Time stepping algorithms for parabolic partial differential equations

Abstract

Abstract : A number of time stepping algorithms for solving parabolic partial differential equations (PDEs) in one and two space dimensions are studied. The parabolic partial differential equation is transformed into a system of ordinary differential equatiors (ODEs) via method of lines approach (MOL). The solution of the system of ODEs satisfies a recurrence relation which involves matrix exponential functions. Padé approximants are empolyed to approximate such exponential terms. It is shown [9] that methods based on (1,1) diagonal Padé aproximant suffer with time steps restriction for problems with high frequency components in the solution when a time discretization is imposed with steps that are "too large" relative to the spatial discretization. A monotonicity property is established for all diagonal Padé approximants from which it is shown that corresponding higher-order methods suffer a similar time step restriction as the (1,1) Padé. Next, various high-order methods based on subdiagonal Padé approximations are presented which, through a partial fraction expansion, are no more complicated to implement than the first order implicit Euler method based on the (1,0) Padé approximation; moreover, the resulting algorithms are free of a time step restriction intrinsic to those based on diagonal Padé approximations [7]. Splitting methods based on the (1,0) and (1,1) Padé approximants are considered in two space dimension with homogeneous and time dependent boundary conditions [9], [18]. All computations are performed on VAX machine, University of Bahrain. Packages, such as, Mathematica and Spyglass are used for symbolic computation and graphics.