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

Jianfang Lu University of Science and Technology of China Chi-Wang Shu Brown University Mengping Zhang University of Science and Technology of China

Numerical Analysis and Scientific Computing mathscidoc:1610.25050

Science China Mathematics, 56, 2645-2676, 2013
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 CFL condition, we obtain $L^2$ 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 $k \geq 1$. 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.
Discontinuous Galerkin method, explicit Runge-Kutta method, stability, error estimates, correlated random walk, positivity-preserving
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@inproceedings{jianfang2013stability,
  title={Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates},
  author={Jianfang Lu, Chi-Wang Shu, and Mengping Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012052228346798168},
  booktitle={Science China Mathematics},
  volume={56},
  pages={2645-2676},
  year={2013},
}
Jianfang Lu, Chi-Wang Shu, and Mengping Zhang. Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates. 2013. Vol. 56. In Science China Mathematics. pp.2645-2676. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012052228346798168.
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