Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently. In the first stage, we develop an efficient algorithm for computing an interpolant to $F$, inspired by the ‘metric polynomial interpolation’, which is based on the theory in Dyn et al. (2014, Approximation of Set-Valued Functions: Adaptation of Classical Approximation Operators. Imperial College Press). By this theory, a ‘metric polynomial interpolant’ is a collection of polynomial interpolants to all the ‘metric chains’ of the given samples of $F$. For set-valued functions whose graphs have nonempty interior, the collection of these ‘metric chains’ can be infinite. Our algorithm computes a small finite subset of ‘significant metric chains’, which is sufficient for approximating $F$. For the class of Lipschitz continuous functions with samples at the roots of the Chebyshev polynomials of the first kind, we prove that the error incurred by our computed interpolant decays with increasing number of interpolation points in the same rate as in the case of interpolation by the metric polynomial interpolant. This is also demonstrated by our numerical examples. For the class of set-valued functions whose graphs have smooth boundaries, we extend our algorithm to achieve a high-precision detection of the points of topology change, followed by a high-order approximation of the boundaries of the graph of F. We further discuss the case of set-valued functions whose graphs have ‘holes’ with Hölder-type singularities at the points of change of topology. To treat this case we apply some special approximation ideas near the singular points of the holes. We analyze the approximation order of the algorithm, including the error in approximating the points of change of topology, and show by several numerical examples the capability of obtaining high-order approximation of the holes.
给定连续集值函数 F 的有限数量样本，将区间映射到实线的紧凑子集，我们就能开发出 F 的良好近似值，并能高效计算。在第一阶段，我们受 "度量多项式插值 "的启发，开发了一种计算 $F$ 插值的高效算法，该算法基于 Dyn 等人（2014，Approximation of Set-Valued Functions：Adaptation of Classical Approximation Operators.帝国学院出版社）。根据这一理论，"度量多项式插值 "是$F$给定样本的所有 "度量链 "的多项式插值的集合。对于图具有非空内部的集值函数，这些 "度量链 "的集合可能是无限的。我们的算法可以计算出一小部分有限的 "重要度量链 "子集，这足以逼近 $F$。对于在切比雪夫多项式第一种的根上有样本的利普齐兹连续函数类，我们证明了我们计算的插值器所产生的误差随着插值点数量的增加而减小，减小的速度与用度量多项式插值器插值时的速度相同。我们的数值示例也证明了这一点。对于图形具有平滑边界的一类集值函数，我们扩展了算法，以实现拓扑变化点的高精度检测，然后对 F 的图形边界进行高阶近似。为了处理这种情况，我们在洞的奇点附近应用了一些特殊的近似思想。我们分析了算法的近似阶数，包括拓扑变化点的近似误差，并通过几个数值示例展示了获得洞的高阶近似的能力。
{"title":"Interpolation of set-valued functions","authors":"Nira Dyn, David Levin, Qusay Muzaffar","doi":"10.1093/imanum/drae031","DOIUrl":"https://doi.org/10.1093/imanum/drae031","url":null,"abstract":"Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently. In the first stage, we develop an efficient algorithm for computing an interpolant to $F$, inspired by the ‘metric polynomial interpolation’, which is based on the theory in Dyn et al. (2014, Approximation of Set-Valued Functions: Adaptation of Classical Approximation Operators. Imperial College Press). By this theory, a ‘metric polynomial interpolant’ is a collection of polynomial interpolants to all the ‘metric chains’ of the given samples of $F$. For set-valued functions whose graphs have nonempty interior, the collection of these ‘metric chains’ can be infinite. Our algorithm computes a small finite subset of ‘significant metric chains’, which is sufficient for approximating $F$. For the class of Lipschitz continuous functions with samples at the roots of the Chebyshev polynomials of the first kind, we prove that the error incurred by our computed interpolant decays with increasing number of interpolation points in the same rate as in the case of interpolation by the metric polynomial interpolant. This is also demonstrated by our numerical examples. For the class of set-valued functions whose graphs have smooth boundaries, we extend our algorithm to achieve a high-precision detection of the points of topology change, followed by a high-order approximation of the boundaries of the graph of F. We further discuss the case of set-valued functions whose graphs have ‘holes’ with Hölder-type singularities at the points of change of topology. To treat this case we apply some special approximation ideas near the singular points of the holes. We analyze the approximation order of the algorithm, including the error in approximating the points of change of topology, and show by several numerical examples the capability of obtaining high-order approximation of the holes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141452781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lina Wang, Mingzhu Zhang, Hongjiong Tian, Lijun Yi
We introduce and analyze an $hp$-version $C^{1}$-continuous Petrov–Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the $L^{2}$-, $L^{infty }$-, $H^{1}$- and $H^{2}$-norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$-version $C^{1}$-CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the $hp$-version $C^{1}$-CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.
{"title":"hp-version C1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations","authors":"Lina Wang, Mingzhu Zhang, Hongjiong Tian, Lijun Yi","doi":"10.1093/imanum/drae036","DOIUrl":"https://doi.org/10.1093/imanum/drae036","url":null,"abstract":"We introduce and analyze an $hp$-version $C^{1}$-continuous Petrov–Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the $L^{2}$-, $L^{infty }$-, $H^{1}$- and $H^{2}$-norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$-version $C^{1}$-CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the $hp$-version $C^{1}$-CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141448749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The minimum action method (MAM) is an effective approach to numerically solving minima and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method with nonuniform mesh, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our main result also reveals the convergence of the stochastic $theta $-method for SDEs with small noise in terms of large deviations. Numerical experiments are reported to verify the theoretical results.
{"title":"Convergence analysis for minimum action methods coupled with a finite difference method","authors":"Jialin Hong, Diancong Jin, Derui Sheng","doi":"10.1093/imanum/drae038","DOIUrl":"https://doi.org/10.1093/imanum/drae038","url":null,"abstract":"The minimum action method (MAM) is an effective approach to numerically solving minima and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method with nonuniform mesh, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our main result also reveals the convergence of the stochastic $theta $-method for SDEs with small noise in terms of large deviations. Numerical experiments are reported to verify the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141452971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $sigma in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $mathcal{O}big (h^{frac{1}{2}}big )$.
{"title":"Precise error bounds for numerical approximations of fractional HJB equations","authors":"Indranil Chowdhury, Espen R Jakobsen","doi":"10.1093/imanum/drae030","DOIUrl":"https://doi.org/10.1093/imanum/drae030","url":null,"abstract":"We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $sigma in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $mathcal{O}big (h^{frac{1}{2}}big )$.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"2014 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Virtual Element Method (VEM) for diffusion-convection-reaction problems is considered. In order to design a quasi-robust scheme also in the convection-dominated regime, a Continuous Interior Penalty approach is employed. Due to the presence of polynomial projection operators, typical of the VEM, the stability and the error analysis requires particular care—especially in treating the advective term. Some numerical tests are presented to support the theoretical results.
{"title":"CIP-stabilized virtual elements for diffusion-convection-reaction problems","authors":"L Beirão da Veiga, C Lovadina, M Trezzi","doi":"10.1093/imanum/drae020","DOIUrl":"https://doi.org/10.1093/imanum/drae020","url":null,"abstract":"The Virtual Element Method (VEM) for diffusion-convection-reaction problems is considered. In order to design a quasi-robust scheme also in the convection-dominated regime, a Continuous Interior Penalty approach is employed. Due to the presence of polynomial projection operators, typical of the VEM, the stability and the error analysis requires particular care—especially in treating the advective term. Some numerical tests are presented to support the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141185236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate stochastic modified equations to explain the mathematical mechanism of symplectic methods applied to rough Hamiltonian systems. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation. For symplectic methods applied to rough Hamiltonian systems, the associated stochastic modified equations are proved to have Hamiltonian formulations. Secondly, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in rough path theory. Thirdly, if increments of noises are simulated by truncated random variables, we show that the error can be made exponentially small with respect to the time step size.
{"title":"Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems","authors":"Chuchu Chen, Jialin Hong, Chuying Huang","doi":"10.1093/imanum/drae019","DOIUrl":"https://doi.org/10.1093/imanum/drae019","url":null,"abstract":"We investigate stochastic modified equations to explain the mathematical mechanism of symplectic methods applied to rough Hamiltonian systems. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation. For symplectic methods applied to rough Hamiltonian systems, the associated stochastic modified equations are proved to have Hamiltonian formulations. Secondly, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in rough path theory. Thirdly, if increments of noises are simulated by truncated random variables, we show that the error can be made exponentially small with respect to the time step size.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"72 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{mathbb{R}}}^{d}$, $dgeq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.
{"title":"Discrete anisotropic curve shortening flow in higher codimension","authors":"Klaus Deckelnick, Robert Nürnberg","doi":"10.1093/imanum/drae015","DOIUrl":"https://doi.org/10.1093/imanum/drae015","url":null,"abstract":"We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{mathbb{R}}}^{d}$, $dgeq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"73 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $mu $ is large, the error is $O(h^{k+1}+varDelta t^{q})$, whereas in the advective regime, $mu ll 1$, the convergence is $O(min (h^{k},frac{h^{k+1} }{varDelta t})+varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $mu ^{-1}$ dependence.
本文介绍了针对平流-扩散方程保守公式的高阶时间和空间拉格朗日-加勒金方法的数值分析。作为时间离散化方案，我们考虑了最高阶数为 $q=5$ 的后向微分公式。方法的开发和分析是在 C. M. Elliot 和 T. Ranner, IMA Journal of Numerical Analysis41, 1696-1845 (2021) 中提出的时间演化有限元框架内进行的。误差估计值通过其对方程参数的依赖性显示出数值解的行为存在不同的体制；即在扩散状态下，即扩散参数 $mu $ 较大时，误差为 $O(h^{k+1}+varDelta t^{q})$ ；而在平流状态下，即 $mu ll 1$，收敛为 $O(min(h^{k},frac{h^{k+1}}{varDelta t})+varDelta t^{q})$ 。值得注意的是，误差常数与 $mu ^{-1}$不呈指数关系。
{"title":"High-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation","authors":"Rodolfo Bermejo, Manuel Colera","doi":"10.1093/imanum/drae018","DOIUrl":"https://doi.org/10.1093/imanum/drae018","url":null,"abstract":"We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $mu $ is large, the error is $O(h^{k+1}+varDelta t^{q})$, whereas in the advective regime, $mu ll 1$, the convergence is $O(min (h^{k},frac{h^{k+1} }{varDelta t})+varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $mu ^{-1}$ dependence.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"127 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.
{"title":"An equilibrated estimator for mixed finite element discretizations of the curl-curl problem","authors":"T Chaumont-Frelet","doi":"10.1093/imanum/drae007","DOIUrl":"https://doi.org/10.1093/imanum/drae007","url":null,"abstract":"We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"126 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $beta =0$ and RPC-SAV-CN when $beta in (0,frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-alpha _{1}varDelta tnu varDelta (widetilde {textbf {u}}^{n+1}-{textbf {u}}^{n})$ and $alpha _{2}varDelta tsigma ^{-1}mbox {curl}mbox {curl} (textbf {H}^{n+1}-textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.
{"title":"Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations","authors":"Xiaojing Dong, Huayi Huang, Yunqing Huang, Xiaojuan Shen, Qili Tang","doi":"10.1093/imanum/drae016","DOIUrl":"https://doi.org/10.1093/imanum/drae016","url":null,"abstract":"In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $beta =0$ and RPC-SAV-CN when $beta in (0,frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-alpha _{1}varDelta tnu varDelta (widetilde {textbf {u}}^{n+1}-{textbf {u}}^{n})$ and $alpha _{2}varDelta tsigma ^{-1}mbox {curl}mbox {curl} (textbf {H}^{n+1}-textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"48 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140949377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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