Abstract:
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. |