Efficient higher order single step methods for parabolic problems. I

Bramble, James H.; Sammon, Peter H.

doi:10.1090/S0025-5718-1980-0572848-X

Efficient higher order single step methods for parabolic problems. I
HTML articles powered by AMS MathViewer

by James H. Bramble and Peter H. Sammon PDF

Math. Comp. 35 (1980), 655-677 Request permission

Abstract:

Some efficient, high order methods are discussed for approximating the solution of an initial boundary value problem for a homogeneous parabolic equation with time dependent coefficients. The methods are based on Galerkin-type approximations in the spacial variables and single step methods in the time variable. The equations defining the time-stepping procedure are solved only approximately, however. A preconditioned iterative technique is used for this purpose. The resulting algorithm is shown to produce optimal order approximations using only the order of work required by the single step method applied to the parabolic problem with time independent coefficients.

References

On Preconditioning and Convergence Acceleration in Sparse Matrix Problems

Garth A. Baker, James H. Bramble, and Vidar Thomée, Single step Galerkin approximations for parabolic problems, Math. Comp. 31 (1977), no. 140, 818–847. MR 448947, DOI 10.1090/S0025-5718-1977-0448947-X
Jim Douglas Jr. and Todd Dupont, Alternating-direction Galerkin methods on rectangles, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 133–214. MR 0273830
Jim Douglas Jr., Todd Dupont, and Richard E. Ewing, Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem, SIAM J. Numer. Anal. 16 (1979), no. 3, 503–522. MR 530483, DOI 10.1137/0716039
Avner Friedman, Partial differential equations, Corrected reprint of the original edition, Robert E. Krieger Publishing Co., Huntington, N.Y., 1976. MR 0454266
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. III, Die Grundlehren der mathematischen Wissenschaften, Band 183, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French by P. Kenneth. MR 0350179, DOI 10.1007/978-3-642-65393-3
Nabil R. Nassif and Jean Descloux, Stability study for time-dependent linear parabolic equations and its application to Hermitian methods, Topics in numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976) Academic Press, London, 1977, pp. 293–316. MR 0657787

Convergence Estimates for Semidiscrete Parabolic Equation Approximations

Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502