NON-UNIFORM MESH ARITHMETIC AVERAGE DISCRETIZATION FOR PARABOLIC INITIAL BOUNDARY VALUE PROBLEMS

摘要

In this article, we report a new implicit difference method of O(k~2h_l~(-1) + kh_l + h_l~3) on a variable mesh based on arithmetic average discretization for the solution of non-linear parabolic partial differential equation εu_(xx)= f(x,t,u,u_x,u_t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions prescribed, where ε > 0 is a small real constant and k > 0, h_l > 0 are grid sizes in time- and space-directions, respectively. We also derive a new explicit variable mesh method of O(kh_l + h_l~3) for the estimates of (partial deriv u / partial deriv x). In both cases, we require only 3-spatial variable grid points. The proposed methods are directly applicable to problems in polar coordinates and we do not require any fictitious points to handle the singular point. The proposed method when applied to a linear parabolic equation is shown to be unconditionally stable. Numerical results are provided to demonstrate the utility of the proposed variable mesh methods derived.