Uncertainty quantification in kinetic and hyperbolic problems

High order DG and WENO methods for correlated random walk with density-dependent turning rates

Chi-Wang Shu

Brown University

We consider high order accurate approximations to the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology. This system involves global integral source terms, making the design and analysis of stable schemes more complicated. We study both Runge-Kutta discontinuous Galerkin (RKDG) schemes, which are suitable for smooth solutions with the need for $h$-$p$ adaptivity, and weighted essentially non-oscillatory (WENO) finite difference schemes, which are suitable when the solution contains discontinuities. Besides the standard $L^2$ stability and error estimates for the RKDG schemes, we also consider two different strategies to obtain positivity-preserving property without compromising accuracy, one for the RKDG schemes and one for the WENO finite difference schemes. Numerical experiments are performed to verify the good performance of the schemes. This is a joint work with Yan Jiang, Jianfang Lu and Mengping Zhang. [1] J. Lu, C.-W. Shu and M. Zhang, Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates, Science China Mathematics, 56 (2013), 2645-2676. [2] Y. Jiang, C.-W. Shu and M. Zhang, High order finite difference WENO schemes with positivity-preserving limiter for correlated random walk with density-dependent turning rates, Mathematical Models and Methods in Applied Sciences ($M^3 AS$), to appear.