Numerical methods for non-homogeneous hyperbolic conservation laws

Abstract

The numerical simulation of hyperbolic conservation equations has reached a level of maturity with the development of several efficient numerical methods in the past three decades. The same level of success is not present in the case of solving hyperbolic conservation equations with source terms. A simple-minded approach in discretizing the source terms often leads to missing the possible multiple steady states, poor convergence, and spurious solutions. This field has been receiving special attention from researchers in CFD in the last few years. This thesis is concerned with the review of these new methods and the development of a new Kinetic Scheme for the numerical simulation of reactive Euler equations. In this thesis, first a review of the recently developed methods for solving hyperbolic conservation equations with source terms is made. Some of these recent methods are presented in considerable detail, with several benchmark problems for their comparison and evaluation, including the methods of Roe, Alouges, Ghidaglia & Tajchmann, LeVeque, Jin, Gascon & Corberan, apart from the standard operator-splitting-based methods. Numerical simulation of hyperbolic conservation equations with stiff source terms representing chemical reactions is a non-trivial problem. Standard approaches to tackle this problem have been associated with failures like the presence of spurious solutions, involving discontinuities moving with incorrect speed of one grid point per time step, in the often-used under-resolved methods. Resolving the chemical reaction scales is often impractical and, therefore, the success of the under-resolved methods in avoiding spurious results is of high importance. The Accurate Deterministic Projection Method of Kurganov and the Random Projection Method of Bao and Jin are two of the successful methods which overcome the problems of spurious solutions associated with the operator-splitting-based methods. In this thesis, these methodologies are coupled with the attractive approach of Kinetic (or Boltzmann) Schemes, and a new Kinetic Scheme is developed to solve the reactive Euler equations, using a new variable to represent the species conservation in the moment function vector and the above-mentioned methodologies for the stiff source terms. Several benchmark problems involving detonation waves are solved with this new method, and the efficiency of this method in solving reactive Euler equations efficiently is demonstrated.