A class of two-level implicit unconditionally stable methods for a fourth order parabolic equation

In this paper, we discuss two-level implicit methods for the solution of a special type of fourth order parabolic partial differential equation of the form uxxxx-2uxxt+utt=f(x,t,u), 00 subject to appropriate initial and Dirichlet boundary conditions by converting the original problem to a coupled system of two second order parabolic equations. We use only three spatial grid points and it is not required to discretize the boundary conditions. The proposed Crank-Nicolson type scheme is second order accurate in both the temporal and spatial dimensions while the compact Crandall's type scheme is second order accurate in temporal and fourth order accurate in spatial dimension. The methods do not require any fictitious nodes outside the solution domain for handling the boundary conditions. For a fixed mesh ratio parameter (∆t/∆x2), the proposed Crandall's type method behaves like a fourth order method in space. Using matrix stability analysis, the proposed methods are shown to be unconditionally stable. The resulting implicit difference formulas gives block tri-diaginal matrix structure which is solved efficiently using block Gauss-Seidel method or block Newton method depending on linear or nonlinear behaviour of the equations. The methods compute the numerical value of u and time-dependent Laplacian uxx−ut, simultaneously. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed methods.