Society for Industrial and Applied Mathematics: SIAM Journal on Numerical Analysis: Table of Contents
Table of Contents for SIAM Journal on Numerical Analysis. List of articles from both the latest and ahead of print issues.
https://epubs.siam.org/loi/sjnaam?ai=s0&mi=3bfys9&af=R
Society for Industrial and Applied Mathematics: SIAM Journal on Numerical Analysis: Table of Contents
Society for Industrial and Applied Mathematics
enUS
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
https://epubs.siam.org/na101/home/literatum/publisher/siam/journals/covergifs/sjnaam/cover.jpg
https://epubs.siam.org/loi/sjnaam?ai=s0&mi=3bfys9&af=R

Coupling Parareal with Optimized Schwarz Waveform Relaxation for Parabolic Problems
https://epubs.siam.org/doi/abs/10.1137/21M1419428?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/3">Volume 60, Issue 3</a>, Page 913939, June 2022. <br/> We propose and analyze a new parallel paradigm that uses both the time and the space directions. The original approach couples the Parareal algorithm with incomplete optimized Schwarz waveform relaxation (OSWR) iterations. The analysis of this coupled method is presented for a onedimensional advectionreactiondiffusion equation. We also prove a general convergence result for this method via energy estimates. Numerical results for twodimensional advectiondiffusion problems and for a diffusion equation with strong heterogeneities are presented to illustrate the performance of the coupled PararealOSWR algorithm.
SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 913939, June 2022. <br/> We propose and analyze a new parallel paradigm that uses both the time and the space directions. The original approach couples the Parareal algorithm with incomplete optimized Schwarz waveform relaxation (OSWR) iterations. The analysis of this coupled method is presented for a onedimensional advectionreactiondiffusion equation. We also prove a general convergence result for this method via energy estimates. Numerical results for twodimensional advectiondiffusion problems and for a diffusion equation with strong heterogeneities are presented to illustrate the performance of the coupled PararealOSWR algorithm.
Coupling Parareal with Optimized Schwarz Waveform Relaxation for Parabolic Problems
10.1137/21M1419428
SIAM Journal on Numerical Analysis
20220503T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Duc Quang Bui
Caroline Japhet
Yvon Maday
Pascal Omnes
Coupling Parareal with Optimized Schwarz Waveform Relaxation for Parabolic Problems
60
3
913
939
20220630T07:00:00Z
20220630T07:00:00Z
10.1137/21M1419428
https://epubs.siam.org/doi/abs/10.1137/21M1419428?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

An Optimal Mass Transport Method for Random Genetic Drift
https://epubs.siam.org/doi/abs/10.1137/20M1389431?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/3">Volume 60, Issue 3</a>, Page 940969, June 2022. <br/> We propose and analyze an optimal mass transport method for a random genetic drift problem driven by a Moran process under weak selection. The continuum limit, formulated as a reactionadvectiondiffusion equation known as the Kimura equation, inherits degenerate diffusion from the discrete stochastic process that conveys to the blowup into Diracdelta singularities and hence brings great challenges to both analytical and numerical studies. The proposed numerical method can quantitatively capture to the fullest possible extent the development of Diracdelta singularities for genetic segregation on the one hand and preserve several sets of biologically relevant and computationally favored properties of the random genetic drift on the other. Moreover, the numerical scheme exponentially converges to the unique numerical stationary state in time at a rate independent of the mesh size up to a mesh error. Numerical evidence is given to illustrate and support these properties and to demonstrate the spatiotemporal dynamics of random genetic drift.
SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 940969, June 2022. <br/> We propose and analyze an optimal mass transport method for a random genetic drift problem driven by a Moran process under weak selection. The continuum limit, formulated as a reactionadvectiondiffusion equation known as the Kimura equation, inherits degenerate diffusion from the discrete stochastic process that conveys to the blowup into Diracdelta singularities and hence brings great challenges to both analytical and numerical studies. The proposed numerical method can quantitatively capture to the fullest possible extent the development of Diracdelta singularities for genetic segregation on the one hand and preserve several sets of biologically relevant and computationally favored properties of the random genetic drift on the other. Moreover, the numerical scheme exponentially converges to the unique numerical stationary state in time at a rate independent of the mesh size up to a mesh error. Numerical evidence is given to illustrate and support these properties and to demonstrate the spatiotemporal dynamics of random genetic drift.
An Optimal Mass Transport Method for Random Genetic Drift
10.1137/20M1389431
SIAM Journal on Numerical Analysis
20220505T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
José A. Carrillo
Lin Chen
Qi Wang
An Optimal Mass Transport Method for Random Genetic Drift
60
3
940
969
20220630T07:00:00Z
20220630T07:00:00Z
10.1137/20M1389431
https://epubs.siam.org/doi/abs/10.1137/20M1389431?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

A New Lagrange Multiplier Approach for Constructing Structure Preserving Schemes, II. Bound Preserving
https://epubs.siam.org/doi/abs/10.1137/21M144877X?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/3">Volume 60, Issue 3</a>, Page 970998, June 2022. <br/> In the second part of this series, we use the Lagrange multiplier approach proposed in the first part [Comput. Methods Appl. Mech. Engr., 391 (2022), 114585] to construct efficient and accurate bound and/or mass preserving schemes for a class of semilinear and quasilinear parabolic equations. We establish stability results under a general setting and carry out an error analysis for a secondorder bound preserving scheme with a hybrid spectral discretization in space. We apply our approach to several typical PDEs which preserve bound and/or mass and also present ample numerical results to validate our approach.
SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 970998, June 2022. <br/> In the second part of this series, we use the Lagrange multiplier approach proposed in the first part [Comput. Methods Appl. Mech. Engr., 391 (2022), 114585] to construct efficient and accurate bound and/or mass preserving schemes for a class of semilinear and quasilinear parabolic equations. We establish stability results under a general setting and carry out an error analysis for a secondorder bound preserving scheme with a hybrid spectral discretization in space. We apply our approach to several typical PDEs which preserve bound and/or mass and also present ample numerical results to validate our approach.
A New Lagrange Multiplier Approach for Constructing Structure Preserving Schemes, II. Bound Preserving
10.1137/21M144877X
SIAM Journal on Numerical Analysis
20220505T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Qing Cheng
Jie Shen
A New Lagrange Multiplier Approach for Constructing Structure Preserving Schemes, II. Bound Preserving
60
3
970
998
20220630T07:00:00Z
20220630T07:00:00Z
10.1137/21M144877X
https://epubs.siam.org/doi/abs/10.1137/21M144877X?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

An Embedded ExponentialType LowRegularity Integrator for mKdV Equation
https://epubs.siam.org/doi/abs/10.1137/21M1408166?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/3">Volume 60, Issue 3</a>, Page 9991025, June 2022. <br/> In this paper, we propose an embedded lowregularity integrator (ELRI) under a new framework for solving the modified Kortewegde Vries (mKdV) equation under rough data. Different from the previous work [Wu and Zhao, BIT, Number. Math., (2021)], the present ELRI scheme is constructed based on an approximation of a scaled Schrödinger operator and a new strategy of iterative regularizing through the inverse Miura transform. Moreover, the ELRI scheme is explicitly defined in the physical space, and it is efficient under the Fourier pseudospectral discretization. By rigorous error analysis, we show that ELRI achieves firstorder accuracy by requiring the boundedness of one additional spatial derivative of the solution. Numerical results are presented to show the accuracy and efficiency of ELRI.
SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 9991025, June 2022. <br/> In this paper, we propose an embedded lowregularity integrator (ELRI) under a new framework for solving the modified Kortewegde Vries (mKdV) equation under rough data. Different from the previous work [Wu and Zhao, BIT, Number. Math., (2021)], the present ELRI scheme is constructed based on an approximation of a scaled Schrödinger operator and a new strategy of iterative regularizing through the inverse Miura transform. Moreover, the ELRI scheme is explicitly defined in the physical space, and it is efficient under the Fourier pseudospectral discretization. By rigorous error analysis, we show that ELRI achieves firstorder accuracy by requiring the boundedness of one additional spatial derivative of the solution. Numerical results are presented to show the accuracy and efficiency of ELRI.
An Embedded ExponentialType LowRegularity Integrator for mKdV Equation
10.1137/21M1408166
SIAM Journal on Numerical Analysis
20220510T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Cui Ning
Yifei Wu
Xiaofei Zhao
An Embedded ExponentialType LowRegularity Integrator for mKdV Equation
60
3
999
1025
20220630T07:00:00Z
20220630T07:00:00Z
10.1137/21M1408166
https://epubs.siam.org/doi/abs/10.1137/21M1408166?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Stability and Error Analysis of IMEX SAV Schemes for the MagnetoHydrodynamic Equations
https://epubs.siam.org/doi/abs/10.1137/21M1430376?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/3">Volume 60, Issue 3</a>, Page 10261054, June 2022. <br/> We construct and analyze first and secondorder implicitexplicit schemes based on the scalar auxiliary variable approach for the magnetohydrodynamic equations. These schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and are unconditionally energy stable. We derive rigorous error estimates for the velocity, pressure, and magnetic field of the firstorder scheme in the twodimensional case without any condition on the time step. Numerical examples are presented to validate the proposed schemes.
SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 10261054, June 2022. <br/> We construct and analyze first and secondorder implicitexplicit schemes based on the scalar auxiliary variable approach for the magnetohydrodynamic equations. These schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and are unconditionally energy stable. We derive rigorous error estimates for the velocity, pressure, and magnetic field of the firstorder scheme in the twodimensional case without any condition on the time step. Numerical examples are presented to validate the proposed schemes.
Stability and Error Analysis of IMEX SAV Schemes for the MagnetoHydrodynamic Equations
10.1137/21M1430376
SIAM Journal on Numerical Analysis
20220510T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Xiaoli Li
Weilong Wang
Jie Shen
Stability and Error Analysis of IMEX SAV Schemes for the MagnetoHydrodynamic Equations
60
3
1026
1054
20220630T07:00:00Z
20220630T07:00:00Z
10.1137/21M1430376
https://epubs.siam.org/doi/abs/10.1137/21M1430376?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Local Convergence of the FEM for the Integral Fractional Laplacian
https://epubs.siam.org/doi/abs/10.1137/20M1343853?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/3">Volume 60, Issue 3</a>, Page 10551082, June 2022. <br/> For firstorder discretizations of the integral fractional Laplacian, we provide sharp local error estimates on proper subdomains in both the local $H^1$norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.
SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 10551082, June 2022. <br/> For firstorder discretizations of the integral fractional Laplacian, we provide sharp local error estimates on proper subdomains in both the local $H^1$norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.
Local Convergence of the FEM for the Integral Fractional Laplacian
10.1137/20M1343853
SIAM Journal on Numerical Analysis
20220512T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Markus Faustmann
Michael Karkulik
Jens Markus Melenk
Local Convergence of the FEM for the Integral Fractional Laplacian
60
3
1055
1082
20220630T07:00:00Z
20220630T07:00:00Z
10.1137/20M1343853
https://epubs.siam.org/doi/abs/10.1137/20M1343853?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Exponential Convolution Quadrature for Nonlinear Subdiffusion Equations with Nonsmooth Initial Data
https://epubs.siam.org/doi/abs/10.1137/21M1421386?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 503528, April 2022. <br/> An exponential type of convolution quadrature is proposed as a timestepping method for the nonlinear subdiffusion equation with bounded measurable initial data. The method combines contour integral representation of the solution, quadrature approximation of contour integrals, multistep exponential integrators for ordinary differential equations, and locally refined stepsizes to resolve the initial singularity. The proposed $k$step exponential convolution quadrature can have $k$thorder convergence for bounded measurable solutions of the nonlinear subdiffusion equation based on natural regularity of the solution with bounded measurable initial data.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 503528, April 2022. <br/> An exponential type of convolution quadrature is proposed as a timestepping method for the nonlinear subdiffusion equation with bounded measurable initial data. The method combines contour integral representation of the solution, quadrature approximation of contour integrals, multistep exponential integrators for ordinary differential equations, and locally refined stepsizes to resolve the initial singularity. The proposed $k$step exponential convolution quadrature can have $k$thorder convergence for bounded measurable solutions of the nonlinear subdiffusion equation based on natural regularity of the solution with bounded measurable initial data.
Exponential Convolution Quadrature for Nonlinear Subdiffusion Equations with Nonsmooth Initial Data
10.1137/21M1421386
SIAM Journal on Numerical Analysis
20220301T08:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Buyang Li
Shu Ma
Exponential Convolution Quadrature for Nonlinear Subdiffusion Equations with Nonsmooth Initial Data
60
2
503
528
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1421386
https://epubs.siam.org/doi/abs/10.1137/21M1421386?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (II). Error and Complexity Analysis
https://epubs.siam.org/doi/abs/10.1137/20M1357524?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 529557, April 2022. <br/> This work is dedicated to a refined error analysis of the highorder transparent boundary conditions for the weighted wave equation on a fractal tree, introduced in the companion work [P. Joly and M. Kachanovska, SIAM J. Sci. Comput., 43 (2021), pp. A3760A3788]. The construction of such boundary conditions relies on truncating the meromorphic series that represents the symbol of the DirichlettoNeumann operator. The error induced by the truncation depends on the behavior of the eigenvalues and the eigenfunctions of the weighted Laplacian on a selfsimilar metric tree. In this work we quantify this error by computing asymptotics of the eigenvalues and bounds for Neumann traces of the eigenfunctions. We prove the sharpness of the obtained bounds for a class of selfsimilar trees.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 529557, April 2022. <br/> This work is dedicated to a refined error analysis of the highorder transparent boundary conditions for the weighted wave equation on a fractal tree, introduced in the companion work [P. Joly and M. Kachanovska, SIAM J. Sci. Comput., 43 (2021), pp. A3760A3788]. The construction of such boundary conditions relies on truncating the meromorphic series that represents the symbol of the DirichlettoNeumann operator. The error induced by the truncation depends on the behavior of the eigenvalues and the eigenfunctions of the weighted Laplacian on a selfsimilar metric tree. In this work we quantify this error by computing asymptotics of the eigenvalues and bounds for Neumann traces of the eigenfunctions. We prove the sharpness of the obtained bounds for a class of selfsimilar trees.
Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (II). Error and Complexity Analysis
10.1137/20M1357524
SIAM Journal on Numerical Analysis
20220307T08:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Patrick Joly
Maryna Kachanovska
Local Transparent Boundary Conditions for Wave Propagation in Fractal Trees (II). Error and Complexity Analysis
60
2
529
557
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/20M1357524
https://epubs.siam.org/doi/abs/10.1137/20M1357524?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Full Discretization of Cauchy's Problem by LavrentievFinite Element Method
https://epubs.siam.org/doi/abs/10.1137/21M1401310?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 558584, April 2022. <br/> We conduct a detailed study of the fully discrete finite element approximation of the data completion problem. This is the continuation of [Numer. Math., 139 (2016), pp. 125], where the variational problem, resulting from the KohnVogelius duplication framed into the SteklovPoincaré condensation approach, was semidiscretized. Under the condition that the problem has a solution, we derive a bound of the error with respect to the meshsize and the Lavrentiev regularization parameter. Sharp local finite element estimates, such as those derived by Nitsche and Schatz [Math. Comp., 28 (1974), pp. 937958], are the central technical tools of the analysis.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 558584, April 2022. <br/> We conduct a detailed study of the fully discrete finite element approximation of the data completion problem. This is the continuation of [Numer. Math., 139 (2016), pp. 125], where the variational problem, resulting from the KohnVogelius duplication framed into the SteklovPoincaré condensation approach, was semidiscretized. Under the condition that the problem has a solution, we derive a bound of the error with respect to the meshsize and the Lavrentiev regularization parameter. Sharp local finite element estimates, such as those derived by Nitsche and Schatz [Math. Comp., 28 (1974), pp. 937958], are the central technical tools of the analysis.
Full Discretization of Cauchy's Problem by LavrentievFinite Element Method
10.1137/21M1401310
SIAM Journal on Numerical Analysis
20220315T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Faker Ben Belgacem
Vivette Girault
Faten Jelassi
Full Discretization of Cauchy's Problem by LavrentievFinite Element Method
60
2
558
584
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1401310
https://epubs.siam.org/doi/abs/10.1137/21M1401310?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Convergence Analysis of the Nonoverlapping RobinRobin Method for Nonlinear Elliptic Equations
https://epubs.siam.org/doi/abs/10.1137/21M1414942?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 585605, April 2022. <br/> We prove convergence for the nonoverlapping RobinRobin method applied to nonlinear elliptic equations with a $p$structure, including degenerate diffusion equations governed by the $p$Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear SteklovPoincaré operators based on the $p$structure and the $L^p$generalization of the LionsMagenes spaces. This framework allows the reformulation of the RobinRobin method into a PeacemanRachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 585605, April 2022. <br/> We prove convergence for the nonoverlapping RobinRobin method applied to nonlinear elliptic equations with a $p$structure, including degenerate diffusion equations governed by the $p$Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. Hence, we develop a new theory for nonlinear SteklovPoincaré operators based on the $p$structure and the $L^p$generalization of the LionsMagenes spaces. This framework allows the reformulation of the RobinRobin method into a PeacemanRachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. The analysis is performed on Lipschitz domains and without restrictive regularity assumptions on the solutions.
Convergence Analysis of the Nonoverlapping RobinRobin Method for Nonlinear Elliptic Equations
10.1137/21M1414942
SIAM Journal on Numerical Analysis
20220321T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Emil Engström
Eskil Hansen
Convergence Analysis of the Nonoverlapping RobinRobin Method for Nonlinear Elliptic Equations
60
2
585
605
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1414942
https://epubs.siam.org/doi/abs/10.1137/21M1414942?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

WellPosedness and Convergence of a Finite Volume Method for Conservation Laws on Networks
https://epubs.siam.org/doi/abs/10.1137/21M145001X?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 606630, April 2022. <br/> We continue the analysis of nonlinear conservation laws on networks initiated in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101128] by extending our analysis to a large class of flux functions which must be neither monotone nor convex/concave. We utilize the framework laid down in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101128] and prove existence and uniqueness within a natural class of entropy solutions via the convergence of an explicit finite volume method. In particular, this leads to the existence of a semigroup of solutions. The theoretical results are supported with numerical experiments including an experimental order of convergence.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 606630, April 2022. <br/> We continue the analysis of nonlinear conservation laws on networks initiated in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101128] by extending our analysis to a large class of flux functions which must be neither monotone nor convex/concave. We utilize the framework laid down in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101128] and prove existence and uniqueness within a natural class of entropy solutions via the convergence of an explicit finite volume method. In particular, this leads to the existence of a semigroup of solutions. The theoretical results are supported with numerical experiments including an experimental order of convergence.
WellPosedness and Convergence of a Finite Volume Method for Conservation Laws on Networks
10.1137/21M145001X
SIAM Journal on Numerical Analysis
20220322T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Ulrik S. Fjordholm
Markus Musch
Nils H. Risebro
WellPosedness and Convergence of a Finite Volume Method for Conservation Laws on Networks
60
2
606
630
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M145001X
https://epubs.siam.org/doi/abs/10.1137/21M145001X?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

A Monolithic DivergenceConforming HDG Scheme for a Linear FluidStructure Interaction Model
https://epubs.siam.org/doi/abs/10.1137/20M1385950?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 631658, April 2022. <br/> We present a novel monolithic divergenceconforming HDG scheme for a linear fluidstructure interaction problem with a thick structure. A pressurerobust optimal energynorm estimate is obtained for the semidiscrete scheme. When combined with a CrankNicolson time discretization, our fully discrete scheme is energy stable and produces an exactly divergencefree fluid velocity approximation. The resulting linear system, which is symmetric and indefinite, is solved using a preconditioned MinRes method with a robust block algebraic multigrid preconditioner.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 631658, April 2022. <br/> We present a novel monolithic divergenceconforming HDG scheme for a linear fluidstructure interaction problem with a thick structure. A pressurerobust optimal energynorm estimate is obtained for the semidiscrete scheme. When combined with a CrankNicolson time discretization, our fully discrete scheme is energy stable and produces an exactly divergencefree fluid velocity approximation. The resulting linear system, which is symmetric and indefinite, is solved using a preconditioned MinRes method with a robust block algebraic multigrid preconditioner.
A Monolithic DivergenceConforming HDG Scheme for a Linear FluidStructure Interaction Model
10.1137/20M1385950
SIAM Journal on Numerical Analysis
20220324T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Guosheng Fu
Wenzheng Kuang
A Monolithic DivergenceConforming HDG Scheme for a Linear FluidStructure Interaction Model
60
2
631
658
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/20M1385950
https://epubs.siam.org/doi/abs/10.1137/20M1385950?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

On the Convergence of Adaptive Stochastic Collocation for Elliptic Partial Differential Equations with Affine Diffusion
https://epubs.siam.org/doi/abs/10.1137/20M1364722?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 659687, April 2022. <br/> Convergence of an adaptive collocation method for the parametric stationary diffusion equation with finitedimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residualbased reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with a hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the now classical approach proposed in [T. Gerstner and M. Griebel, Computing, 71 (2003), pp. 6587]) are explored.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 659687, April 2022. <br/> Convergence of an adaptive collocation method for the parametric stationary diffusion equation with finitedimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residualbased reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with a hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the now classical approach proposed in [T. Gerstner and M. Griebel, Computing, 71 (2003), pp. 6587]) are explored.
On the Convergence of Adaptive Stochastic Collocation for Elliptic Partial Differential Equations with Affine Diffusion
10.1137/20M1364722
SIAM Journal on Numerical Analysis
20220324T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Martin Eigel
Oliver G. Ernst
Björn Sprungk
Lorenzo Tamellini
On the Convergence of Adaptive Stochastic Collocation for Elliptic Partial Differential Equations with Affine Diffusion
60
2
659
687
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/20M1364722
https://epubs.siam.org/doi/abs/10.1137/20M1364722?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

A SpaceTime Trefftz Discontinuous Galerkin Method for the Linear Schrödinger Equation
https://epubs.siam.org/doi/abs/10.1137/21M1426079?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 688714, April 2022. <br/> A spacetime Trefftz discontinuous Galerkin method for the Schrödinger equation with piecewiseconstant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by nonpolynomial complex wave functions that satisfy the Schrödinger equation locally on each element of the spacetime mesh. This allows for a significant reduction in the number of degrees of freedom in comparison with full polynomial spaces. We prove wellposedness and stability of the method and, for the one and twodimensional cases, optimal, highorder, $h$convergence error estimates in a skeleton norm. Some numerical experiments validate the theoretical results presented.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 688714, April 2022. <br/> A spacetime Trefftz discontinuous Galerkin method for the Schrödinger equation with piecewiseconstant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by nonpolynomial complex wave functions that satisfy the Schrödinger equation locally on each element of the spacetime mesh. This allows for a significant reduction in the number of degrees of freedom in comparison with full polynomial spaces. We prove wellposedness and stability of the method and, for the one and twodimensional cases, optimal, highorder, $h$convergence error estimates in a skeleton norm. Some numerical experiments validate the theoretical results presented.
A SpaceTime Trefftz Discontinuous Galerkin Method for the Linear Schrödinger Equation
10.1137/21M1426079
SIAM Journal on Numerical Analysis
20220331T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Sergio Gómez
Andrea Moiola
A SpaceTime Trefftz Discontinuous Galerkin Method for the Linear Schrödinger Equation
60
2
688
714
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1426079
https://epubs.siam.org/doi/abs/10.1137/21M1426079?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Approximations of Energy Minimization in CellInduced Phase Transitions of Fibrous Biomaterials: $\Gamma$Convergence Analysis
https://epubs.siam.org/doi/abs/10.1137/20M137286X?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 715750, April 2022. <br/> We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rankone convex nonlinear elasticity for phase transitions. We study appropriate numerical approximations based on the discontinuous Galerkin treatment of higher gradients and used succesfully in numerical simulations of experiments. We show that the discrete minimizers converge in the limit to minimizers of the continuous problem. This is achieved by employing the theory of $\Gamma$convergence of the approximate energy functionals to the continuous model when the discretization parameter tends to zero. The analysis is involved due to the structure of numerical approximations which are defined in spaces with lower regularity than the space where the minimizers of the continuous variational problem are sought. This fact leads to the development of a new approach to $\Gamma$convergence, appropriate for discontinuous finite element discretizations, which can be applied to quite general energy minimization problems. Furthermore, the adoption of exponential terms penalizing the interpenetration of matter requires a new framework based on Orlicz spaces for discontinuous Galerkin methods which is developed in this paper as well.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 715750, April 2022. <br/> We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rankone convex nonlinear elasticity for phase transitions. We study appropriate numerical approximations based on the discontinuous Galerkin treatment of higher gradients and used succesfully in numerical simulations of experiments. We show that the discrete minimizers converge in the limit to minimizers of the continuous problem. This is achieved by employing the theory of $\Gamma$convergence of the approximate energy functionals to the continuous model when the discretization parameter tends to zero. The analysis is involved due to the structure of numerical approximations which are defined in spaces with lower regularity than the space where the minimizers of the continuous variational problem are sought. This fact leads to the development of a new approach to $\Gamma$convergence, appropriate for discontinuous finite element discretizations, which can be applied to quite general energy minimization problems. Furthermore, the adoption of exponential terms penalizing the interpenetration of matter requires a new framework based on Orlicz spaces for discontinuous Galerkin methods which is developed in this paper as well.
Approximations of Energy Minimization in CellInduced Phase Transitions of Fibrous Biomaterials: $\Gamma$Convergence Analysis
10.1137/20M137286X
SIAM Journal on Numerical Analysis
20220404T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Georgios Grekas
Konstantinos Koumatos
Charalambos Makridakis
Phoebus Rosakis
Approximations of Energy Minimization in CellInduced Phase Transitions of Fibrous Biomaterials: $\Gamma$Convergence Analysis
60
2
715
750
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/20M137286X
https://epubs.siam.org/doi/abs/10.1137/20M137286X?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Stochastic Convergence of Regularized Solutions and Their Finite Element Approximations to Inverse Source Problems
https://epubs.siam.org/doi/abs/10.1137/21M1409779?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 751780, April 2022. <br/> In this work, we investigate the regularized solutions and their finite element solutions to the inverse source problems governed by partial differential equations, and we establish the stochastic convergence and optimal finite element convergence rates of these solutions under pointwise measurement data with random noise. The regularization error estimates and the finite element error estimates are derived with explicit dependence on the noise level, regularization parameter, mesh size, and time step size, which can guide practical choices among these key parameters in real applications. The error estimates also suggest an iterative algorithm for determining an optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the analytical results.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 751780, April 2022. <br/> In this work, we investigate the regularized solutions and their finite element solutions to the inverse source problems governed by partial differential equations, and we establish the stochastic convergence and optimal finite element convergence rates of these solutions under pointwise measurement data with random noise. The regularization error estimates and the finite element error estimates are derived with explicit dependence on the noise level, regularization parameter, mesh size, and time step size, which can guide practical choices among these key parameters in real applications. The error estimates also suggest an iterative algorithm for determining an optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the analytical results.
Stochastic Convergence of Regularized Solutions and Their Finite Element Approximations to Inverse Source Problems
10.1137/21M1409779
SIAM Journal on Numerical Analysis
20220405T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Zhiming Chen
Wenlong Zhang
Jun Zou
Stochastic Convergence of Regularized Solutions and Their Finite Element Approximations to Inverse Source Problems
60
2
751
780
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1409779
https://epubs.siam.org/doi/abs/10.1137/21M1409779?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Convergence Analysis of the Variational Operator Splitting Scheme for a ReactionDiffusion System with Detailed Balance
https://epubs.siam.org/doi/abs/10.1137/21M1421283?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 781803, April 2022. <br/> We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu, C. Wang, and Y. Wang, J. Comput. Phys., 436 (2021), 110253] for a reactiondiffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivitypreserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms. In addition, a combination of rough error estimate and refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting procedure results in the convergence estimate of the numerical scheme for the full reactiondiffusion system. The convergence analysis technique could be extended to a more general class of dissipative reaction mechanisms. As an example, we also consider a nearequilibrium reaction kinetics, which was derived by the linear response assumption on the reaction trajectory. Although the reaction rate is more complicated in terms of concentration variables, we show that the numerical approach and the convergence analysis also work in this case.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 781803, April 2022. <br/> We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu, C. Wang, and Y. Wang, J. Comput. Phys., 436 (2021), 110253] for a reactiondiffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation, in which the reaction part is reformulated in terms of the reaction trajectory, and both the reaction and diffusion parts dissipate the same free energy. The scheme is energy stable and positivitypreserving. In this paper, the detailed convergence analysis and error estimate are performed for the operator splitting scheme. The nonlinearity in the reaction trajectory equation, as well as the implicit treatment of nonlinear and singular logarithmic terms, impose challenges in numerical analysis. To overcome these difficulties, we make use of the convex nature of the logarithmic nonlinear terms. In addition, a combination of rough error estimate and refined error estimate leads to a desired bound of the numerical error in the reaction stage, in the discrete maximum norm. Furthermore, a discrete maximum principle yields the evolution bound of the numerical error function at the diffusion stage. As a direct consequence, a combination of the numerical error analysis at different stages and the consistency estimate for the operator splitting procedure results in the convergence estimate of the numerical scheme for the full reactiondiffusion system. The convergence analysis technique could be extended to a more general class of dissipative reaction mechanisms. As an example, we also consider a nearequilibrium reaction kinetics, which was derived by the linear response assumption on the reaction trajectory. Although the reaction rate is more complicated in terms of concentration variables, we show that the numerical approach and the convergence analysis also work in this case.
Convergence Analysis of the Variational Operator Splitting Scheme for a ReactionDiffusion System with Detailed Balance
10.1137/21M1421283
SIAM Journal on Numerical Analysis
20220413T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Chun Liu
Cheng Wang
Yiwei Wang
Steven M. Wise
Convergence Analysis of the Variational Operator Splitting Scheme for a ReactionDiffusion System with Detailed Balance
60
2
781
803
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1421283
https://epubs.siam.org/doi/abs/10.1137/21M1421283?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Exponential Convergence of Perfectly Matched Layers for Scattering Problems with Periodic Surfaces
https://epubs.siam.org/doi/abs/10.1137/21M1439043?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 804823, April 2022. <br/> The main task of this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter for scattering problems with periodic surfaces. In [S. N. ChandlerWilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 21312154], a linear convergence is proved for the PML method for scattering problems with rough surfaces. At the end of that paper, three important questions are asked, and the third question is whether exponential convergence holds locally. In our paper, we answer this open question for a special case, when the rough surface is actually periodic. Due to technical reasons, we have to exclude all the wavenumbers which are half integers. The main idea of the proof is to apply the FloquetBloch transform to rewrite the problem as an equivalent family of quasiperiodic problems, and then study the analytic extension of the quasiperiodic problems with respect to the FloquetBloch parameters. Then the Cauchy integral formula is applied to avoid linear convergent points. Finally, the exponential convergence is proved from the inverse FloquetBloch transform. Numerical results are also presented at the end of this paper.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 804823, April 2022. <br/> The main task of this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter for scattering problems with periodic surfaces. In [S. N. ChandlerWilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 21312154], a linear convergence is proved for the PML method for scattering problems with rough surfaces. At the end of that paper, three important questions are asked, and the third question is whether exponential convergence holds locally. In our paper, we answer this open question for a special case, when the rough surface is actually periodic. Due to technical reasons, we have to exclude all the wavenumbers which are half integers. The main idea of the proof is to apply the FloquetBloch transform to rewrite the problem as an equivalent family of quasiperiodic problems, and then study the analytic extension of the quasiperiodic problems with respect to the FloquetBloch parameters. Then the Cauchy integral formula is applied to avoid linear convergent points. Finally, the exponential convergence is proved from the inverse FloquetBloch transform. Numerical results are also presented at the end of this paper.
Exponential Convergence of Perfectly Matched Layers for Scattering Problems with Periodic Surfaces
10.1137/21M1439043
SIAM Journal on Numerical Analysis
20220413T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Ruming Zhang
Exponential Convergence of Perfectly Matched Layers for Scattering Problems with Periodic Surfaces
60
2
804
823
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1439043
https://epubs.siam.org/doi/abs/10.1137/21M1439043?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Efficient Approximation of SDEs Driven by Countably Dimensional Wiener Process and Poisson Random Measure
https://epubs.siam.org/doi/abs/10.1137/21M1442747?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 824855, April 2022. <br/> In this paper we deal with pointwise approximation of solutions of stochastic differential equations (SDEs) driven by an infinite dimensional Wiener process, with additional jumps generated by a Poisson random measure. Further investigations contain upper error bounds for the proposed truncated dimension randomized Euler scheme. We also establish matching (up to constants) upper and lower bounds for $\varepsilon$complexity and show that the defined algorithm is optimal in the informationbased complexity (IBC) sense. Finally, results of numerical experiments performed via GPU architecture are also reported.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 824855, April 2022. <br/> In this paper we deal with pointwise approximation of solutions of stochastic differential equations (SDEs) driven by an infinite dimensional Wiener process, with additional jumps generated by a Poisson random measure. Further investigations contain upper error bounds for the proposed truncated dimension randomized Euler scheme. We also establish matching (up to constants) upper and lower bounds for $\varepsilon$complexity and show that the defined algorithm is optimal in the informationbased complexity (IBC) sense. Finally, results of numerical experiments performed via GPU architecture are also reported.
Efficient Approximation of SDEs Driven by Countably Dimensional Wiener Process and Poisson Random Measure
10.1137/21M1442747
SIAM Journal on Numerical Analysis
20220418T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Paweł Przybyłowicz
Michał Sobieraj
Łukasz Stȩpień
Efficient Approximation of SDEs Driven by Countably Dimensional Wiener Process and Poisson Random Measure
60
2
824
855
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1442747
https://epubs.siam.org/doi/abs/10.1137/21M1442747?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

GammaConvergent ProjectionFree Finite Element Methods for Nematic Liquid Crystals: The Ericksen Model
https://epubs.siam.org/doi/abs/10.1137/21M1407495?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 856887, April 2022. <br/> The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation of the problem that can be implemented easily within standard finite element packages. Our scheme is projectionfree and thus circumvents the use of weakly acute meshes, which are quite restrictive in three dimensions but are required by recent algorithms for convergence. We prove stability and $\Gamma$convergence properties of the new method in the presence of defects. We also design an effective nested gradient flow algorithm for computing minimizers that controls the violation of the unitlength constraint of the director. We present several simulations in two and three dimensions that document the performance of the proposed scheme and its ability to capture quite intriguing defects.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 856887, April 2022. <br/> The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation of the problem that can be implemented easily within standard finite element packages. Our scheme is projectionfree and thus circumvents the use of weakly acute meshes, which are quite restrictive in three dimensions but are required by recent algorithms for convergence. We prove stability and $\Gamma$convergence properties of the new method in the presence of defects. We also design an effective nested gradient flow algorithm for computing minimizers that controls the violation of the unitlength constraint of the director. We present several simulations in two and three dimensions that document the performance of the proposed scheme and its ability to capture quite intriguing defects.
GammaConvergent ProjectionFree Finite Element Methods for Nematic Liquid Crystals: The Ericksen Model
10.1137/21M1407495
SIAM Journal on Numerical Analysis
20220419T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
Ricardo H. Nochetto
Michele Ruggeri
Shuo Yang
GammaConvergent ProjectionFree Finite Element Methods for Nematic Liquid Crystals: The Ericksen Model
60
2
856
887
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1407495
https://epubs.siam.org/doi/abs/10.1137/21M1407495?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics

Uniformly Accurate Low Regularity Integrators for the KleinGordon Equation from the Classical to NonRelativistic Limit Regime
https://epubs.siam.org/doi/abs/10.1137/21M1415030?ai=s0&mi=3bfys9&af=R
SIAM Journal on Numerical Analysis, <a href="https://epubs.siam.org/toc/sjnaam/60/2">Volume 60, Issue 2</a>, Page 888912, April 2022. <br/> We propose a novel class of uniformly accurate integrators for the KleinGordon equation which capture classical $c=1$ as well as highly oscillatory nonrelativistic regimes $c\gg1$ and, at the same time, allow for low regularity approximations. In particular, the schemes converge with order $\tau$ and $\tau^2$, respectively, under lower regularity assumptions than classical schemes, such as splitting or exponential integrator methods, require. The new schemes in addition preserve the nonlinear Schrödinger (NLS) limit on the discrete level. More precisely, we will design our schemes in such a way that in the limit $c\to \infty$ they converge to a recently introduced class of low regularity integrators for NLS.
SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 888912, April 2022. <br/> We propose a novel class of uniformly accurate integrators for the KleinGordon equation which capture classical $c=1$ as well as highly oscillatory nonrelativistic regimes $c\gg1$ and, at the same time, allow for low regularity approximations. In particular, the schemes converge with order $\tau$ and $\tau^2$, respectively, under lower regularity assumptions than classical schemes, such as splitting or exponential integrator methods, require. The new schemes in addition preserve the nonlinear Schrödinger (NLS) limit on the discrete level. More precisely, we will design our schemes in such a way that in the limit $c\to \infty$ they converge to a recently introduced class of low regularity integrators for NLS.
Uniformly Accurate Low Regularity Integrators for the KleinGordon Equation from the Classical to NonRelativistic Limit Regime
10.1137/21M1415030
SIAM Journal on Numerical Analysis
20220421T07:00:00Z
© 2022, Society for Industrial and Applied Mathematics
María Cabrera Calvo
Katharina Schratz
Uniformly Accurate Low Regularity Integrators for the KleinGordon Equation from the Classical to NonRelativistic Limit Regime
60
2
888
912
20220430T07:00:00Z
20220430T07:00:00Z
10.1137/21M1415030
https://epubs.siam.org/doi/abs/10.1137/21M1415030?ai=s0&mi=3bfys9&af=R
© 2022, Society for Industrial and Applied Mathematics