An Alternative Formulation Of Finite Difference WENO Scheme And Numerical Simulation Of Random Walk Model

This dissertation introduces an alternative formulation of high order finite differ-ence weighted essentially non-oscillatory (WENO) scheme and numerical simulation of a biological random walk model. The dissertation is mainly divided into the following two parts:In the first part, the high order WENO interpolation procedure is applied to the so-lution itself to construct the conservative finite difference numerical flux, instead of the usual practice of reconstructing the flux functions. Compared with the standard formu-lation, it does have several advantages. The first advantage is that arbitrary monotone fluxes and Riemann solvers can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting to achieve nonlinear stability and high order accuracy. Hence, not all monotone fluxes or Riemann solvers can be used. The second advantage is that a narrower effective stencil is used compared with standard high order finite difference WENO schemes based on the re-construction of flux functions, with a Lax-Wendroff time discretization. We describe the scheme formulation of fifth order in space and fourth order in time, showing that the effective stencil used in the alternative scheme is the same as the semi-discrete scheme. And numerical tests demonstrate that the scheme can achieve the designed high order accuracy in smooth regions while maintaining stable, non-oscillatory and sharp discon-tinuity transitions. The errors of the standard WENO scheme are significantly bigger than those for the schemes designed here when compared on the same mesh, indicat-ing that the narrower effective stencil and flexibility in choosing numerical fluxes are helping to improve the magnitude of the errors. The third advantage is the exact preser-vation of free-stream solutions on curvilinear meshes, while this property is difficult to fulfill for standard high order finite difference WENO schemes. Based on a numerical technique for the metric terms, theoretical derivation and numerical results show that the finite difference WENO schemes based on the alternative flux formulation can pre-serve free-stream and vortex solutions on both stationary and dynamically generalized coordinate systems. While the technique for metric terms is difficult to be applied to the standard finite difference WENO formulation.In the second part, we discuss the numerical method to solve the semi-linear hy-perbolic system of a correlated random walk model in biology. This model describes the densities over time to reflect global movement of animals or cells. We assume the particles move in a constant speed and change their directions when interacting with their neighbors. Here, we consider the interaction with three social interactions:attrac-tion, repulsion and alignment. Solutions should be non-negative since they present the densities of particles. Standard high order finite difference WENO schemes, coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration is used to solve the model. Since the solutions to this system are non-negative, we discuss a positivity-preserving limiter without compromising accuracy. Analysis is performed to justify the maintenance of third order spatial/temporal accuracy when the limiters are applied to a third order finite difference scheme and third order TVD-RK time dis-cretization for solving this model. Numerical results are also provided to demonstrate these methods up to fifth order accuracy.