{"title":"用局部 DG 预测器解决一阶常微分方程系统初值问题的任意高阶 ADER-DG 方法","authors":"I. S. Popov","doi":"arxiv-2409.09933","DOIUrl":null,"url":null,"abstract":"An adaptation of the arbitrary high order ADER-DG numerical method with local\nDG predictor for solving the IVP for a first-order non-linear ODE system is\nproposed. The proposed numerical method is a completely one-step ODE solver\nwith uniform steps, and is simple in algorithmic and software implementations.\nIt was shown that the proposed version of the ADER-DG numerical method is\nA-stable and L-stable. The ADER-DG numerical method demonstrates\nsuperconvergence with convergence order 2N+1 for the solution at grid nodes,\nwhile the local solution obtained using the local DG predictor has convergence\norder N+1. It was demonstrated that an important applied feature of this\nimplementation of the numerical method is the possibility of using the local\nsolution as a solution with a subgrid resolution, which makes it possible to\nobtain a detailed solution even on very coarse coordinate grids. The scale of\nthe error of the local solution, when calculating using standard\nrepresentations of single or double precision floating point numbers, using\nlarge values of the degree N, practically does not differ from the error of the\nsolution at the grid nodes. The capabilities of the ADER-DG method for solving\nstiff ODE systems characterized by extreme stiffness are demonstrated.\nEstimates of the computational costs of the ADER-DG numerical method are\nobtained.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Arbitrary high order ADER-DG method with local DG predictor for solutions of initial value problems for systems of first-order ordinary differential equations\",\"authors\":\"I. S. Popov\",\"doi\":\"arxiv-2409.09933\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"An adaptation of the arbitrary high order ADER-DG numerical method with local\\nDG predictor for solving the IVP for a first-order non-linear ODE system is\\nproposed. The proposed numerical method is a completely one-step ODE solver\\nwith uniform steps, and is simple in algorithmic and software implementations.\\nIt was shown that the proposed version of the ADER-DG numerical method is\\nA-stable and L-stable. The ADER-DG numerical method demonstrates\\nsuperconvergence with convergence order 2N+1 for the solution at grid nodes,\\nwhile the local solution obtained using the local DG predictor has convergence\\norder N+1. It was demonstrated that an important applied feature of this\\nimplementation of the numerical method is the possibility of using the local\\nsolution as a solution with a subgrid resolution, which makes it possible to\\nobtain a detailed solution even on very coarse coordinate grids. The scale of\\nthe error of the local solution, when calculating using standard\\nrepresentations of single or double precision floating point numbers, using\\nlarge values of the degree N, practically does not differ from the error of the\\nsolution at the grid nodes. The capabilities of the ADER-DG method for solving\\nstiff ODE systems characterized by extreme stiffness are demonstrated.\\nEstimates of the computational costs of the ADER-DG numerical method are\\nobtained.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"40 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09933\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09933","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
提出了一种带有局部DG预测器的任意高阶ADER-DG数值方法,用于求解一阶非线性ODE系统的IVP。结果表明,所提出的 ADER-DG 数值方法具有 A 稳定性和 L 稳定性。ADER-DG 数值方法对网格节点上的解具有超收敛性,收敛阶数为 2N+1,而使用局部 DG 预测器得到的局部解的收敛阶数为 N+1。结果表明,这种数值方法的一个重要应用特征是可以将局部解用作具有子网格分辨率的解,这使得即使在非常粗糙的坐标网格上也能获得详细的解。当使用单精度或双精度浮点数的标准表示,并使用较大的阶数 N 值进行计算时,局部解的误差范围实际上与网格节点上的解的误差并无差别。演示了 ADER-DG 方法求解以极端刚度为特征的刚性 ODE 系统的能力,并估算了 ADER-DG 数值方法的计算成本。
Arbitrary high order ADER-DG method with local DG predictor for solutions of initial value problems for systems of first-order ordinary differential equations
An adaptation of the arbitrary high order ADER-DG numerical method with local
DG predictor for solving the IVP for a first-order non-linear ODE system is
proposed. The proposed numerical method is a completely one-step ODE solver
with uniform steps, and is simple in algorithmic and software implementations.
It was shown that the proposed version of the ADER-DG numerical method is
A-stable and L-stable. The ADER-DG numerical method demonstrates
superconvergence with convergence order 2N+1 for the solution at grid nodes,
while the local solution obtained using the local DG predictor has convergence
order N+1. It was demonstrated that an important applied feature of this
implementation of the numerical method is the possibility of using the local
solution as a solution with a subgrid resolution, which makes it possible to
obtain a detailed solution even on very coarse coordinate grids. The scale of
the error of the local solution, when calculating using standard
representations of single or double precision floating point numbers, using
large values of the degree N, practically does not differ from the error of the
solution at the grid nodes. The capabilities of the ADER-DG method for solving
stiff ODE systems characterized by extreme stiffness are demonstrated.
Estimates of the computational costs of the ADER-DG numerical method are
obtained.