基于动力学关系的非压缩激波的Godunov型格式
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摘要
非压缩激波产生于大量的物理问题,如相变动力学、磁流体动力学、Camassa-Holm模型和燃烧系统等。与压缩激波不同的是,它不仅满足单个的熵不等式,还能满足一些动力学关系。为了计算非压缩激波的数值解,本文设计了一种Godunov型格式,包括函数重构、发展和求网格平均三个步骤。在函数重构时,先对非压缩激波的位置进行预估,在其相邻网格上利用动力学关系进行重构,而在其它网格上采用数值解和数值熵进行重构。数值实验表明,此格式不仅对非压缩激波有较好的分辨率,而且对经典波也有较高的精度。
Undercompressive shock waves arise from many physical problems: phase transitions, magnetohydrodynamics,Camassa-Holm model, combustion theory, etc. Different from compressive shock waves, it satisfies single entropy inequality and some kinetic relations. In order to computing undercompressive shock waves, we design a Godunov-type scheme, including reconstruction, evolution, cell-averaging. We forecast the position of undercompressive shocks and use kinetic relations to reconstruct functions near undercompressive shocks. In other cells, we use numerical solutions and numerical entropy to reconstruct functions. Numerical experiments show that our scheme has not only good resolution for undercompressive shocks, but also better accuracy for classical waves.
Undercompressive shock waves arise from many physical problems: phase transitions, magnetohydrodynamics,Camassa-Holm model, combustion theory, etc. Different from compressive shock waves, it satisfies single entropy inequality and some kinetic relations. In order to computing undercompressive shock waves, we design a Godunov-type scheme, including reconstruction, evolution, cell-averaging. We forecast the position of undercompressive shocks and use kinetic relations to reconstruct functions near undercompressive shocks. In other cells, we use numerical solutions and numerical entropy to reconstruct functions. Numerical experiments show that our scheme has not only good resolution for undercompressive shocks, but also better accuracy for classical waves.
引文
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[9]Ernest J,Le Floch P G,Mishra S.Schemes with Well Controlled Dissipation[J].SIAM J.Numer.Anal.2015,53(1):674-699.
[10]Hou T Y,Le Floch P G,Zhong X.Converging method for the computation of propagating solid-solid phas boundaries[J].J.Comput.Phys.,1996,124:192-216.
[11]Amadori D,Baiti P,Le Floch P G,Piccoli B.Nonclassi cal shocks and the Cauchy problem for non-convex conservation laws[J].J.Differential Equations,1999151:345-372
[12]Hou T Y,Le Floch P G,Rosakis P.A level-set approach to the computation of twinning and phase transition dy namics[J].J Comput.Phys.,1999,150:302-331.
[13]Chalons C,Le Floch P G.Computing undercompres sive waves with the random choice scheme[J].Non classical shock waves,Interfaces Free Boundaries2003,5:129-158.
[14]Boutin B,Chalons C,Lagoutiere F A.Convergent and conservative schemes for nonclassical solutions based on kinetic relations[J].Interfaces and Free Boundaries2008,10(3):399-421.
[15]Chen R S,Mao D K.Entropy-TVD scheme for nonlin ear scalar conservation Laws[J].Journal of Scientifi Computing,2011,47(2):150-169.
[2]Oleinik O.Discontinuous solutions of nonlinear differ ential equations[J].Transl.Amer.Math.Soc.,196326:95-172.
[3]Kruzkov S N,First order quasilinear equations with several independent variables[J].Mat.Sb.N.S.,197081:228-235.
[4]Lefloch P G.Hyperbolic systems of conservation laws the theory of classical and nonclassical shock wave[M].Lecture Notes in Mathematics,ETH Zurich Birkhauser,2002.
[5]Hayes B T,Le Floch P G.Nonclassical shocks and ki netic relations:fi nite difference schemes[J].SIAM JNumer.Anal.,1998,35:2169-2194.
[6]Le Floch P G,Rohde C.High-order schemes,entropy inequalities,and nonclassical shocks[J].SIAM J.Num er.Anal.,2000,37:2023-2060.
[7]Chalons C,Le Floch P G.A fully discrete scheme fo diffusive-dispersive conservation laws[J].Numerisch Math.,2001,89:493-509.
[8]Chalons C,Le Floch P G.High-order entropy conserva tive schemes and kinetic relations for van der waals fl uids[J].J Comput.Phys.,2001,167:1-23.
[9]Ernest J,Le Floch P G,Mishra S.Schemes with Well Controlled Dissipation[J].SIAM J.Numer.Anal.2015,53(1):674-699.
[10]Hou T Y,Le Floch P G,Zhong X.Converging method for the computation of propagating solid-solid phas boundaries[J].J.Comput.Phys.,1996,124:192-216.
[11]Amadori D,Baiti P,Le Floch P G,Piccoli B.Nonclassi cal shocks and the Cauchy problem for non-convex conservation laws[J].J.Differential Equations,1999151:345-372
[12]Hou T Y,Le Floch P G,Rosakis P.A level-set approach to the computation of twinning and phase transition dy namics[J].J Comput.Phys.,1999,150:302-331.
[13]Chalons C,Le Floch P G.Computing undercompres sive waves with the random choice scheme[J].Non classical shock waves,Interfaces Free Boundaries2003,5:129-158.
[14]Boutin B,Chalons C,Lagoutiere F A.Convergent and conservative schemes for nonclassical solutions based on kinetic relations[J].Interfaces and Free Boundaries2008,10(3):399-421.
[15]Chen R S,Mao D K.Entropy-TVD scheme for nonlin ear scalar conservation Laws[J].Journal of Scientifi Computing,2011,47(2):150-169.