Generalized trapezoidal formulas for parabolic equations

Abstract

We investigate the application of the one-parameter family of generalized trapezoidal formulas (GTFs) introduced in Chawla et al. [2] for the time-integration of parabolic equations. The resulting GTF finite-difference schemes (GTF-FDS) are, in general, second order in both time and space and unconditionally stable. Interestingly, there exists a method of the family which is third order in time. Unlike the popular Crank -Nicolson scheme, our present GTF-FDS can cope with discontinuities in the boundary conditions and the initial conditions. We consider extensions of the GTF-FDS for equations with derivative boundary conditions and to a nonlinear problem. Numerical experiments demonstrate the superiority of the present GTF-FDS, especially for the case of problems with discontinuities in the boundary and the initial conditions.