High Order Difference Schemes For The First Order Hyperbolic Equations With Variable Coefficients

The first order hyperbolic equation with variable coefficients has a broad application background in the natural sciences. This article is devoted to constructing high accurate numerical methods for the first order hyperbolic equation with variable coefficients in one and two dimensions and establishing the corresponding error estimates.This article is divided into two parts.The box scheme is a well-known difference scheme for the one dimensional hyperbolic equation. It only includes two-level four-point and is a semi-implicit scheme. Its trun-cation error is second order both in space and in time. It is unconditionally stable and convergent. But the existing theoretical results are obtained only in L2 norm. In the first part of this thesis, a priori estimate for the box scheme in L∞, norm is derived. Then the asymptotic expansion of the difference solution is given and a Richardson extrapolation is constructed, the accuracy of the numerical solution is improved greatly. Then the tech-nique in [Zhou, Tian, Deng, J. Sci. Comput,56 (2013),45-66] is used to establish a high accuracy compact difference scheme. A priori estimate in the L2 norm of this difference scheme is derived by the energy method, then it is used to show that the difference scheme is unconditionally stable and convergent in the L2 norm with the convergence order of O(τ2+h4). The asymptotic expansion of the difference solution is given and a Richardson extrapolation is constructed. At last some numerical examples are given to demonstrate the theoretical results.In the second part, at first a second order difference scheme is obtained for the two-dimensional problem with variable coefficients. The convergence and stability of this difference scheme is proved by the energy method. The convergence order is two in both time and space in the L2 norm. Then similarly to the compact difference scheme for one dimensional problem, a compact difference scheme is constructed, and a compact ADI difference scheme is also proposed. The unique solvability is analyzed. With the constant coefficients, the unconditional stability and convergence of the difference scheme are proved by the Von Neumann method. At last some numerical examples is provided to verify the theoretical results.