Locally One Dimensional Finite Difference Schemes For A Class Of Viscous Wave Equations

The study of numerical solutions of partial differential equations plays an important rule in computational mathematics field. Finite difference method is one mean of some important methods at present. In some difference schemes, the explicit scheme is easy to be computed, but it has the limitation of stability. Generally, the implicit schemes are unconditionally stable, but on every time level, we must solve linear systems. When we deal with high dimensional problems, the computational cost will be very expensive.In this paper, we consider the alternating direction finite difference method for a class of viscous wave equations. First we reduce the order of equation by replacement of variable to construct C-N difference scheme in the direction of time, then add some perturbed terms and decompose the operator to get a new LOD difference scheme.We present an extended locally one-dimensional finite difference scheme for two and three dimensional viscous wave equations with nonhomogeneous boundary condition in§2 and§3. This scheme can decompose high dimensional problems to one dimension problems completely. It overcomes the defect that the source term is hard to decompose and the intermediate boundary conditions are difficulty to determine. The convergence order of the LOD scheme is O(△t~2+h~2) in a discrete L~2 norm.We get a compact LOD difference scheme by improving the result of the section 2. This scheme not only preserves the advantages of foregoing schemes but also improves the accuracy of space direction to O(h~4). Numerical example indicates the computing efficiency is good.