Finite volume approximations for some equations arising in mathematical physics

Abstract

In this thesis we design and implement finite volume schemes to approximate the solution of 1-dimensional (partial) differential equations. Most of these partial differential equations (PDEs) are made up of not only mathematically interesting but also physically relevant terms such as the hyperbolic convective and parabolic diffusive operators. The coupling of higher order, linear and nonlinear operators and the presence of a small parameter multiplying the highest derivative imposes some stiffness into the equations thereby making both their numerical and mathematical analysis interesting but very challenging. For example, singularly perturbed second order ordinary differential equations (ODEs) possess boundary layers and/or oscillatory solutions which make their numerical approximation by difference-type schemes expensive. We design two uniformly convergent finite volume schemes for a singularly perturbed ODE: the Schr¨odinger equation. The first scheme is based on the nonstandard finite difference (NSFD) method which is known to preserve the qualitative properties of the physical model and the second is based on boundary layer analysis. We employ fractional splitting method for the analysis of higher order equations in order to isolate the linear and nonlinear terms thereby resolving the stiffness in the equation. The nonlinear hyperbolic term is solved by shock capturing schemes while the fourth order linear parabolic term is handled by A-stable schemes. We also utilize the idea of the NSFD method to design a scheme for the hyperbolic, nonlinear parabolic and the linear fourth order PDEs. Each of the terms is solved sequentially within every time step and their solutions are pieced together in such a way as to preserve the properties of the original equations. We observe uniform convergence with better approximation at relatively low computation cost when the Schr¨odinger equation was solved by the proposed schemes. We also examined the computational strength of our schemes on two fourth order equations: the Kuramoto- Sivashinsky equation and Cahn-Hilliard equation. We studied the effect of combining different schemes for each of the split sub-problems on the convergence of the fractional splitting scheme. We are able to reproduce all the expected properties of the selected equations. We observed a better convergence when the nonstandard finite volume method was applied to these PDEs. Throughout this work, numerical simulations are provided to validate the computational power of the proposed schemes.