearly implicit method is easier to implement than another one (semi-implicit method), but less efficient based on two types of batch sedimentation tests.

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Yi, Wenfan, and Yongyong Cai. "Optimal error estimates of finite difference time domain methods for the Klein–Gordon–Dirac system." IMA Journal of Numerical Analysis 40, no. 2 (December 5, 2018): 1266–93. http://dx.doi.org/10.1093/imanum/dry084.

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Abstract We propose and analyze finite difference methods for solving the Klein–Gordon–Dirac (KGD) system. Due to the nonlinear coupling between the complex Dirac ‘wave function’ and the real Klein–Gordon field, it is a great challenge to design and analyze numerical methods for KGD. To overcome the difficulty induced by the nonlinearity, four implicit/semi-implicit/explicit finite difference time domain (FDTD) methods are presented, which are time symmetric or time reversible. By rigorous error estimates, the FDTD methods converge with second-order accuracy in both spatial and temporal discretizations, and numerical results in one dimension are reported to support our conclusion. The error analysis relies on the energy method, the special nonlinear structure in KGD and the mathematical induction. Thanks to tensor grids and discrete Sobolev inequalities, our approach and convergence results are valid in higher dimensions under minor modifications.

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Zhang, Yong. "Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System." Communications in Computational Physics 13, no. 5 (May 2013): 1357–88. http://dx.doi.org/10.4208/cicp.251011.270412a.

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AbstractWe study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave function and external potential V(x). The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order Ҩ(h4 + τ2) in discrete l2,H1 and l∞ norms with mesh size h and time step t. For the errors ofcompact finite difference approximation to the second derivative andPoisson potential are nonlocal, thus besides the standard energy method and mathematical induction method, the key technique in analysisis to estimate the nonlocal approximation errors in discrete l∞ and H1 norm by discrete maximum principle of elliptic equation and properties of some related matrix. Also some useful inequalities are established in this paper. Finally, extensive numerical re-sults are reported to support our error estimates of the numerical methods.

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Benacchio, Tommaso, Warren P. O’Neill, and Rupert Klein. "A Blended Soundproof-to-Compressible Numerical Model for Small- to Mesoscale Atmospheric Dynamics." Monthly Weather Review 142, no. 12 (December 1, 2014): 4416–38. http://dx.doi.org/10.1175/mwr-d-13-00384.1.

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Abstract A blended model for atmospheric flow simulations is introduced that enables seamless transition from fully compressible to pseudo-incompressible dynamics. The model equations are written in nonperturbation form and integrated using a well-balanced second-order finite-volume discretization. The semi-implicit scheme combines an explicit predictor for advection with elliptic corrections for the pressure field. Compressibility is implemented in the elliptic equations through a diagonal term. The compressible/pseudo-incompressible transition is realized by suitably weighting the term and provides a mechanism for removing unwanted acoustic imbalances in compressible runs. As the gradient of the pressure is used instead of the Exner pressure in the momentum equation, the influence of perturbation pressure on buoyancy must be included to ensure thermodynamic consistency. With this effect included, the thermodynamically consistent model is equivalent to Durran’s original pseudo-incompressible model, which uses the Exner pressure. Numerical experiments demonstrate quadratic convergence and competitive solution quality for several benchmarks. With the inclusion of an additional buoyancy term required for thermodynamic consistency, the “p–ρ formulation” of the pseudo-incompressible model closely reproduces the compressible results. The proposed unified approach offers a framework for models that are largely free of the biases that can arise when different discretizations are used. With data assimilation applications in mind, the seamless compressible/pseudo-incompressible transition mechanism is also shown to enable the flattening of acoustic imbalances in initial data for which balanced pressure distributions are unknown.

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Shen Yijiang, 沈逸江, 王小朋 Wang Xiaopeng, 周延周 Zhou Yanzhou, and 张振荣 Zhang Zhenrong. "基于半隐式离散化的局部水平集掩模优化." Acta Optica Sinica 41, no. 9 (2021): 0911004. http://dx.doi.org/10.3788/aos202141.0911004.

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Zajkani, Asghar, Abolfazl Darvizeh, and Mansour Darvizeh. "Analytical modelling of high-rate elasto-viscoplastic deformation of circular plates subjected to impulsive loads using pseudo-spectral collocation method." Journal of Strain Analysis for Engineering Design 48, no. 2 (October 29, 2012): 126–49. http://dx.doi.org/10.1177/0309324712460359.

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An analytical methodology is developed to study dynamic elasto-viscoplastic behaviour of moderately thick circular plates subjected to high-intensity impulsive loads, comprehensively. First, incremental kinematic formulations are derived based on the first-order shear deformation theory to take into account viscous damping and rotary inertia. Geometrical and material non-linearities are applied by the complete von Kármán system and a mixed strain hardening law coupled with a physically based viscoplastic model, respectively. A semi-implicit scheme of return-mapping is employed by the cutting-plane algorithm to obtain effective plastic strains apart from satisfying the consistency condition. The subsequent part is devoted to the transformation of this boundary value problem into an initial value problem, to evaluate displacement fields. Spatial and temporal discretizations are carried out by the Chebyshev pseudo-spectral collocation method and Houbolt time-marching scheme, respectively. This transformation has been handled in the compact matrix forms to stabilize the solution and to make it more convenient. Influence of impulsive load and other parameters on plate deflections, effective strain and stress, temperature rise and stresses are considered. Ultimately, good accuracy is achieved through comparison between results and existing experimental data and finite element simulation from the literature. In addition, some challengeable aspects for the modelling are discussed.

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Degond, Pierre, and Min Tang. "All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations." Communications in Computational Physics 10, no. 1 (July 2011): 1–31. http://dx.doi.org/10.4208/cicp.210709.210610a.

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AbstractAn all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests are displayed to demonstrate its performances.

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QUAINI, A., and A. QUARTERONI. "A SEMI-IMPLICIT APPROACH FOR FLUID-STRUCTURE INTERACTION BASED ON AN ALGEBRAIC FRACTIONAL STEP METHOD." Mathematical Models and Methods in Applied Sciences 17, no. 06 (June 2007): 957–83. http://dx.doi.org/10.1142/s0218202507002170.

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We address the numerical simulation of fluid-structure interaction problems characterized by a strong added-mass effect. We propose a semi-implicit coupling scheme based on an algebraic fractional-step method. The basic idea of a semi-implicit scheme consists in coupling implicitly the added-mass effect, while the other terms (dissipation, convection and geometrical nonlinearities) are treated explicitly. Thanks to this kind of explicit–implicit splitting, computational costs can be reduced (in comparison to fully implicit coupling algorithms) and the scheme remains stable for a wide range of discretization parameters. In this paper we derive this kind of splitting from the algebraic formulation of the coupled fluid-structure problem (after finite-element space discretization). From our knowledge, it is the first time that algebraic fractional step methods, used thus far only for fluid problems in computational domains with rigid boundaries, are applied to fluid-structure problems. In particular, for the specific semi-implicit method presented in this work, we adapt the Yosida scheme to the case of a coupled fluid-structure problem. This scheme relies on an approximate LU block factorization of the matrix obtained after the discretization in time and space of the fluid-structure system. We analyze the numerical performances of this scheme on 2D fluid-structure simulations performed with a simple 1D structure model.

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Busto, Saray, Michael Dumbser, and Laura Río-Martín. "Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows." Mathematics 9, no. 22 (November 21, 2021): 2972. http://dx.doi.org/10.3390/math9222972.

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This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

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Ethier, Marc, and Yves Bourgault. "Semi-Implicit Time-Discretization Schemes for the Bidomain Model." SIAM Journal on Numerical Analysis 46, no. 5 (January 2008): 2443–68. http://dx.doi.org/10.1137/070680503.

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Zhang, Jing-Jing, Xiang-Gui Li, and Jing-Fang Shao. "Implicit integration factor method for the nonlinear Dirac equation." International Journal of Modeling, Simulation, and Scientific Computing 09, no. 02 (March 20, 2018): 1850019. http://dx.doi.org/10.1142/s1793962318500198.

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A high-order accuracy time discretization method is developed in this paper to solve the one-dimensional nonlinear Dirac (NLD) equation. Based on the implicit integration factor (IIF) method, two schemes are proposed. Central differences are applied to the spatial discretization. The semi-discrete scheme keeps the conservation of the charge and energy. For the temporal discretization, second-order IIF method and fourth-order IIF method are applied respectively to the nonlinear system arising from the spatial discretization. Numerical experiments are given to validate the accuracy of these schemes and to discuss the interaction dynamics of the NLD solitary waves.

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Zhang, Guo-Dong, and Yinnian He. "Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 8 (November 2, 2015): 1912–23. http://dx.doi.org/10.1108/hff-08-2014-0257.

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Purpose – The purpose of this paper is to consider the numerical implementation of the Euler semi-implicit scheme for three-dimensional non-stationary magnetohydrodynamics (MHD) equations. The Euler semi-implicit scheme is used for time discretization and (P 1b , P 1, P 1) finite element for velocity, pressure and magnet is used for the spatial discretization. Design/methodology/approach – Several numerical experiments are provided to show this scheme is unconditional stability and unconditional L2−H2 convergence with the L2−H2 optimal error rates for solving the non-stationary MHD flows. Findings – In this paper, the authors mainly focus on the numerical investigation of the Euler semi-implicit scheme for MHD flows. First, the unconditional stability and the L2−H2 unconditional convergence with optimal L2−H2 error rates of this scheme are validated through our numerical tests. Some interesting phenomenons are presented. Originality/value – The Euler semi-implicit scheme is used to simulate a practical physics model problem to investigate the interaction of fluid and induced magnetic field. Some interesting phenomenons are presented.

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Yang, XueSheng, JiaBin Chen, JiangLin Hu, DeHui Chen, XueShun Shen, and HongLiang Zhang. "A semi-implicit semi-Lagrangian global nonhydrostatic model and the polar discretization scheme." Science in China Series D: Earth Sciences 50, no. 12 (December 2007): 1885–91. http://dx.doi.org/10.1007/s11430-007-0124-7.

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BOFFI, DANIELE, LUCIA GASTALDI, and LUCA HELTAI. "NUMERICAL STABILITY OF THE FINITE ELEMENT IMMERSED BOUNDARY METHOD." Mathematical Models and Methods in Applied Sciences 17, no. 10 (October 2007): 1479–505. http://dx.doi.org/10.1142/s0218202507002352.

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The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully nonlinearly coupled formulation for the study of fluid structure interactions. Many numerical methods have been introduced to reduce the difficulties related to the nonlinear coupling between the structure and the fluid evolution. However numerical instabilities arise when explicit or semi-implicit methods are considered. In this work we present a stability analysis based on energy estimates of the variational formulation of the immersed boundary method. A two-dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. We use a linearization of the Navier–Stokes equations and a linear elasticity model to prove the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution, and a CFL condition for the semi-implicit method where the fluid terms are treated implicitly while the structure is treated explicitly. We present some numerical tests that show good accordance between the observed stability behavior and the one predicted by our results.

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Wong, May, William C. Skamarock, Peter H. Lauritzen, Joseph B. Klemp, and Roland B. Stull. "Testing of a Cell-Integrated Semi-Lagrangian Semi-Implicit Nonhydrostatic Atmospheric Solver (CSLAM-NH) with Idealized Orography." Monthly Weather Review 143, no. 4 (March 31, 2015): 1382–98. http://dx.doi.org/10.1175/mwr-d-14-00059.1.

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Abstract A recently developed cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic atmospheric solver that uses the conservative semi-Lagrangian multitracer (CSLAM) transport scheme is extended to include orographic influences. With the introduction of a new semi-implicit CISL discretization of the continuity equation, the nonhydrostatic solver, called CSLAM-NH, has been shown to ensure inherently conservative and numerically consistent transport of air mass and other scalar variables, such as moisture and passive tracers. The extended CSLAM-NH presented here includes two main modifications: transformation of the equation set to a terrain-following height coordinate to incorporate orography and an iterative centered-implicit time-stepping scheme to enhance the stability of the scheme associated with gravity wave propagation at large time steps. CSLAM-NH is tested for a suite of idealized 2D flows, including linear mountain waves (dry), a downslope windstorm (dry), and orographic cloud formation.

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Zhang, Rongpei, Xijun Yu, Jiang Zhu, Abimael F. D. Loula, and Xia Cui. "Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation." Communications in Computational Physics 14, no. 5 (November 2013): 1287–303. http://dx.doi.org/10.4208/cicp.190612.010313a.

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AbstractWeighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

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Xu, Xiaojing, and Xiaoping Xie. "Robust Semi-Discrete and Fully Discrete Hybrid Stress Finite Element Methods for Elastodynamic Problems." Advances in Applied Mathematics and Mechanics 9, no. 2 (January 9, 2017): 324–48. http://dx.doi.org/10.4208/aamm.2015.m1326.

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AbstractThis paper analyzes semi-discrete and fully discrete hybrid stress quadrilateral finite element methods for 2-dimensional linear elastodynamic problems. The methods use a 4 node hybrid stress quadrilateral element in the space discretization. In the fully discrete scheme, an implicit second-order scheme is adopted in the time discretization. We derive optimal a priori error estimates for the two schemes and an unconditional stability result for the fully discrete scheme. Numerical experiments confirm the theoretical results.

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Michel, Loïc, Malek Ghanes, Franck Plestan, Yannick Aoustin, and Jean-Pierre Barbot. "Semi-Implicit Euler Discretization for Homogeneous Observer-based Control: one dimensional case." IFAC-PapersOnLine 53, no. 2 (2020): 5135–40. http://dx.doi.org/10.1016/j.ifacol.2020.12.1152.

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Talay Akyildiz, F., and K. Vajravelu. "Galerkin-Chebyshev Pseudo Spectral Method and a Split Step New Approach for a Class of Two dimensional Semi-linear Parabolic Equations of Second Order." Applied Mathematics and Nonlinear Sciences 3, no. 1 (June 8, 2018): 255–64. http://dx.doi.org/10.21042/amns.2018.1.00019.

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AbstractIn this paper, we use a time splitting method with higher-order accuracy for the solutions (in space variables) of a class of two-dimensional semi-linear parabolic equations. Galerkin-Chebyshev pseudo spectral method is used for discretization of the spatial derivatives, and implicit Euler method is used for temporal discretization. In addition, we use this novel method to solve the well-known semi-linear Poisson-Boltzmann (PB) model equation and obtain solutions with higher-order accuracy. Furthermore, we compare the results obtained by our method for the semi-linear parabolic equation with the available analytical results in the literature for some special cases, and found excellent agreement. Furthermore, our new technique is also applicable for three-dimensional problems.

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32

Payne, T. J. "A linear-stability analysis of the semi-implicit semi-Lagrangian discretization of the fully-compressible equations." Quarterly Journal of the Royal Meteorological Society 134, no. 632 (2008): 779–86. http://dx.doi.org/10.1002/qj.227.

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He, Guoliang, and Yong Zhang. "The Optimal Error Estimate of the Fully Discrete Locally Stabilized Finite Volume Method for the Non-Stationary Navier-Stokes Problem." Entropy 24, no. 6 (May 30, 2022): 768. http://dx.doi.org/10.3390/e24060768.

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This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem. In this paper, the semi-implicit scheme based on Euler method is adopted for time discretization, while the special finite volume scheme is adopted for space discretization. Specifically, the spatial discretization adopts the traditional triangle P1−P0 trial function pair, combined with macro element form to ensure local stability. The theoretical analysis results show that under certain conditions, the full discretization proposed here has the characteristics of local stability, and we can indeed obtain the optimal theoretic and numerical order error estimation of velocity and pressure. This helps to enrich the corresponding theoretical results.

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Wood, Nigel, Andrew Staniforth, Andy White, Thomas Allen, Michail Diamantakis, Markus Gross, Thomas Melvin, et al. "An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the deep-atmosphere global non-hydrostatic equations." Quarterly Journal of the Royal Meteorological Society 140, no. 682 (December 4, 2013): 1505–20. http://dx.doi.org/10.1002/qj.2235.

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Shi, Kaiwen, Haiyan Su, and Xinlong Feng. "Error Analysis of a PFEM Based on the Euler Semi-Implicit Scheme for the Unsteady MHD Equations." Entropy 24, no. 10 (September 30, 2022): 1395. http://dx.doi.org/10.3390/e24101395.

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In this article, we mainly consider a first order penalty finite element method (PFEM) for the 2D/3D unsteady incompressible magnetohydrodynamic (MHD) equations. The penalty method applies a penalty term to relax the constraint ``∇·u=0", which allows us to transform the saddle point problem into two smaller problems to solve. The Euler semi-implicit scheme is based on a first order backward difference formula for time discretization and semi-implicit treatments for nonlinear terms. It is worth mentioning that the error estimates of the fully discrete PFEM are rigorously derived, which depend on the penalty parameter ϵ, the time-step size τ, and the mesh size h. Finally, two numerical tests show that our scheme is effective.

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Henríquez, Fernando, and Carlos Jerez-Hanckes. "Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 2 (March 2018): 659–703. http://dx.doi.org/10.1051/m2an/2018019.

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We model the electrical behavior of several biological cells under external stimuli by extending and computationally improving the multiple traces formulation introduced in Henríquez et al. [Numer. Math. 136 (2016) 101–145]. Therein, the electric potential and current for a single cell are retrieved through the coupling of boundary integral operators and non-linear ordinary differential systems of equations. Yet, the low-order discretization scheme presented becomes impractical when accounting for interactions among multiple cells. In this note, we consider multi-cellular systems and show existence and uniqueness of the resulting non-linear evolution problem in finite time. Our main tools are analytic semigroup theory along with mapping properties of boundary integral operators in Sobolev spaces. Thanks to the smoothness of cellular shapes, solutions are highly regular at a given time. Hence, spectral spatial discretization can be employed, thereby largely reducing the number of unknowns. Time-space coupling is achieved via a semi-implicit time-stepping scheme shown to be stable and second order convergent. Numerical results in two dimensions validate our claims and match observed biological behavior for the Hodgkin–Huxley dynamical model.

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37

Subich, Christopher. "Instabilities in the Shallow-Water System with a Semi-Lagrangian, Time-Centered Discretization." Monthly Weather Review 150, no. 3 (March 2022): 467–80. http://dx.doi.org/10.1175/mwr-d-21-0054.1.

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Abstract Conventional wisdom suggests that the combination of semi-Lagrangian advection and an implicit treatment of gravity wave terms should result in a combined scheme for the shallow-water equations stable for high Courant numbers. This wisdom is well justified by linear analysis of the system about a uniform reference state with constant fluid depth and velocity, but it is only assumed to hold true in more complex scenarios. This work finds that this conventional wisdom no longer holds in more complicated flow regimes, in particular when the background state is given by steady-state flow past topography. Instead, this background state admits a wide range of instabilities that can lead to noise in atmospheric forecasts. Significance Statement This work shows that solutions to the shallow-water equations with a semi-Lagrangian treatment of advection and an implicit, time-centered treatment of gravity wave terms can be unstable when there is a background state of flow over topography. This basic algorithm is used by many operational weather-forecasting models to simulate the meteorological equations, and showing an instability in the simplified, shallow-water system suggests that a similar mechanism may be responsible for “noise” in operational weather forecasts under some circumstances. If this problem can be addressed, it could allow numerical weather models to operate with less dissipation, improving forecast quality.

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Wu, Pan, Fei Chao, Dan Wu, Jianqiang Shan, and Junli Gou. "Implementation and Comparison of High-Resolution Spatial Discretization Schemes for Solving Two-Fluid Seven-Equation Two-Pressure Model." Science and Technology of Nuclear Installations 2017 (2017): 1–14. http://dx.doi.org/10.1155/2017/4252975.

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As compared to the two-fluid single-pressure model, the two-fluid seven-equation two-pressure model has been proved to be unconditionally well-posed in all situations, thus existing with a wide range of industrial applications. The classical 1st-order upwind scheme is widely used in existing nuclear system analysis codes such as RELAP5, CATHARE, and TRACE. However, the 1st-order upwind scheme possesses issues of serious numerical diffusion and high truncation error, thus giving rise to the challenge of accurately modeling many nuclear thermal-hydraulics problems such as long term transients. In this paper, a semi-implicit algorithm based on the finite volume method with staggered grids is developed to solve such advanced well-posed two-pressure model. To overcome the challenge from 1st-order upwind scheme, eight high-resolution total variation diminishing (TVD) schemes are implemented in such algorithm to improve spatial accuracy. Then the semi-implicit algorithm with high-resolution TVD schemes is validated on the water faucet test. The numerical results show that the high-resolution semi-implicit algorithm is robust in solving the two-pressure two-fluid two-phase flow model; Superbee scheme and Koren scheme give two highest levels of accuracy while Minmod scheme is the worst one among the eight TVD schemes.

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Liang, Zongqi. "Large Time-Stepping Spectral Methods for the Semiclassical Limit of the Defocusing Nonlinear Schrödinger Equation." Discrete Dynamics in Nature and Society 2009 (2009): 1–27. http://dx.doi.org/10.1155/2009/283959.

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We analyze a class of large time-stepping Fourier spectral methods for the semiclassical limit of the defocusing Nonlinear Schrödinger equation and provide highly stable methods which allow much larger time step than for a standard implicit-explicit approach. An extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Meanwhile, the first-order and second-order semi-implicit schemes are constructed and analyzed. Finally the numerical experiments are performed to demonstrate the effectiveness of the large time-stepping approaches.

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Alvarez, Jorge, Mikel Zatarain, David Barrenetxea, Jose Ignacio Marquinez, and Borja Izquierdo. "Implicit Subspace Iteration to Improve the Stability Analysis in Grinding Processes." Applied Sciences 10, no. 22 (November 19, 2020): 8203. http://dx.doi.org/10.3390/app10228203.

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An alternative method is devised for calculating dynamic stability maps in cylindrical and centerless infeed grinding processes. The method is based on the application of the Floquet theorem by repeated time integrations. Without the need of building the transition matrix, this is the most efficient calculation in terms of computation effort compared to previously presented time-domain stability analysis methods (semi-discretization or time-domain simulations). In the analyzed cases, subspace iteration has been up to 130 times faster. One of the advantages of these time-domain methods to the detriment of frequency domain ones is that they can analyze the stability of regenerative chatter with the application of variable workpiece speed, a well-known technique to avoid chatter vibrations in grinding processes so the optimal combination of amplitude and frequency can be selected. Subspace iteration methods also deal with this analysis, providing an efficient solution between 27 and 47 times faster than the abovementioned methods. Validation of this method has been carried out by comparing its accuracy with previous published methods such as semi-discretization, frequency and time-domain simulations, obtaining good correlation in the results of the dynamic stability maps and the instability reduction ratio maps due to the application of variable speed.

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BEIKIRCH, MAXIMILIAN, SIMON CRAMER, MARTIN FRANK, PHILIPP OTTE, EMMA PABICH, and TORSTEN TRIMBORN. "ROBUST MATHEMATICAL FORMULATION AND PROBABILISTIC DESCRIPTION OF AGENT-BASED COMPUTATIONAL ECONOMIC MARKET MODELS." Advances in Complex Systems 23, no. 06 (September 2020): 2050017. http://dx.doi.org/10.1142/s0219525920500174.

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In science and especially in economics, agent-based modeling has become a widely used modeling approach. These models are often formulated as a large system of difference equations. In this study, we discuss two aspects, numerical modeling and the probabilistic description for two agent-based computational economic market models: the Levy–Levy–Solomon model and the Franke–Westerhoff model. We derive time-continuous formulations of both models, and in particular, we discuss the impact of the time-scaling on the model behavior for the Levy–Levy–Solomon model. For the Franke–Westerhoff model, we proof that a constraint required in the original model is not necessary for stability of the time-continuous model. It is shown that a semi-implicit discretization of the time-continuous system preserves this unconditional stability. In addition, this semi-implicit discretization can be computed at cost comparable to the original model. Furthermore, we discuss possible probabilistic descriptions of time-continuous agent-based computational economic market models. Especially, we present the potential advantages of kinetic theory in order to derive mesoscopic descriptions of agent-based models. Exemplified, we show two probabilistic descriptions of the Levy–Levy–Solomon and Franke–Westerhoff model.

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Bi, Xiaolei, Shanjun Mu, Qingxia Liu, Quanzhen Liu, Baoquan Liu, Pinghui Zhuang, Jian Gao, Hui Jiang, Xin Li, and Bochen Li. "Advanced Implicit Meshless Approaches for the Rayleigh–Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative." International Journal of Computational Methods 15, no. 05 (June 5, 2018): 1850032. http://dx.doi.org/10.1142/s0219876218500329.

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To solve the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in a bounded domain is important in the research for diffusion processes. In this paper, novel implicit meshless approaches based on the moving least squares (MLS) approximation for spatial discretization and two different time discrete schemes, which are the first-order semi-discrete scheme and the second-order semi-discrete scheme for time, are developed for the numerical simulation of the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in a bounded domain. Based on these two time discretization schemes, the newly developed meshless approaches will have the first-order and the second-order accuracy in time, respectively. The stability and convergence of the implicit MLS meshless approaches are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approaches. It has found that the newly developed meshless approaches are accurate and convergent for fractional partial differential equations (FPDEs). Most importantly, the meshless approaches are robust for arbitrarily distributed nodes and complex domains.

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43

Taheri, Sina, Jacob R. King, and Uri Shumlak. "Time-discretization of a plasma-neutral MHD model with a semi-implicit leapfrog algorithm." Computer Physics Communications 274 (May 2022): 108288. http://dx.doi.org/10.1016/j.cpc.2022.108288.

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Simarro, Juan, and Mariano Hortal. "A semi-implicit non-hydrostatic dynamical kernel using finite elements in the vertical discretization." Quarterly Journal of the Royal Meteorological Society 138, no. 664 (October 31, 2011): 826–39. http://dx.doi.org/10.1002/qj.952.

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45

Melvin, Thomas, Tommaso Benacchio, Ben Shipway, Nigel Wood, John Thuburn, and Colin Cotter. "A mixed finite‐element, finite‐volume, semi‐implicit discretization for atmospheric dynamics: Cartesian geometry." Quarterly Journal of the Royal Meteorological Society 145, no. 724 (March 31, 2019): 2835–53. http://dx.doi.org/10.1002/qj.3501.

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Tamai, Tasuku, and Seiichi Koshizuka. "Convergence study of Poisson equation utilizing discretization schemes of Least Squares Moving Particle Semi-implicit method." Proceedings of The Computational Mechanics Conference 2014.27 (2014): 663–65. http://dx.doi.org/10.1299/jsmecmd.2014.27.663.

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47

Zhang, Rongpei, Xijun Yu, Mingjun Li, and Zhen Wang. "A semi-implicit integration factor discontinuous Galerkin method for the non-linear heat equation." Thermal Science 23, no. 3 Part A (2019): 1623–28. http://dx.doi.org/10.2298/tsci180921232z.

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In this paper, a new discontinuous Galerkin method is employed to study the non-linear heat conduction equation with temperature dependent thermal conductivity. We present practical implementation of the new discontinuous Galerkin scheme with weighted flux averages. The second-order implicit integration factor for time discretization method is applied to the semi discrete form. We obtain the L2 stability of the discontinuous Galerkin scheme. Numerical examples show that the error estimates are of second order when linear element approximations are applied. The method is applied to the non-linear heat conduction equations with source term.

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48

Benacchio, Tommaso, and Rupert Klein. "A Semi-Implicit Compressible Model for Atmospheric Flows with Seamless Access to Soundproof and Hydrostatic Dynamics." Monthly Weather Review 147, no. 11 (November 1, 2019): 4221–40. http://dx.doi.org/10.1175/mwr-d-19-0073.1.

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Abstract When written in conservation form for mass, momentum, and density-weighted potential temperature, and with Exner pressure in the momentum equation, the pseudoincompressible model and the hydrostatic model only differ from the full compressible equations by some additive terms. This structural proximity is transferred here to a numerical discretization providing seamless access to all three analytical models. The semi-implicit second-order scheme discretizes the rotating compressible equations by evolving full variables, and, optionally, with two auxiliary fields that facilitate the construction of an implicit pressure equation. Time steps are constrained by the advection speed only as a result. Borrowing ideas on forward-in-time differencing, the algorithm reframes the authors’ previously proposed schemes into a sequence of implicit midpoint step, advection step, and implicit trapezoidal step. Compared with existing approaches, results on benchmarks of nonhydrostatic- and hydrostatic-scale dynamics are competitive. The tests include a new planetary-scale gravity wave test that highlights the scheme’s ability to run with large time steps and to access multiple models. The advancement represents a sizeable step toward generalizing the authors’ acoustics-balanced initialization strategy to also cover the hydrostatic case in the framework of an all-scale blended multimodel solver.

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Kühnlein, Christian, Willem Deconinck, Rupert Klein, Sylvie Malardel, Zbigniew P. Piotrowski, Piotr K. Smolarkiewicz, Joanna Szmelter, and Nils P. Wedi. "FVM 1.0: a nonhydrostatic finite-volume dynamical core for the IFS." Geoscientific Model Development 12, no. 2 (February 13, 2019): 651–76. http://dx.doi.org/10.5194/gmd-12-651-2019.

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Abstract. We present a nonhydrostatic finite-volume global atmospheric model formulation for numerical weather prediction with the Integrated Forecasting System (IFS) at ECMWF and compare it to the established operational spectral-transform formulation. The novel Finite-Volume Module of the IFS (henceforth IFS-FVM) integrates the fully compressible equations using semi-implicit time stepping and non-oscillatory forward-in-time (NFT) Eulerian advection, whereas the spectral-transform IFS solves the hydrostatic primitive equations (optionally the fully compressible equations) using a semi-implicit semi-Lagrangian scheme. The IFS-FVM complements the spectral-transform counterpart by means of the finite-volume discretization with a local low-volume communication footprint, fully conservative and monotone advective transport, all-scale deep-atmosphere fully compressible equations in a generalized height-based vertical coordinate, and flexible horizontal meshes. Nevertheless, both the finite-volume and spectral-transform formulations can share the same quasi-uniform horizontal grid with co-located arrangement of variables, geospherical longitude–latitude coordinates, and physics parameterizations, thereby facilitating their comparison, coexistence, and combination in the IFS. We highlight the advanced semi-implicit NFT finite-volume integration of the fully compressible equations of IFS-FVM considering comprehensive moist-precipitating dynamics with coupling to the IFS cloud parameterization by means of a generic interface. These developments – including a new horizontal–vertical split NFT MPDATA advective transport scheme, variable time stepping, effective preconditioning of the elliptic Helmholtz solver in the semi-implicit scheme, and a computationally efficient implementation of the median-dual finite-volume approach – provide a basis for the efficacy of IFS-FVM and its application in global numerical weather prediction. Here, numerical experiments focus on relevant dry and moist-precipitating baroclinic instability at various resolutions. We show that the presented semi-implicit NFT finite-volume integration scheme on co-located meshes of IFS-FVM can provide highly competitive solution quality and computational performance to the proven semi-implicit semi-Lagrangian integration scheme of the spectral-transform IFS.

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Boscheri, Walter, Maurizio Tavelli, and Nicola Paoluzzi. "High order Finite Difference/Discontinuous Galerkin schemes for the incompressible Navier-Stokes equations with implicit viscosity." Communications in Applied and Industrial Mathematics 13, no. 1 (January 1, 2022): 21–38. http://dx.doi.org/10.2478/caim-2022-0003.

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Abstract In this work we propose a novel numerical method for the solution of the incompressible Navier-Stokes equations on Cartesian meshes in 3D. The semi-discrete scheme is based on an explicit discretization of the nonlinear convective flux tensor and an implicit treatment of the pressure gradient and viscous terms. In this way, the momentum equation is formally substituted into the divergence-free constraint, thus obtaining an elliptic equation on the pressure which eventually maintains at the discrete level the involution on the divergence of the velocity field imposed by the governing equations. This makes our method belonging to the class of so-called structure-preserving schemes. High order of accuracy in space is achieved using an efficient CWENO reconstruction operator that is exploited to devise a conservative finite difference scheme for the convective terms. Implicit central finite differences are used to remove the numerical dissipation in the pressure gradient discretization. To avoid the severe time step limitation induced by the viscous eigenvalues related to the parabolic terms in the governing equations, we propose to devise an implicit local discontinuous Galerkin (DG) solver. The resulting viscous sub-system is symmetric and positive definite, therefore it can be efficiently solved at the aid of a matrix-free conjugate gradient method. High order in time is granted by a semi-implicit IMEX time stepping technique. Convergence rates up to third order of accuracy in space and time are proven, and a suite of academic benchmarks is shown in order to demonstrate the robustness and the validity of the novel schemes, especially in the context of high viscosity coefficients.

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Journal articles on the topic 'Semi- implicit discretizations'

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