Exact and Approximation Riemann Solutions of Nonlinear Hyperbolic Conservation Laws

Abstract

This thesis project focuses on the numerical solutions of selected nonlinear hyperbolic sys tems of partial differential equations (PDEs) describing incompressible and compressible flows. Such type of PDEs are used to simulate various flows in science and engineering. The underlying physics of such systems of PDEs is very complex and some mathematical and computational issues are associated with them. For instance, they may contain non conservative terms or may be weakly hyperbolic. The strong nonlinearity of the systems could generate sharp fronts in the solutions in a finite time interval, even for smooth initial data. Moreover, accurate discretization of the non-conservative terms is a challenge task for the numerical solution techniques. In the presence of non-conservative terms, well balancing, positivity preservation and capturing of steady states demand special attention during the application of a numerical algorithm. In this thesis project, we develop exact Riemann solvers for the one-dimensional Ripa model, containing shallow water equations that incorporate horizontal temperature gradients and considering both flat and non flat bottom topographies. Such Riemann solvers are helpful for understanding the behavior of solutions, as these solutions contain fundamental physical and mathematical characters of the set of conservation laws. Such solvers are also very helpful for evaluating performance of the numerical schemes for more complex models. Afterwards, third order well-balanced finite volume weighted essentially non-oscillatory (FV WENO) schemes are applied to solve the same model equations in one and two space dimensions and a Runge-Kutta discontin uous Galerkin (RKDG) finite element method is applied to solve this model in one space dimension. In the case of compressible fluid flow models, an upwind conservation element and solution element (CE/SE) method and third order finite volume WENO schemes are applied to solve the dusty gas and two-phase flow models. The suggested numerical schemes are able to tackle the above mentioned associated difficulties in a more efficient manner. The accuracy and order of convergence of the proposed numerical schemes are analyzed qualitatively and quantitatively. A number of numerical test problems are considered and results of the suggested numerical schemes are compared with the derived exact Riemann solutions, results available in the literature, and with the results of a high resolution central upwind (CUP) scheme.