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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

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Abstract

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 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 ⩾ 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.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China

    JianFang Lu & MengPing Zhang

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Chi-Wang Shu

Authors
  1. JianFang Lu
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  2. Chi-Wang Shu
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  3. MengPing Zhang
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Correspondence to Chi-Wang Shu.

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Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

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Lu, J., Shu, CW. & Zhang, M. Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates. Sci. China Math. 56, 2645–2676 (2013). https://doi.org/10.1007/s11425-013-4739-1

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  • DOI: https://doi.org/10.1007/s11425-013-4739-1

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