| The thesis consists of two major parts. In the first part, we propose a new finite volume method for solving general multidimensional hyperbolic systems of conservation laws. Our method is based on an appropriate numerical flux and a high-order piecewise polynomial reconstruction. The latter is utilized without any computationally expensive nonlinear limiters, which are typically needed to guarantee nonlinear stability of the scheme. Instead, we enforce stability of the proposed method by adding a new adaptive artificial viscosity, whose coefficients are proportional to the size of the weak local residual, which is sufficiently large (∼Delta, where Delta is a discrete small scale) at the shock regions, much smaller (∼Deltaalpha, where alpha is close to 2) near the contact waves, and very small (∼Delta4) in the smooth parts of the computed solution. We test the proposed scheme on a number of benchmarks for both scalar conservation laws and for one- and two-dimensional Euler equations of gas dynamics and shallow water equations. The obtained numerical results clearly demonstrate the robustness and high accuracy of the new method.;In the second part of the thesis, we introduce a central-upwind scheme for one- and two-dimensional systems of shallow-water equations with thermodynamics (the Ripa system). The scheme is well-balanced, positivity preserving and does not develop spurious pressure oscillations in the neighborhood of temperature jumps, that is, near the contact waves. Such oscillations would typically appear when a conventional Godunov-type finite volume method is applied to the Ripa system, and the nature of the oscillation is similar to the ones appearing at material interfaces in compressible multifluid computations. The idea behind the proposed approach is to utilize the interface tracking method, originally developed in [A. CHERTOCK, S. KARNI, A. KURGANOV, M2AN Math. Model. Numer. Anal., 42(2008), PP. 991-1019] for compressible multifluids. The resulting scheme is highly accurate, preserves two types of "lake at rest" steady states, and is oscillation free across the temperature jumps, as it is illustrated in a number of numerical experiments. |
