We present exponential wave integrator Fourier pseudospectral (EWI‐FP) methods and establish their error estimates of the fully discrete schemes for the Dirac equation in the massless and nonrelativistic regime. This regime involves a small dimensionless parameter where , and is inversely proportional to the speed of light. The solution exhibits highly oscillatory behavior in time and rapid wave propagation in space in this regime. Specifically, the time oscillations have a wavelength of , while the spatial oscillations have a wavelength of , with a wave speed of . We employ (symmetric) exponential wave integrators for temporal derivatives and Fourier spectral discretization for spatial derivatives. We rigorously derive the error bounds which explicitly depend on the mesh size , the time step and the small dimensionless parameter . The error estimates for the EWI‐FP methods demonstrate that their meshing strategy requirement (‐scalability) necessitates setting and when . Finally, some numerical examples are provided to validate the error bounds.
{"title":"Error estimates of exponential wave integrators for the Dirac equation in the massless and nonrelativistic regime","authors":"Ying Ma, Lizhen Chen","doi":"10.1002/num.23128","DOIUrl":"https://doi.org/10.1002/num.23128","url":null,"abstract":"We present exponential wave integrator Fourier pseudospectral (EWI‐FP) methods and establish their error estimates of the fully discrete schemes for the Dirac equation in the massless and nonrelativistic regime. This regime involves a small dimensionless parameter where , and is inversely proportional to the speed of light. The solution exhibits highly oscillatory behavior in time and rapid wave propagation in space in this regime. Specifically, the time oscillations have a wavelength of , while the spatial oscillations have a wavelength of , with a wave speed of . We employ (symmetric) exponential wave integrators for temporal derivatives and Fourier spectral discretization for spatial derivatives. We rigorously derive the error bounds which explicitly depend on the mesh size , the time step and the small dimensionless parameter . The error estimates for the EWI‐FP methods demonstrate that their meshing strategy requirement (‐scalability) necessitates setting and when . Finally, some numerical examples are provided to validate the error bounds.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141925068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss the identification of a time‐dependent potential in a time‐fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem. Numerically, we develop an easily implementable iterative algorithm to recover the unknown coefficient, and also derive rigorous error bounds for the discrete reconstruction. These results are attained by leveraging the (discrete) solution theory of direct problems, and applying error estimates that are optimal with respect to problem data regularity. Numerical simulations are provided to demonstrate the theoretical results.
{"title":"Determining a time‐varying potential in time‐fractional diffusion from observation at a single point","authors":"Siyu Cen, Kwancheol Shin, Zhi Zhou","doi":"10.1002/num.23136","DOIUrl":"https://doi.org/10.1002/num.23136","url":null,"abstract":"We discuss the identification of a time‐dependent potential in a time‐fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem. Numerically, we develop an easily implementable iterative algorithm to recover the unknown coefficient, and also derive rigorous error bounds for the discrete reconstruction. These results are attained by leveraging the (discrete) solution theory of direct problems, and applying error estimates that are optimal with respect to problem data regularity. Numerical simulations are provided to demonstrate the theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
While there exists a rich array of matrix column subset selection problem (CSSP) algorithms for use with interpolative and CUR‐type decompositions, their use can often become prohibitive as the size of the input matrix increases. In an effort to address these issues, in earlier work we developed a general framework that pairs a column‐partitioning routine with a column‐selection algorithm. Two of the four algorithms presented in that work paired the Centroidal Voronoi Orthogonal Decomposition (CVOD; Centroidal Voronoi Tessellation Based Proper Orthogonal Decomposition Analysis, 2003, 137–150) and an adaptive variant (adaptCVOD) with the Discrete Empirical Interpolation Method (DEIM; SIAM J. Sci. Computer. 38 (2016), no. 3, A1454–A1482). In this work, we extend this framework and pair the CVOD‐type algorithms with any CSSP algorithm that returns linearly independent columns. Our results include detailed error bounds for the solutions provided by these paired algorithms, as well as expressions that explicitly characterize how the quality of the selected column partition affects the resulting CSSP solution. In addition to examples involving matrix approximation, we test several of our partition‐based constructions on tasks commonly encountered in model order reduction (MOR).
虽然有一系列丰富的矩阵列子集选择问题（CSSP）算法可用于内插法和 CUR 型分解，但随着输入矩阵的增大，这些算法的使用往往会变得令人望而却步。为了解决这些问题，我们在早期的工作中开发了一个通用框架，将列分割例程与列选择算法配对使用。在该工作中提出的四种算法中，有两种将中心伏罗诺正交分解法（CVOD；Centroidal Voronoi Tessellation Based Proper Orthogonal Decomposition Analysis，2003，137-150）和自适应变体（adaptCVOD）与离散经验插值法（DEIM；SIAM J. Sci. Computer.38 (2016), no.3，A1454-A1482）。在这项工作中，我们扩展了这一框架，并将 CVOD 类型算法与任何返回线性独立列的 CSSP 算法配对。我们的结果包括这些配对算法所提供解的详细误差边界，以及明确描述所选列分割质量如何影响 CSSP 解的表达式。除了涉及矩阵逼近的示例外，我们还在模型阶次缩减 (MOR) 中经常遇到的任务上测试了我们的几种基于分区的构造。
{"title":"On the optimality of voronoi‐based column selection","authors":"Maria Emelianenko, Guy B. Oldaker","doi":"10.1002/num.23137","DOIUrl":"https://doi.org/10.1002/num.23137","url":null,"abstract":"While there exists a rich array of matrix column subset selection problem (CSSP) algorithms for use with interpolative and CUR‐type decompositions, their use can often become prohibitive as the size of the input matrix increases. In an effort to address these issues, in earlier work we developed a general framework that pairs a column‐partitioning routine with a column‐selection algorithm. Two of the four algorithms presented in that work paired the Centroidal Voronoi Orthogonal Decomposition (<jats:styled-content>CVOD</jats:styled-content>; <jats:italic>Centroidal Voronoi Tessellation Based Proper Orthogonal Decomposition Analysis</jats:italic>, 2003, 137–150) and an adaptive variant (<jats:styled-content>adaptCVOD</jats:styled-content>) with the Discrete Empirical Interpolation Method (<jats:styled-content>DEIM; <jats:italic>SIAM J. Sci. Computer</jats:italic>. 38 (2016), no. 3, A1454–A1482</jats:styled-content>). In this work, we extend this framework and pair the <jats:styled-content>CVOD</jats:styled-content>‐type algorithms with any CSSP algorithm that returns linearly independent columns. Our results include detailed error bounds for the solutions provided by these paired algorithms, as well as expressions that explicitly characterize how the quality of the selected column partition affects the resulting CSSP solution. In addition to examples involving matrix approximation, we test several of our partition‐based constructions on tasks commonly encountered in model order reduction (MOR).","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For the nonseparable noncanonical Hamiltonian systems, we propose efficient K‐symplectic‐like methods which are semiexplicit and energy‐preserving. By introducing two copies of the phase space and constructing an augmented Hamiltonian, we can separate the noncanonical Hamiltonian system into two explicitly integrable parts. Subsequently, explicit K‐symplectic methods can be constructed by using the splitting and composing method. To enforce constraints on the two copies of the phase space, we provide two transformations with energy conservation property. This enables us to obtain semiexplicit K‐symplectic‐like methods that preserve energy. Two algorithms are provided to implement the semiexplicit K‐symplectic‐like methods with energy conservation and their convergence has been proved. Numerical results on two noncanonical Hamiltonian systems demonstrate that the energy errors of our proposed methods remain bounded within machine precision over long time without exhibiting energy drift. Furthermore, the proposed methods exhibit superior computational efficiency compared to the canonicalized symplectic methods of the same order.
对于不可分离的非经典哈密顿系统，我们提出了高效的类 K 交映方法，这些方法是半显式和能量守恒的。通过引入两份相空间并构建增强哈密顿，我们可以将非经典哈密顿系统分离成两个显式可积分部分。随后，我们就可以利用拆分和组合方法构建显式 K 交映方法。为了对相空间的两个副本施加约束，我们提供了两种具有能量守恒性质的变换。这样，我们就能得到能量守恒的半显式 K-symplectic 样方法。我们提供了两种算法来实现能量守恒的半显式 K-symplectic-like 方法，并证明了它们的收敛性。两个非对称哈密顿系统的数值结果表明，我们提出的方法的能量误差长期保持在机器精度范围内，不会出现能量漂移。此外，与同阶的典型化交映方法相比，我们提出的方法具有更高的计算效率。
{"title":"Semiexplicit K‐symplectic‐like methods with energy conservation for noncanonical Hamiltonian systems","authors":"Beibei Zhu, Ran Gu","doi":"10.1002/num.23138","DOIUrl":"https://doi.org/10.1002/num.23138","url":null,"abstract":"For the nonseparable noncanonical Hamiltonian systems, we propose efficient K‐symplectic‐like methods which are semiexplicit and energy‐preserving. By introducing two copies of the phase space and constructing an augmented Hamiltonian, we can separate the noncanonical Hamiltonian system into two explicitly integrable parts. Subsequently, explicit K‐symplectic methods can be constructed by using the splitting and composing method. To enforce constraints on the two copies of the phase space, we provide two transformations with energy conservation property. This enables us to obtain semiexplicit K‐symplectic‐like methods that preserve energy. Two algorithms are provided to implement the semiexplicit K‐symplectic‐like methods with energy conservation and their convergence has been proved. Numerical results on two noncanonical Hamiltonian systems demonstrate that the energy errors of our proposed methods remain bounded within machine precision over long time without exhibiting energy drift. Furthermore, the proposed methods exhibit superior computational efficiency compared to the canonicalized symplectic methods of the same order.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ernesto Pimentel‐García, Manuel J. Castro, Christophe Chalons, Carlos Parés
In this work, we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in‐cell discontinuous reconstruction operator are the key points to develop a new family of high‐order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two‐Layer Shallow Water system.
{"title":"High‐order in‐cell discontinuous reconstruction path‐conservative methods for nonconservative hyperbolic systems–DR.MOOD method","authors":"Ernesto Pimentel‐García, Manuel J. Castro, Christophe Chalons, Carlos Parés","doi":"10.1002/num.23133","DOIUrl":"https://doi.org/10.1002/num.23133","url":null,"abstract":"In this work, we develop a new framework to deal numerically with discontinuous solutions in nonconservative hyperbolic systems. First an extension of the MOOD methodology to nonconservative systems based on Taylor expansions is presented. This extension combined with an in‐cell discontinuous reconstruction operator are the key points to develop a new family of high‐order methods that are able to capture exactly isolated shocks. Several test cases are proposed to validate these methods for the Modified Shallow Water equations and the Two‐Layer Shallow Water system.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.
{"title":"Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator","authors":"Yongyong Cai, Yue Feng, Yichen Guo, Zhiguo Xu","doi":"10.1002/num.23135","DOIUrl":"https://doi.org/10.1002/num.23135","url":null,"abstract":"We propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the finite element approximation of a coupled fluid‐structure interaction (FSI) system, which comprises a three‐dimensional (3D) Stokes flow and a two‐dimensional (2D) fourth‐order Euler–Bernoulli or Kirchhoff plate. The interaction of these parabolic and hyperbolic partial differential equations (PDE) occurs at the boundary interface which is assumed to be fixed. The vertical displacement of the plate dynamics evolves on the flat portion of the boundary where the coupling conditions are implemented via the matching velocities of the plate and fluid flow, as well as the Dirichlet boundary trace of the pressure. This pressure term also acts as a coupling agent, since it appears as a forcing term on the flat, elastic plate domain. Our main focus in this work is to generate some numerical results concerning the approximate solutions to the FSI model. For this, we propose a numerical algorithm that sequentially solves the fluid and plate subsystems through an effective decoupling approach. Numerical results of test problems are presented to illustrate the performance of the proposed method.
{"title":"Partitioning method for the finite element approximation of a 3D fluid‐2D plate interaction system","authors":"Pelin G. Geredeli, Hemanta Kunwar, Hyesuk Lee","doi":"10.1002/num.23132","DOIUrl":"https://doi.org/10.1002/num.23132","url":null,"abstract":"We consider the finite element approximation of a coupled fluid‐structure interaction (FSI) system, which comprises a three‐dimensional (3D) Stokes flow and a two‐dimensional (2D) fourth‐order Euler–Bernoulli or Kirchhoff plate. The interaction of these parabolic and hyperbolic partial differential equations (PDE) occurs at the boundary interface which is assumed to be fixed. The vertical displacement of the plate dynamics evolves on the flat portion of the boundary where the coupling conditions are implemented via the matching velocities of the plate and fluid flow, as well as the Dirichlet boundary trace of the pressure. This pressure term also acts as a coupling agent, since it appears as a forcing term on the flat, elastic plate domain. Our main focus in this work is to generate some numerical results concerning the approximate solutions to the FSI model. For this, we propose a numerical algorithm that sequentially solves the fluid and plate subsystems through an effective decoupling approach. Numerical results of test problems are presented to illustrate the performance of the proposed method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness.
{"title":"A type of multigrid method for semilinear elliptic problems based on symmetric interior penalty discontinuous Galerkin method","authors":"Fan Chen, Ming Cui, Chenguang Zhou","doi":"10.1002/num.23130","DOIUrl":"https://doi.org/10.1002/num.23130","url":null,"abstract":"This article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings.
{"title":"Optimal control of variably distributed‐order time‐fractional diffusion equation: Analysis and computation","authors":"Xiangcheng Zheng, Huan Liu, Hong Wang, Xu Guo","doi":"10.1002/num.23134","DOIUrl":"https://doi.org/10.1002/num.23134","url":null,"abstract":"Fractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we propose two procedures focusing on the computation of the time‐dependent convected wave equation in a free field with a uniform background flow. Both procedures are based on a framework, expended from Du et al. (SIAM J. Sci. Comput. 40 (2018), A1430–A1445.), of constructing the Dirichlet‐to‐Dirichlet (DtD)‐type discrete absorbing boundary conditions (ABCs). The first procedure is dedicated to reducing the infinite problem into a finite problem by a direct application of the framework on the finite difference discretization of the convected wave equation. However, the presence of convection terms makes the stability analysis hard to implement, which motivates us to develop the second procedure. First, the convected wave equation is transformed into a standard wave equation by using the Prandtl‐Glauert‐Lorentz transformation. After that, we obtain the DtD‐type ABC by using the above framework, and on this basis, derive an equivalent Dirichlet‐to‐Neumann‐type ABCs, which makes stability and convergence analysis easy to be obtained by the classical energy method. The effectiveness and comparison of these two procedures are investigated through numerical experiments.
在本文中，我们提出了两种程序，重点计算自由场中均匀背景流的时变对流波方程。这两个程序都基于 Du 等人（SIAM J. Sci. Comput.40 (2018), A1430-A1445.）的框架，构建了Dirichlet-to-Dirichlet（DtD）型离散吸收边界条件（ABC）。第一个过程致力于通过直接应用对流波方程有限差分离散化框架，将无限问题还原为有限问题。然而，对流项的存在使得稳定性分析难以实现，这促使我们开发第二种程序。首先，利用普朗特-格劳尔特-洛伦茨变换将对流波方程转换为标准波方程。然后，我们利用上述框架得到 DtD 型 ABC，并在此基础上推导出等效的 Dirichlet 到 Neumann 型 ABC，从而使经典能量法易于得到稳定性和收敛性分析。通过数值实验研究了这两种程序的有效性和比较。
{"title":"Numerical solution of convected wave equation in free field using artificial boundary method","authors":"Xin Wang, Jihong Wang, Yana Di, Jiwei Zhang","doi":"10.1002/num.23131","DOIUrl":"https://doi.org/10.1002/num.23131","url":null,"abstract":"In this article, we propose two procedures focusing on the computation of the time‐dependent convected wave equation in a free field with a uniform background flow. Both procedures are based on a framework, expended from Du et al. (SIAM J. Sci. Comput. 40 (2018), A1430–A1445.), of constructing the Dirichlet‐to‐Dirichlet (DtD)‐type discrete absorbing boundary conditions (ABCs). The first procedure is dedicated to reducing the infinite problem into a finite problem by a direct application of the framework on the finite difference discretization of the convected wave equation. However, the presence of convection terms makes the stability analysis hard to implement, which motivates us to develop the second procedure. First, the convected wave equation is transformed into a standard wave equation by using the Prandtl‐Glauert‐Lorentz transformation. After that, we obtain the DtD‐type ABC by using the above framework, and on this basis, derive an equivalent Dirichlet‐to‐Neumann‐type ABCs, which makes stability and convergence analysis easy to be obtained by the classical energy method. The effectiveness and comparison of these two procedures are investigated through numerical experiments.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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