High Resolution Schemes for Conservation Laws With Source Terms.

• Main Author: Martínez i Gavara, Anna
• Other Authors: Donat Beneito, Rosa M.
• Format: Doctoral Thesis
• Language: English
• Published: Universitat de València 2008
• Subjects: Facultat de Matemàtiques; 51
• Online Access: http://hdl.handle.net/10803/10012; http://nbn-resolving.de/urn:isbn:9788437074283

This memoir is devoted to the study of the numerical treatment ofsource terms in hyperbolic conservation laws and systems. In particular,we study two types of situations that are particularly delicate fromthe point of view of their numerical approximation: The case of balancelaws, with the shallow water system as the main example, and the case ofhyperbolic equations with stiff source terms.In this work, we concentrate on the theoretical foundations of highresolutiontotal variation diminishing (TVD) schemes for homogeneousscalar conservation laws, firmly established. We analyze the propertiesof a second order, flux-limited version of the Lax-Wendroff scheme whichavoids oscillations around discontinuities, while preserving steady states.When applied to homogeneous conservation laws, TVD schemes preventan increase in the total variation of the numerical solution, hence guaranteeingthe absence of numerically generated oscillations. They are successfullyimplemented in the form of flux-limiters or slope limiters forscalar conservation laws and systems. Our technique is based on a fluxlimiting procedure applied only to those terms related to the physicalflow derivative/Jacobian. We also extend the technique developed by Chiavassaand Donat to hyperbolic conservation laws with source terms andapply the multilevel technique to the shallow water system.With respect to the numerical treatment of stiff source terms, we takethe simple model problem considered by LeVeque and Yee. We studythe properties of the numerical solution obtained with different numericaltechniques. We are able to identify the delay factor, which is responsiblefor the anomalous speed of propagation of the numerical solutionon coarse grids. The delay is due to the introduction of non equilibrium values through numerical dissipation, and can only be controlledby adequately reducing the spatial resolution of the simulation.Explicit schemes suffer from the same numerical pathology, even after reducingthe time step so that the stability requirements imposed by thefastest scales are satisfied. We study the behavior of Implicit-Explicit(IMEX) numerical techniques, as a tool to obtain high resolution simulationsthat incorporate the stiff source term in an implicit, systematic,manner.