SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 73-92, February 2024. Abstract. Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical methods for DLRA need to be carefully constructed to guarantee stability while preserving crucial properties of the original problem. Important physical effects that one likes to preserve with DLRA include capturing the diffusion limit in the high-scattering regimes as well as dissipating energy. In this work we propose and analyze a dynamical low-rank method based on the “unconventional” basis update & Galerkin step integrator. We show that this method is asymptotic preserving, i.e., it captures the diffusion limit, and energy stable under a CFL condition. The derived CFL condition captures the transition from the hyperbolic to the parabolic regime when approaching the diffusion limit.
{"title":"Asymptotic-Preserving and Energy Stable Dynamical Low-Rank Approximation","authors":"Lukas Einkemmer, Jingwei Hu, Jonas Kusch","doi":"10.1137/23m1547603","DOIUrl":"https://doi.org/10.1137/23m1547603","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 73-92, February 2024. <br/> Abstract. Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical methods for DLRA need to be carefully constructed to guarantee stability while preserving crucial properties of the original problem. Important physical effects that one likes to preserve with DLRA include capturing the diffusion limit in the high-scattering regimes as well as dissipating energy. In this work we propose and analyze a dynamical low-rank method based on the “unconventional” basis update & Galerkin step integrator. We show that this method is asymptotic preserving, i.e., it captures the diffusion limit, and energy stable under a CFL condition. The derived CFL condition captures the transition from the hyperbolic to the parabolic regime when approaching the diffusion limit.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139407651","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}
Abhijit Biswas, David Ketcheson, Benjamin Seibold, David Shirokoff
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 48-72, February 2024. Abstract. Runge–Kutta (RK) methods may exhibit order reduction when applied to stiff problems. For linear problems with time-independent operators, order reduction can be avoided if the method satisfies certain weak stage order (WSO) conditions, which are less restrictive than traditional stage order conditions. This paper outlines the first algebraic theory of WSO, and establishes general order barriers that relate the WSO of a RK scheme to its order and number of stages for both fully-implicit and DIRK schemes. It is shown in several scenarios that the constructed bounds are sharp. The theory characterizes WSO in terms of orthogonal invariant subspaces and associated minimal polynomials. The resulting necessary conditions on the structure of RK methods with WSO are then shown to be of practical use for the construction of such schemes.
{"title":"Algebraic Structure of the Weak Stage Order Conditions for Runge–Kutta Methods","authors":"Abhijit Biswas, David Ketcheson, Benjamin Seibold, David Shirokoff","doi":"10.1137/22m1483943","DOIUrl":"https://doi.org/10.1137/22m1483943","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 48-72, February 2024. <br/> Abstract. Runge–Kutta (RK) methods may exhibit order reduction when applied to stiff problems. For linear problems with time-independent operators, order reduction can be avoided if the method satisfies certain weak stage order (WSO) conditions, which are less restrictive than traditional stage order conditions. This paper outlines the first algebraic theory of WSO, and establishes general order barriers that relate the WSO of a RK scheme to its order and number of stages for both fully-implicit and DIRK schemes. It is shown in several scenarios that the constructed bounds are sharp. The theory characterizes WSO in terms of orthogonal invariant subspaces and associated minimal polynomials. The resulting necessary conditions on the structure of RK methods with WSO are then shown to be of practical use for the construction of such schemes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139101454","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 1-24, February 2024. Abstract. One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus are inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs’ model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration’s FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics previously developed in [J. Han, R. Hu, and J. Long, Stochastic Process. Appl., 164 (2023), pp. 242–287]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker–Smale model, the cost of which depends on the full distribution of the forward process.
{"title":"Learning High-Dimensional McKean–Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence","authors":"Jiequn Han, Ruimeng Hu, Jihao Long","doi":"10.1137/22m151861x","DOIUrl":"https://doi.org/10.1137/22m151861x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 1-24, February 2024. <br/> Abstract. One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus are inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs’ model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration’s FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics previously developed in [J. Han, R. Hu, and J. Long, Stochastic Process. Appl., 164 (2023), pp. 242–287]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker–Smale model, the cost of which depends on the full distribution of the forward process.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139101458","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 25-47, February 2024. Abstract. Lattice Green’s functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order discretization and operate on domains with free-space boundary conditions in all directions. Under these conditions, fast expansion methods are available that enable precomputation of 2-dimensional or 3-dimensional (3D) LGFs in linear time, avoiding the need for brute-force multidimensional quadrature of numerically unstable integrals. Here we focus on higher-order discretizations of the Laplace operator on domains with more general boundary conditions, by (1) providing an algorithm for fast and accurate evaluation of the LGFs associated with high-order dimension-split centered finite differences on unbounded domains, and (2) deriving closed-form expressions for the LGFs associated with both dimension-split and Mehrstellen discretizations on domains with one unbounded dimension. Through numerical experiments we demonstrate that these techniques provide LGF evaluations with near machine-precision accuracy, and that the resulting LGFs allow for numerically consistent solutions to high-order discretizations of the Poisson’s equation on fully or partially unbounded 3D domains.
{"title":"Lattice Green’s Functions for High-Order Finite Difference Stencils","authors":"James Gabbard, Wim M. van Rees","doi":"10.1137/23m1573872","DOIUrl":"https://doi.org/10.1137/23m1573872","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 25-47, February 2024. <br/> Abstract. Lattice Green’s functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order discretization and operate on domains with free-space boundary conditions in all directions. Under these conditions, fast expansion methods are available that enable precomputation of 2-dimensional or 3-dimensional (3D) LGFs in linear time, avoiding the need for brute-force multidimensional quadrature of numerically unstable integrals. Here we focus on higher-order discretizations of the Laplace operator on domains with more general boundary conditions, by (1) providing an algorithm for fast and accurate evaluation of the LGFs associated with high-order dimension-split centered finite differences on unbounded domains, and (2) deriving closed-form expressions for the LGFs associated with both dimension-split and Mehrstellen discretizations on domains with one unbounded dimension. Through numerical experiments we demonstrate that these techniques provide LGF evaluations with near machine-precision accuracy, and that the resulting LGFs allow for numerically consistent solutions to high-order discretizations of the Poisson’s equation on fully or partially unbounded 3D domains.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139101294","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}
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2994-3013, December 2023. Abstract. We consider a Markov chain Monte Carlo method, known as Hamiltonian Monte Carlo (HMC), on compact manifolds in Euclidean space. It utilizes Hamiltonian dynamics to generate samples approximating a target distribution in high dimensions efficiently. The efficiency of HMC is characterized by its convergence property, called geometric ergodicity. This property is important to generate low-correlated samples. It also plays a crucial role in establishing the error estimate for the quadrature of bounded functions by HMC sampling, referred to as the Hoeffding-type inequality. While the geometric ergodicity has been proved for HMC on Euclidean space, it has not been established on manifolds. In this paper, we prove the geometric ergodicity for HMC on compact manifolds. As an example to confirm the efficiency of the proposed HMC method, we consider a sampling problem associated with the [math]-vortex problem on the unit sphere, which is a statistical model of two-dimensional turbulence. We apply HMC to approximate the statistical quantities with respect to the invariant measure of the [math]-vortex problem, called the Gibbs measure. We observe the organization of large vortex structures as seen in two-dimensional turbulence.
{"title":"Geometric Ergodicity for Hamiltonian Monte Carlo on Compact Manifolds","authors":"Kota Takeda, Takashi Sakajo","doi":"10.1137/22m1543550","DOIUrl":"https://doi.org/10.1137/22m1543550","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2994-3013, December 2023. <br/> Abstract. We consider a Markov chain Monte Carlo method, known as Hamiltonian Monte Carlo (HMC), on compact manifolds in Euclidean space. It utilizes Hamiltonian dynamics to generate samples approximating a target distribution in high dimensions efficiently. The efficiency of HMC is characterized by its convergence property, called geometric ergodicity. This property is important to generate low-correlated samples. It also plays a crucial role in establishing the error estimate for the quadrature of bounded functions by HMC sampling, referred to as the Hoeffding-type inequality. While the geometric ergodicity has been proved for HMC on Euclidean space, it has not been established on manifolds. In this paper, we prove the geometric ergodicity for HMC on compact manifolds. As an example to confirm the efficiency of the proposed HMC method, we consider a sampling problem associated with the [math]-vortex problem on the unit sphere, which is a statistical model of two-dimensional turbulence. We apply HMC to approximate the statistical quantities with respect to the invariant measure of the [math]-vortex problem, called the Gibbs measure. We observe the organization of large vortex structures as seen in two-dimensional turbulence.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138550748","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}
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2940-2966, December 2023. Abstract. The discontinuous Galerkin approximation of the grad-div and curl-curl problems formulated in conservative first-order form is investigated. It is shown that the approximation is spectrally correct, thereby confirming numerical observations made by various authors in the literature. This result hinges on the existence of discrete involutions which are formulated as discrete orthogonality properties. The involutions are crucial to establish discrete versions of weak Poincaré–Steklov inequalities that hold true at the continuous level.
{"title":"The Discontinuous Galerkin Approximation of the Grad-Div and Curl-Curl Operators in First-Order Form Is Involution-Preserving and Spectrally Correct","authors":"Alexandre Ern, Jean-Luc Guermond","doi":"10.1137/23m1555235","DOIUrl":"https://doi.org/10.1137/23m1555235","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2940-2966, December 2023. <br/> Abstract. The discontinuous Galerkin approximation of the grad-div and curl-curl problems formulated in conservative first-order form is investigated. It is shown that the approximation is spectrally correct, thereby confirming numerical observations made by various authors in the literature. This result hinges on the existence of discrete involutions which are formulated as discrete orthogonality properties. The involutions are crucial to establish discrete versions of weak Poincaré–Steklov inequalities that hold true at the continuous level.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138544630","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}
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2967-2993, December 2023. Abstract. We provide the convergence analysis for a [math]-Galerkin method to solve the fractional Dirichlet problem. This can be understood as a follow-up of [H. Antil, P. Dondl, and L. Striet, SIAM J. Sci. Comput., 43 (2021), pp. A2897–A2922], where the authors presented a [math]-function based method to solve fractional PDEs. While the original method was formulated as a collocation method, we show that the same method can be interpreted as a nonconforming Galerkin method, giving access to abstract error estimates. Optimal order of convergence is shown without any unrealistic regularity assumptions on the solution.
SIAM 数值分析期刊》，第 61 卷，第 6 期，第 2967-2993 页，2023 年 12 月。 摘要。我们提供了求解分数 Dirichlet 问题的 [math]-Galerkin 方法的收敛性分析。这可以理解为 [H. Antil, P. Dondl] 方法的后续。Antil, P. Dondl, and L. Striet, SIAM J. Sci. Comput., 43 (2021), pp.虽然最初的方法是作为配位法提出的，但我们证明，同样的方法可以解释为非顺应 Galerkin 方法，从而获得抽象误差估计。在不对解作任何不切实际的正则性假设的情况下，我们展示了最佳收敛阶次。
{"title":"Analysis of a sinc-Galerkin Method for the Fractional Laplacian","authors":"Harbir Antil, Patrick W. Dondl, Ludwig Striet","doi":"10.1137/22m1542374","DOIUrl":"https://doi.org/10.1137/22m1542374","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2967-2993, December 2023. <br/> Abstract. We provide the convergence analysis for a [math]-Galerkin method to solve the fractional Dirichlet problem. This can be understood as a follow-up of [H. Antil, P. Dondl, and L. Striet, SIAM J. Sci. Comput., 43 (2021), pp. A2897–A2922], where the authors presented a [math]-function based method to solve fractional PDEs. While the original method was formulated as a collocation method, we show that the same method can be interpreted as a nonconforming Galerkin method, giving access to abstract error estimates. Optimal order of convergence is shown without any unrealistic regularity assumptions on the solution.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138550813","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}
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2917-2939, December 2023. Abstract. The perfectly matched layer (PML) is a very popular tool in the truncation of wave scattering in unbounded domains. In [S. N. Chandler-Wilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 2131–2154], the author proposed a conjecture that for scattering problems with rough surfaces, the PML converges exponentially with respect to the PML parameter in any compact subset. In the author’s previous paper [R. Zhang, SIAM J. Numer. Math., 60 (2022), pp. 804–823], this result has been proved for periodic surfaces in two-dimensional spaces, when the wave number is not a half-integer. In this paper, we prove that the method has a high-order convergence rate in the three-dimensional biperiodic surface scattering problems. We extend the two-dimensional results and prove that the exponential convergence still holds when the wave number is smaller than 0.5. For larger wave numbers, although exponential convergence is no longer proved, we are able to prove a higher-order convergence for the PML method.
SIAM数值分析学报，61卷，第6期，2917-2939页，2023年12月。摘要。完美匹配层(PML)是一种非常流行的截断无界域波散射的工具。在[S。N.钱德勒-王尔德和P.蒙克，苹果。号码。数学。， 59 (2009)， pp. 2131-2154]，作者提出了一个猜想，对于粗糙表面的散射问题，PML在任意紧子集中相对于PML参数呈指数收敛。在作者之前的论文中[R]。张siam J.数字。数学。[j]， 60 (2022)， pp. 804-823]，当波数不是半整数时，该结果已在二维空间中的周期曲面上得到证明。在本文中，我们证明了该方法在三维双周期表面散射问题中具有高阶收敛速度。推广了二维结果，证明了当波数小于0.5时，指数收敛性仍然成立。对于较大的波数，虽然不再证明指数收敛性，但我们能够证明PML方法的高阶收敛性。
{"title":"Higher-Order Convergence of Perfectly Matched Layers in Three-Dimensional Biperiodic Surface Scattering Problems","authors":"Ruming Zhang","doi":"10.1137/22m1532615","DOIUrl":"https://doi.org/10.1137/22m1532615","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2917-2939, December 2023. <br/> Abstract. The perfectly matched layer (PML) is a very popular tool in the truncation of wave scattering in unbounded domains. In [S. N. Chandler-Wilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 2131–2154], the author proposed a conjecture that for scattering problems with rough surfaces, the PML converges exponentially with respect to the PML parameter in any compact subset. In the author’s previous paper [R. Zhang, SIAM J. Numer. Math., 60 (2022), pp. 804–823], this result has been proved for periodic surfaces in two-dimensional spaces, when the wave number is not a half-integer. In this paper, we prove that the method has a high-order convergence rate in the three-dimensional biperiodic surface scattering problems. We extend the two-dimensional results and prove that the exponential convergence still holds when the wave number is smaller than 0.5. For larger wave numbers, although exponential convergence is no longer proved, we are able to prove a higher-order convergence for the PML method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492086","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}
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2859-2886, December 2023. Abstract. In this paper we consider the application of implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations. The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly. Both the second-order backward differentiation and the Crank–Nicolson methods are considered for time discretization, resulting in a scheme similar to Gear’s method on the one hand and to the Adams–Bashforth of second order on the other. For the discretization in space, we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal-order interpolation and robustness at a high Reynolds number. Under suitable Courant conditions we prove stability of Gear’s scheme in this regime. The stabilization allows us to prove error estimates of order [math]. Here [math] is the mesh parameter, [math] is the polynomial order, and [math] the time step. Finally we show that for inviscid flow (or underresolved viscous flow) the IMEX scheme can be written as a fractional step method in which only a mass matrix is inverted for each velocity component and a Poisson-type equation is solved for the pressure.
{"title":"Implicit-Explicit Time Discretization for Oseen’s Equation at High Reynolds Number with Application to Fractional Step Methods","authors":"Erik Burman, Deepika Garg, Johnny Guzman","doi":"10.1137/23m1547573","DOIUrl":"https://doi.org/10.1137/23m1547573","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2859-2886, December 2023. <br/> Abstract. In this paper we consider the application of implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations. The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly. Both the second-order backward differentiation and the Crank–Nicolson methods are considered for time discretization, resulting in a scheme similar to Gear’s method on the one hand and to the Adams–Bashforth of second order on the other. For the discretization in space, we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal-order interpolation and robustness at a high Reynolds number. Under suitable Courant conditions we prove stability of Gear’s scheme in this regime. The stabilization allows us to prove error estimates of order [math]. Here [math] is the mesh parameter, [math] is the polynomial order, and [math] the time step. Finally we show that for inviscid flow (or underresolved viscous flow) the IMEX scheme can be written as a fractional step method in which only a mass matrix is inverted for each velocity component and a Poisson-type equation is solved for the pressure.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138454856","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}
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2887-2916, December 2023. Abstract. We design a finite element method for a membrane model of liquid crystal polymer networks. This model consists of a minimization problem of a nonconvex stretching energy. We discuss properties of this energy functional such as lack of weak lower semicontinuity. We devise a discretization with regularization, propose a novel iterative scheme to solve the nonconvex discrete minimization problem, and prove stability of the scheme and convergence of discrete minimizers. We present numerical simulations to illustrate convergence properties of our algorithm and features of the model.
{"title":"Convergent FEM for a Membrane Model of Liquid Crystal Polymer Networks","authors":"Lucas Bouck, Ricardo H. Nochetto, Shuo Yang","doi":"10.1137/22m1521584","DOIUrl":"https://doi.org/10.1137/22m1521584","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2887-2916, December 2023. <br/> Abstract. We design a finite element method for a membrane model of liquid crystal polymer networks. This model consists of a minimization problem of a nonconvex stretching energy. We discuss properties of this energy functional such as lack of weak lower semicontinuity. We devise a discretization with regularization, propose a novel iterative scheme to solve the nonconvex discrete minimization problem, and prove stability of the scheme and convergence of discrete minimizers. We present numerical simulations to illustrate convergence properties of our algorithm and features of the model.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138454857","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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