Numerical transport in diffusive regimes

Abstract

In highly scattering regimes, the transport equation with anisotropic boundary conditions has a limit in which the leading behavior of its solution is determined by the solution of a diffusion equation with associated boundary conditions. In order for a numerical scheme to be effective in these regimes, it must have both a correct interior diffusion limit and a correct boundary condition limit. The behavior of several numerical methods are studied in these limits and formulas for the resulting diffusion equations and its boundary conditions are derived. Theoretic and numerical results show that with correct diffusion limits, the numerical methods will give promising results with coarse grids throughout the domain, even if the boundary layers are not resolved. We also prove that with correct diffusion limits, the numerical solutions will converge to the transport solution uniformly in ε, although the collision operators have a ε⁻¹ contribution to the truncation error that generally gives rise to a nonuniform consistency with the transport equation for small ε. In last part of this dissertation we study numerical methods for the hyperbolic systems with long time parabolic behavior. In this regime the lower order terms of the hyperbolic systems break the conservation law and the systems become parabolic. Most of the numerical methods for conservation laws fail to capture this long time behavior, as shown in our analysis. We will solve the general Riemann problem of the shallow water equations and use it to modified higher order Godunov schemes in order to capture the long time behavior of the nonlinear river equations.