Abstract
In this paper we study in an abstract setting the structure of estimates for the global (accumulated) error in semilinear finite-difference methods. We derive error estimates, which are the most refined ones (in a sense specified precisely in this paper) that are possible for the difference methods considered. Applications and (numerical) examples are presented in the following fields: 1. Numerical solution of ordinary as well as partial differential equations with prescribed initial or boundary values. 2. Accumulation of local round-off error as well as of local discretization error. 3. The problem of fixing which methods out of a given class of finite-difference methods are “most stable”. 4. The construction of finite-difference methods which are convergent but not consistent with respect to a given differential equation.
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References
Ceschino, F., Kuntzmann, J.: Problèmes différentiels de conditions initiales. Paris: Dunod 1963.
Forsythe, G. E., Wasow, W. R.: Finite-difference methods for partial differential equations. New York: J. Wiley & Sons 1960.
Godunov, S. K., Ryabenki, V. S.: Theory of difference schemes. Amsterdam: North-Holland Publishing Company 1964.
Gragg, W. B., Stetter, H. J.: Generalized multistep predictor-corrector methods. J. Assoc. Comput. Mach.11, 188–209 (1964).
Henrici, P.: Discrete variable methods in ordinary differential equations. New York: J. Wiley & Sons 1962.
Hull, T. E., Luxemburg, W. A. J.: Numerical methods and existence theorems for ordinary differential equations. Numer. Math.2, 30–41 (1960).
Isaacson, E., Keller, H. B.: Analysis of numerical methods. New York: J. Wiley & Sons 1966.
Lees, M.: Discrete methods for nonlinear two-point boundary value problems. In: Numerical solution of partial differential equations, e. d. J. H. Bramble. New York: Academic Press 1966.
Metté, A.: Essai de résolution du problème de Goursat par la methode de Runge-Kutta pour une equation aux dérivées partielles du type hyperbolique. Rev. Francaise Informat. Recherche Opérationnelle1, 67–90 (1967).
Spijker, M. N.: Convergence and stability of step-by-step methods for the numerical solution of initial-value problems. Numer. Math.8, 161–177 (1966).
Spijker, M. N.: Stability and convergence of finite-difference methods (Thesis). Leiden University 1968.
Spijker, M. N.: Reduction of round-off error by splitting of difference formulae. J. Soc. Indust. Appl. Math. Ser. B. Num. Anal., to appear.
Stetter, H. J.: A study of strong and weak stability in discretization algorithms. J. Soc. Indust. Appl. Math. Ser. B. Numer. Anal.2, 265–280 (1965).
—: Stability of nonlinear discretization algorithms. In: Numerical solution of partial differential equations, ed. J. H. Bramble. New York: Academic Press 1966.
—: Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations. Numer. Math.7, 18–31 (1965).
—: Instability and non-monotonicity phenomena in discretizations to boundaryvalue problems. Numer. Math.12, 139–145 (1968).
—, Törnig, W.: General multistep finite difference methods for the solution ofu xy =f(x, y, u, u x ,u y ). Rend. Circ. Mat. Palermo12, 281–298 (1963).
Törnig, W.: Zur numerischen Behandlung von Anfangswertproblemen partieller hyperbolischer Differentialgleichungen zweiter Ordnung in zwei unabhängigen Veränderlichen. Arch. Rat. Mech. Anal.4, 428–466 (1960).
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Spijker, M.N. On the structure of error estimates for finite-difference methods. Numer. Math. 18, 73–100 (1971). https://doi.org/10.1007/BF01398460
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DOI: https://doi.org/10.1007/BF01398460