Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates  

作　　者：LU JianFang SHU Chi-Wang ZHANG MengPing 

机构地区：[1]School of Mathematical Sciences, University of Science and Technology of China [2]Division of Applied Mathematics, Brown University,Providence, RI 02912, USA

出　　处：《Science China Mathematics》2013年第12期2645-2676,共32页中国科学：数学（英文版）

基　　金：supported by the University of Science and Technology of China Special Grant for Postgraduate Research;Innovation and Practice;the Chinese Academy of Science Special Grant for Postgraduate Research;Innovation and Practice;Department of Energy of USA(Grant No.DE-FG02-08ER25863);National Science Foundation of USA(Grant No.DMS-1112700);National Natural Science Foundation of China(Grant Nos.11071234;91130016 and 91024025)

摘　　要：In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.In this paper, we analyze the explicit Runge-Kutta discontinuous Galerkin （RKDG） methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta （TVDRK3） time discretization and upwinding numerical fluxes. By using the energy method, under a standard Courant- Friedrichs-Lewy （CFL） condition, we obtain L2 stability for general solutions and a priori error estimates when the solutions are smooth enough. The theoretical results are proved for piecewise polynomials with any degree k≥ 1. Finally, since the solutions to this system are non-negative, we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy. Numerical results are provided to demonstrate these RKDG methods.

关 键 词：discontinuous Galerkin method explicit Runge-Kutta method stability error estimates corre-lated random walk positivity-preserving 

分 类 号：O241.82[理学—计算数学]