High-order central schemes for balance laws and Hamilton-Jacobi equations

We are interested in central schemes, which are a sub-class of Godunov-type schemes for approximating solutions of hyperbolic conservation laws. Central schemes do not require solving any Riemann problems and are therefore suitable for multi-dimensional problems, systems of equations, and problems with complicated geometries. This thesis extends the development of central schemes for conservation laws and Hamilton-Jacobi equations in several ways.; For Hamilton-Jacobi equations, the first high-order, fully- and semi-discrete central schemes are developed, with order of accuracy ranging from second- through fifth-order. Critical to this work are reconstructions based on new non-oscillatory (or essentially non-oscillatory) interpolants. These schemes are tested using the standard examples, and we prove the monotonicity of the semi-discrete flux.; We also develop central schemes for balance laws, i.e. conservation laws with source terms, in cases where the balance law must be satisfied numerically. First, a method for solving the two-dimensional shallow water equations with bottom topography is developed for conformal triangular meshes. This work extends a semi-discrete central scheme recently developed for conservation laws on triangles, as well as a balanced method for shallow water equations with bottom topography on Cartesian meshes. It is shown that the existing method on triangles cannot be balanced except in the case of very special meshes. A new second-order method is then developed and shown to be balanced on any conformal triangular mesh. The balance and accuracy of this method is demonstrated with various examples.; The second balance law application is the study of the response of the solar atmosphere in vertical flux tubes to forcing at the surface, modeled by the one-dimensional Euler equations in gravity. In this case the required balance is an initial hydrostatic equilibrium. In our simulation the resulting phenomena are dominated by a strong upward propagating shocks. The simulation is performed with second- and third-order semi-discrete central schemes. New interpolants for computational meshes with variable grid spacing are developed via slope-limiting and WENO techniques. The simulations use various observational data for initial and boundary conditions. The results show a quantitative match with recent observational data of the solar atmosphere.