Abstract
Time dependent problems in Partial Differential Equations (PDEs) are often solved by the Method Of Lines (MOL). For linear parabolic PDEs, the exact solution of the resulting system of first order Ordinary Differential Equations (ODEs) satisfies a recurrence relation involving the matrix exponential function. In this paper, we consider the development of a fourth order rational approximant to the matrix exponential function possessing real and distinct poles which, consequently, readily admits a partial fraction expansion, thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. The resulting parallel algorithm possesses appropriate stability properties, and is implemented on various parabolic PDEs from the literature including the forced heat equation and the advection-diffusion equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Brenner, M. Crouzeix and V. Thomée, Single step methods for inhomogeneous linear differential equations in a Banach space, RAIRO Anal. Numér. 16 (1982) 5–26.
G. Fairweather, A note on the efficient implementation of certain Padé methods for linear parabolic problems, BIT 18 (1978) 106–109.
E. Gallopoulos and Saad, On the parallel solution of parabolic equations, CSRD Report No. 854 (1989).
M. K. Jain, R. K. Jain and R. K. Mohanty, A fourth order difference method for the one-dimensional general quasilinear parabolic partial differential equation, Numer. Methods Partial Differential Equations 6 (1990) 311–319.
A. Q. M. Khaliq, E. H. Twizell and D. A. Voss, On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations, Numer. Methods Partial Differential Equations 9 (1993) 107–116.
J. D. Lawson and J. Ll. Morris, The extrapolation of first-order methods for parabolic partial differential equations I, SIAM J. Numer. Anal. 15 (1978) 1212–1224.
J. D. Lawson and D. A. Swayne, A simple efficient algorithm for the solution of heat conduction problems, in:Proc. 6th Manitoba Conf. Numer. Math. (1976) pp. 239–250.
S. P. Nørsett and A. Wolfbrandt, Attainable order of rational approximations to the exponential function with only real poles, BIT 17 (1977) 200–208.
B. Orel, Real pole approximations to the exponential function, BIT 17 (1991) 144–159.
M. F. Reusch, L. Ratzan, N. Pomphrey and W. Park, Diagonal Padé approximations for initial value problems, SIAM J. Sci. Statist. Comput. 9 (1988) 829–838.
S. M. Serbin, A scheme for parallelizing certain algorithms for the linear inhomogeneous heat equation, SIAM J. Sci. Statist. Comput. 13 (1992) 449–458.
L. F. Shampine, ODE solvers and the Method of Lines, Numer. Methods Partial Differential Equations 10 (1994) 739–755.
G. D. Smith,Numerical Solution of Partial Differential Equations: Finite Difference Methods (Oxford University Press, London, 1985).
D. A. Swayne, Time-dependent boundary and interior forcing in locally one-dimensional schemes, SIAM J. Sci. Statist. Comput. 8 (1987) 755–767.
D. A. Voss and A. Q. M. Khaliq, Parallel LOD methods for second order time dependent PDEs, Comput. Math. Appl. 30 (1995) 739–755.
Author information
Authors and Affiliations
Additional information
Dedicated to Professor J. Crank on the occasion of his 80th birthday
Rights and permissions
About this article
Cite this article
Voss, D.A., Khaliq, A.Q.M. Time-stepping algorithms for semidiscretized linear parabolic PDEs based on rational approximants with distinct real poles. Adv Comput Math 6, 353–363 (1996). https://doi.org/10.1007/BF02127713
Issue Date:
DOI: https://doi.org/10.1007/BF02127713