{"title":"低马赫数等熵欧拉方程的渐近保线性隐含加法 IMEX-RK 有限体积方案","authors":"Saurav Samantaray","doi":"arxiv-2409.05854","DOIUrl":null,"url":null,"abstract":"We consider the compressible Euler equations of gas dynamics with isentropic\nequation of state. In the low Mach number regime i.e. when the fluid velocity\nis very very small in comparison to the sound speed in the medium, the solution\nof the compressible system converges to the solution of its incompressible\ncounter part. Standard numerical schemes fail to respect this transition\nproperty and hence are plagued with inaccuracies as well as instabilities. In\nthis paper we introduce an extra flux term to the momentum flux. This extra\nterm is brought to fore by looking at the incompressibility constraints of the\nasymptotic limit system. This extra flux term enables us to get a suitable flux\nsplitting, so that an additive IMEX-RK scheme could be applied. Using an\nelliptic reformulation the scheme boils down to just solving a linear elliptic\nproblem for the density and then explicit updates for the momentum. The IMEX\nschemes developed are shown to be formally asymptotically consistent with the\nlow Mach number limit of the Euler equations. A second order space time fully\ndiscrete scheme is obtained in the finite volume framework using a combination\nof Rusanov flux for the explicit part and simple central differences for the\nimplicit part. Numerical results are reported which elucidate the theoretical\nassertions regarding the scheme and its robustness.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"389 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Asymptotic Preserving Linearly Implicit Additive IMEX-RK Finite Volume Schemes for Low Mach Number Isentropic Euler Equations\",\"authors\":\"Saurav Samantaray\",\"doi\":\"arxiv-2409.05854\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the compressible Euler equations of gas dynamics with isentropic\\nequation of state. In the low Mach number regime i.e. when the fluid velocity\\nis very very small in comparison to the sound speed in the medium, the solution\\nof the compressible system converges to the solution of its incompressible\\ncounter part. Standard numerical schemes fail to respect this transition\\nproperty and hence are plagued with inaccuracies as well as instabilities. In\\nthis paper we introduce an extra flux term to the momentum flux. This extra\\nterm is brought to fore by looking at the incompressibility constraints of the\\nasymptotic limit system. This extra flux term enables us to get a suitable flux\\nsplitting, so that an additive IMEX-RK scheme could be applied. Using an\\nelliptic reformulation the scheme boils down to just solving a linear elliptic\\nproblem for the density and then explicit updates for the momentum. The IMEX\\nschemes developed are shown to be formally asymptotically consistent with the\\nlow Mach number limit of the Euler equations. A second order space time fully\\ndiscrete scheme is obtained in the finite volume framework using a combination\\nof Rusanov flux for the explicit part and simple central differences for the\\nimplicit part. Numerical results are reported which elucidate the theoretical\\nassertions regarding the scheme and its robustness.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"389 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-09\",\"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-2409.05854\",\"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-2409.05854","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Asymptotic Preserving Linearly Implicit Additive IMEX-RK Finite Volume Schemes for Low Mach Number Isentropic Euler Equations
We consider the compressible Euler equations of gas dynamics with isentropic
equation of state. In the low Mach number regime i.e. when the fluid velocity
is very very small in comparison to the sound speed in the medium, the solution
of the compressible system converges to the solution of its incompressible
counter part. Standard numerical schemes fail to respect this transition
property and hence are plagued with inaccuracies as well as instabilities. In
this paper we introduce an extra flux term to the momentum flux. This extra
term is brought to fore by looking at the incompressibility constraints of the
asymptotic limit system. This extra flux term enables us to get a suitable flux
splitting, so that an additive IMEX-RK scheme could be applied. Using an
elliptic reformulation the scheme boils down to just solving a linear elliptic
problem for the density and then explicit updates for the momentum. The IMEX
schemes developed are shown to be formally asymptotically consistent with the
low Mach number limit of the Euler equations. A second order space time fully
discrete scheme is obtained in the finite volume framework using a combination
of Rusanov flux for the explicit part and simple central differences for the
implicit part. Numerical results are reported which elucidate the theoretical
assertions regarding the scheme and its robustness.