High Resolution Schemes for Conservation Laws With Source Terms.

Abstract

This memoir is devoted to the study of the numerical treatment of<br/>source terms in hyperbolic conservation laws and systems. In particular,<br/>we study two types of situations that are particularly delicate from<br/>the point of view of their numerical approximation: The case of balance<br/>laws, with the shallow water system as the main example, and the case of<br/>hyperbolic equations with stiff source terms.<br/>In this work, we concentrate on the theoretical foundations of highresolution<br/>total variation diminishing (TVD) schemes for homogeneous<br/>scalar conservation laws, firmly established. We analyze the properties<br/>of a second order, flux-limited version of the Lax-Wendroff scheme which<br/>avoids oscillations around discontinuities, while preserving steady states.<br/>When applied to homogeneous conservation laws, TVD schemes prevent<br/>an increase in the total variation of the numerical solution, hence guaranteeing<br/>the absence of numerically generated oscillations. They are successfully<br/>implemented in the form of flux-limiters or slope limiters for<br/>scalar conservation laws and systems. Our technique is based on a flux<br/>limiting procedure applied only to those terms related to the physical<br/>flow derivative/Jacobian. We also extend the technique developed by Chiavassa<br/>and Donat to hyperbolic conservation laws with source terms and<br/>apply the multilevel technique to the shallow water system.<br/>With respect to the numerical treatment of stiff source terms, we take<br/>the simple model problem considered by LeVeque and Yee. We study<br/>the properties of the numerical solution obtained with different numerical<br/>techniques. We are able to identify the delay factor, which is responsible<br/>for the anomalous speed of propagation of the numerical solution<br/>on coarse grids. The delay is due to the introduction of non equilibrium values through numerical dissipation, and can only be controlled<br/>by adequately reducing the spatial resolution of the simulation.<br/>Explicit schemes suffer from the same numerical pathology, even after reducing<br/>the time step so that the stability requirements imposed by the<br/>fastest scales are satisfied. We study the behavior of Implicit-Explicit<br/>(IMEX) numerical techniques, as a tool to obtain high resolution simulations<br/>that incorporate the stiff source term in an implicit, systematic,<br/>manner.