Pub Date : 2022-12-05DOI: 10.48550/arXiv.2212.02426
Wasilij Barsukow, J. P. Berberich
Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely, to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.
{"title":"A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying","authors":"Wasilij Barsukow, J. P. Berberich","doi":"10.48550/arXiv.2212.02426","DOIUrl":"https://doi.org/10.48550/arXiv.2212.02426","url":null,"abstract":"Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely, to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"67 1","pages":"1-46"},"PeriodicalIF":1.6,"publicationDate":"2022-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78649495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-01DOI: 10.48550/arXiv.2212.00823
Yifan Chen, T. Hou, Yixuan Wang
We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n -width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.
本文简要介绍了指数收敛多尺度有限元法(ExpMsFEM)在非均质介质无尺度分离和高频波传播中的有效模型缩减。ExpMsFEM建立在经典MsFEM的无重叠区域分解的基础上，系统地丰富了逼近空间，在基函数数量上实现了接近指数的收敛速度。与文献中大多数MsFEM的推广不同，ExpMsFEM不依赖于单位函数的任何划分。一般来说，有必要使用依赖于右侧的函数表示来打破代数Kolmogorov n -width障碍以实现指数收敛。实际上，ExpMsFEM提供的函数表示中有在线部分和离线部分。在线部分局部依赖于右侧，可以有效地并行计算。离线部分包含在伽辽金法中用于装配刚度矩阵的基函数;它们都独立于右侧，因此刚度矩阵可以在多查询场景中重复使用。
{"title":"Exponentially Convergent Multiscale Finite Element Method","authors":"Yifan Chen, T. Hou, Yixuan Wang","doi":"10.48550/arXiv.2212.00823","DOIUrl":"https://doi.org/10.48550/arXiv.2212.00823","url":null,"abstract":"We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n -width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"1 1","pages":"1-17"},"PeriodicalIF":1.6,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90316726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-01DOI: 10.1007/s42967-022-00229-7
A. Hidalgo, M. Dumbser, E. Toro
{"title":"Preface to the Focused Issue on High-Order Numerical Methods for Evolutionary PDEs","authors":"A. Hidalgo, M. Dumbser, E. Toro","doi":"10.1007/s42967-022-00229-7","DOIUrl":"https://doi.org/10.1007/s42967-022-00229-7","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"4 1","pages":"529-531"},"PeriodicalIF":1.6,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78788968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-28DOI: 10.1007/s42967-022-00230-0
Jiequan Li, Wancheng Sheng, Chi-Wang Shu, P. Zhang, Yuxi Zheng
{"title":"Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday","authors":"Jiequan Li, Wancheng Sheng, Chi-Wang Shu, P. Zhang, Yuxi Zheng","doi":"10.1007/s42967-022-00230-0","DOIUrl":"https://doi.org/10.1007/s42967-022-00230-0","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"1 1","pages":"965 - 966"},"PeriodicalIF":1.6,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86675649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-23DOI: 10.1007/s42967-022-00235-9
Sethupathy S., D. Balsara, Deepak Bhoriya, Harish Kumar
{"title":"Techniques, Tricks, and Algorithms for Efficient GPU-Based Processing of Higher Order Hyperbolic PDEs","authors":"Sethupathy S., D. Balsara, Deepak Bhoriya, Harish Kumar","doi":"10.1007/s42967-022-00235-9","DOIUrl":"https://doi.org/10.1007/s42967-022-00235-9","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"14 1","pages":"1-49"},"PeriodicalIF":1.6,"publicationDate":"2022-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78782111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-14DOI: 10.1007/s42967-023-00255-z
Hongxing Wang, P. Huang
{"title":"Characterizations and Properties of Dual Matrix Star Orders","authors":"Hongxing Wang, P. Huang","doi":"10.1007/s42967-023-00255-z","DOIUrl":"https://doi.org/10.1007/s42967-023-00255-z","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"78 1","pages":"1-24"},"PeriodicalIF":1.6,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88565049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-10DOI: 10.48550/arXiv.2211.05600
Fangyao Zhu, Juntao Huang, Yang Yang
In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for chemical reactive flows. There are several difficulties in constructing suitable numerical schemes. First of all, the density and internal energy are positive, and the mass fraction of each species is between 0 and 1. Second, due to the rapid reaction rate, the system may contain stiff sources, and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes. To obtain physically relevant numerical approximations, we apply the bound-preserving technique to the DG methods. Though traditional positivity-preserving techniques can successfully yield positive density, internal energy, and mass fractions, they may not enforce the upper bound 1 of the mass fractions. To solve this problem, we need to (i) make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density; (ii) choose conservative time integrations, such that the summation of the mass fractions is preserved. With the above two conditions, the positive mass fractions have summation 1, and then, they are all between 0 and 1. For time discretization, we apply the modified Runge-Kutta/multi-step Patankar methods, which are explicit for the flux while implicit for the source. Such methods can handle stiff sources with relatively large time steps, preserve the positivity of the target variables, and keep the summation of the mass fractions to be 1. Finally, it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations. The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces, while Patankar methods can keep the positivity of the pre-selected point values of the target variables. To match the degree of freedom, we use $$Q^k$$ Q k polynomials on rectangular meshes for problems in two space dimensions. To evolve in time, we first read the polynomials at the Gaussian points. Then, suitable slope limiters can be applied to enforce the positivity of the solutions at those points, which can be preserved by the Patankar methods, leading to positive updated numerical cell averages. In addition, we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux. Numerical examples are given to demonstrate the good performance of the proposed schemes.
{"title":"Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows","authors":"Fangyao Zhu, Juntao Huang, Yang Yang","doi":"10.48550/arXiv.2211.05600","DOIUrl":"https://doi.org/10.48550/arXiv.2211.05600","url":null,"abstract":"In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for chemical reactive flows. There are several difficulties in constructing suitable numerical schemes. First of all, the density and internal energy are positive, and the mass fraction of each species is between 0 and 1. Second, due to the rapid reaction rate, the system may contain stiff sources, and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes. To obtain physically relevant numerical approximations, we apply the bound-preserving technique to the DG methods. Though traditional positivity-preserving techniques can successfully yield positive density, internal energy, and mass fractions, they may not enforce the upper bound 1 of the mass fractions. To solve this problem, we need to (i) make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density; (ii) choose conservative time integrations, such that the summation of the mass fractions is preserved. With the above two conditions, the positive mass fractions have summation 1, and then, they are all between 0 and 1. For time discretization, we apply the modified Runge-Kutta/multi-step Patankar methods, which are explicit for the flux while implicit for the source. Such methods can handle stiff sources with relatively large time steps, preserve the positivity of the target variables, and keep the summation of the mass fractions to be 1. Finally, it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations. The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces, while Patankar methods can keep the positivity of the pre-selected point values of the target variables. To match the degree of freedom, we use $$Q^k$$ Q k polynomials on rectangular meshes for problems in two space dimensions. To evolve in time, we first read the polynomials at the Gaussian points. Then, suitable slope limiters can be applied to enforce the positivity of the solutions at those points, which can be preserved by the Patankar methods, leading to positive updated numerical cell averages. In addition, we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux. Numerical examples are given to demonstrate the good performance of the proposed schemes.","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"47 1","pages":"1-28"},"PeriodicalIF":1.6,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84145003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-09DOI: 10.48550/arXiv.2211.04900
Bo Dong, Wei Wang
In this paper, numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin (DG) methods (Dong et al. in J Sci Comput 66: 321–345, 2016; Dong and Wang in J Comput Appl Math 380: 1–11, 2020) for a one-dimensional stationary Schrödinger equation. Previous work showed that penalty parameters were required to be positive in error analysis, but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes. In this work, by performing extensive numerical experiments, we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods, and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.
本文通过数值实验研究了数值迹线中惩罚参数对高阶多尺度不连续伽辽金(DG)方法共振误差的影响(Dong et al. J Sci computer 66: 321-345, 2016;王晓东，王晓东，王晓东，等。数学学报(自然科学版)，2016，第1期，第11期。以往的研究表明，在误差分析中，惩罚参数必须为正，但在粗网格的数值模拟中，零惩罚参数的方法效果良好。通过大量的数值实验，我们发现在多尺度DG方法中，零惩罚参数会导致共振误差，而采用正惩罚参数可以有效地减小共振误差，使全局线性系统中的矩阵具有更好的条件数。
{"title":"Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation","authors":"Bo Dong, Wei Wang","doi":"10.48550/arXiv.2211.04900","DOIUrl":"https://doi.org/10.48550/arXiv.2211.04900","url":null,"abstract":"In this paper, numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin (DG) methods (Dong et al. in J Sci Comput 66: 321–345, 2016; Dong and Wang in J Comput Appl Math 380: 1–11, 2020) for a one-dimensional stationary Schrödinger equation. Previous work showed that penalty parameters were required to be positive in error analysis, but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes. In this work, by performing extensive numerical experiments, we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods, and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"216 1","pages":"1-14"},"PeriodicalIF":1.6,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91388700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-08DOI: 10.1007/s42967-022-00215-z
Xin He Miao, Zhenghai Huang
{"title":"Norms of Dual Complex Vectors and Dual Complex Matrices","authors":"Xin He Miao, Zhenghai Huang","doi":"10.1007/s42967-022-00215-z","DOIUrl":"https://doi.org/10.1007/s42967-022-00215-z","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"28 1","pages":"1484 - 1508"},"PeriodicalIF":1.6,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74343877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-02DOI: 10.1007/s42967-022-00220-2
Yaoqi Li
{"title":"Existence of Two Limit Cycles in Zeeman’s Class 30 for 3D Lotka-Volterra Competitive System","authors":"Yaoqi Li","doi":"10.1007/s42967-022-00220-2","DOIUrl":"https://doi.org/10.1007/s42967-022-00220-2","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"9 1","pages":"1584-1590"},"PeriodicalIF":1.6,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82743894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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