Numerical methods for the evolution of compressible fluids on a curved geometry

Overview & Synopsis

This thesis is devoted to the study of a class of nonlinear hyperbolic balance laws in one or two space dimensions, which describe the evolution of compressible fluids on a curved geometry. In Chapter 1, we study the initial value problem for an Euler model with source term posed on a Schwarzschild spacetime. We prove the existence of a global-in-time weak solution by using the Glimm method based on generalized Riemann solutions. In Chapter 2, we consider a Burgers model evolving on a cosmological background of expanding or contracting type. We design a numerical algorithm based on finite volume methodology and we compute weak solutions containing shock waves. Our scheme is fourth-order in time and second-order in space. We then study the late-time asymptotic structure of the solutions as the time variable approaches infinity (in the expanding case) or approaches zero (in the contracting case). A generalization of the numerical method and numerical experiments to the more complete cosmological Euler model of compressible fluids is presented in Chapter 3 and 4.