Pub Date : 2023-03-21DOI: 10.1007/s42967-023-00254-0
Michael Franco, P. Persson, Will Pazner
{"title":"Iterative Subregion Correction Preconditioners with Adaptive Tolerance for Problems with Geometrically Localized Stiffness","authors":"Michael Franco, P. Persson, Will Pazner","doi":"10.1007/s42967-023-00254-0","DOIUrl":"https://doi.org/10.1007/s42967-023-00254-0","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"15 1","pages":"1-26"},"PeriodicalIF":1.6,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72630720","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 : 2023-03-20DOI: 10.1007/s42967-022-00247-5
Mengjiao Jiao, Yan Jiang, Mengping Zhang
{"title":"A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations","authors":"Mengjiao Jiao, Yan Jiang, Mengping Zhang","doi":"10.1007/s42967-022-00247-5","DOIUrl":"https://doi.org/10.1007/s42967-022-00247-5","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"1 1","pages":"1-32"},"PeriodicalIF":1.6,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82338688","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 : 2023-03-17DOI: 10.1007/s42967-022-00243-9
Lin Mu, X. Ye, Shangyou Zhang, P. Zhu
{"title":"A DG Method for the Stokes Equations on Tensor Product Meshes with $$[P_k]^d-P_{k-1}$$ Element","authors":"Lin Mu, X. Ye, Shangyou Zhang, P. Zhu","doi":"10.1007/s42967-022-00243-9","DOIUrl":"https://doi.org/10.1007/s42967-022-00243-9","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"58 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84155496","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 : 2023-03-08DOI: 10.1007/s42967-022-00248-4
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":"10.1007/s42967-022-00248-4","DOIUrl":"https://doi.org/10.1007/s42967-022-00248-4","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":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136244718","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 : 2023-03-08DOI: 10.1007/s42967-022-00244-8
Y. Wang, Min Cai
{"title":"Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions","authors":"Y. Wang, Min Cai","doi":"10.1007/s42967-022-00244-8","DOIUrl":"https://doi.org/10.1007/s42967-022-00244-8","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"30 1","pages":"1674-1696"},"PeriodicalIF":1.6,"publicationDate":"2023-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83401547","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 : 2023-03-06DOI: 10.1007/s42967-022-00228-8
Tingting Li, Jianfang Lu, Pengde Wang
{"title":"Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes","authors":"Tingting Li, Jianfang Lu, Pengde Wang","doi":"10.1007/s42967-022-00228-8","DOIUrl":"https://doi.org/10.1007/s42967-022-00228-8","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"40 1","pages":"1-48"},"PeriodicalIF":1.6,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79647903","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 : 2023-02-16DOI: 10.1007/s42967-022-00227-9
Hao Li, Xiangxiong Zhang
{"title":"A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow","authors":"Hao Li, Xiangxiong Zhang","doi":"10.1007/s42967-022-00227-9","DOIUrl":"https://doi.org/10.1007/s42967-022-00227-9","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"135 1","pages":"1-29"},"PeriodicalIF":1.6,"publicationDate":"2023-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88705144","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 : 2023-02-13DOI: 10.1007/s42967-022-00240-y
Yue Liu, Jun Xie, Hay-Oak Park, W. Lo
{"title":"Mathematical Modeling of Cell Polarity Establishment of Budding Yeast","authors":"Yue Liu, Jun Xie, Hay-Oak Park, W. Lo","doi":"10.1007/s42967-022-00240-y","DOIUrl":"https://doi.org/10.1007/s42967-022-00240-y","url":null,"abstract":"","PeriodicalId":29916,"journal":{"name":"Communications on Applied Mathematics and Computation","volume":"39 1","pages":"1-18"},"PeriodicalIF":1.6,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75085386","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 : 2023-02-06DOI: 10.1007/s42967-022-00231-z
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 $$Q^k$$ 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":"10.1007/s42967-022-00231-z","DOIUrl":"https://doi.org/10.1007/s42967-022-00231-z","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 $$Q^k$$ 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":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134915866","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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