Linearized Alternating Direction Implicit Difference Schemes For Two-dimensional Semilinear Evolution Equations

The study of numerical solutions of partial differential equations plays an important rule in computational mathematics. Finite difference is one of the main methods. For the semilinear evolution equations, one discretizing approach is to use an explicit difference scheme which is easy to be computed, but conditionally stable. The other one is the implicit scheme which is unconditionally stable, but at each time level, we must solve linear systems. When we deal with high dimensional problems, the computational cost becomes very large.In this paper, we construct linearized alternating direction implicit difference methods for two-dimensional semilinear evolution equations. First, based on the Crank-Nicolson difference discretizing idea, we discretize the semi-linear equations. By adding some perturbing terms and making full use of the derivative information of the semilinear source term, we construct a class of linearized two level unconditionally stable implicit difference scheme possessing second order accuracy both in temporal and spatial directions.In section 2, we propose a class of linearized two level Peaceman-Rachford type alternating direction implicit difference scheme for semilinear reaction-diffusion equations. By making full use of the P-R method, the scheme has the advantages of concise form and easy to use. This section also proves that the scheme has second order accuracy with respect to discreteL2 norm both in space and time directions by using discrete energy method. Numerical examples illustrate the correctness of the theoretical analysis and the effectiveness of the scheme.The section 3, we presents a P-R type alternating direction difference scheme for viscous wave equations, which are a special class of semi-linear hyperbolic equations. First, we reduce the order of the equations by replacement of variable, then by using P-R discretizing idea, we derive the computational scheme. This section also proves that the scheme has second order accuracy with respect to discrete L2 norm and discrete H1 norm both in spatial and temporal directions. Numerical examples show that the scheme is effective.