Summary
The error of the approximate solution obtained by discretising a functional equation can be shown under certain conditions to possess an asymptotic expansion in terms of some parameter which is usually a representative step-length. We consider the case of two-parameter expansions, which is particularly relevant to parabolic equations. We derive results for the existence of the expansion and for the application of the classical difference correction and of defect correction. The theory is illustrated by the discussion of a simple parabolic problem
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References
Frank, R.: The method of iterated defect correction and its application to two-point boundary value problems. Part I. Numer. Math.25, 409–419 (1976). Part II. Numer. Math.27, 407–420 (1977)
Frank, R., Ueberhuber, C.W.: Iterated defect correction for differential equations. Report No. 29/77. Inst. f. Numer. Math., Technical University of Vienna 1977
Hanson, P.M., Walsh, J.E.: Asymptotic theory of the global error and some techniques of error estimation for parabolic equations. Report No. 43. Department of Mathematics, University of Manchester 1979
Hildebrand, F.B.: Introduction to Numerical Analysis. New York: McGraw-Hill, 1956
Lindberg, B.: Compact deferred correction formulas. In: Numerical integration of differential equations and large linear systems (J. Hinze, ed). Lecture Notes in Mathematics, No. 968. Berlin, Heidelberg, New York: Springer 1982
Pereyra, V.: On improving an approximate solution of a functional equation by deferred corrections. Numer. Math.8, 376–391 (1966)
Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Berlin, Heidelberg, New York: Springer 1973
Stetter, H.J.: Global error estimation in O.D.E. solvers. Proceedings of Dundee Conference on Numerical Analysis (G.A. Watson, ed.) No. 630. Berlin, Heidelberg, New York: Springer 1977
Stetter, H.J.: The defect correction principle and discretization methods. Numer. Math.29, 425–443 (1978)
Zadunaisky, P.E.: On the estimation of errors propagated in the numerical integration of ordinary differential equations. Numer. Math.27, 21–39 (1976)