Abstract
In this paper we derive high order implicit difference methods for large systems of ODE. The methods are based on the deferred correction principle, yielding accuracy of order p by applying the trapezoidal rule p/2 times in each timestep. Numerical experiments demonstrate the efficiency of the method.
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REFERENCES
M. Carpenter, D. Gottlieb, S. Abarbanel, and W.-H. Don, The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: A study of the boundary error, SIAM J. Sci. Comput., 16 (1995), pp. 1241–1252.
G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), pp. 27–43.
J. Daniel, V. Pereyra, and L. Schumaker, Iterated deferred corrections for initial value problems, Acta Cient. Venezolana, 19 (1968), pp. 128–135.
A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT, 40 (2000), pp. 241–266.
B. Gustafsson and L. Hemmingsson-Frändén, A fast domain decomposition high order Poisson solver, J. Sci. Comput., 14 (1999), pp. 223–243.
B. Gustafsson and L. Hemmingsson-Frändén, Implicit high-order difference methods and domain decomposition, Appl. Numer. Math., 33 (2000), pp. 493–500.
B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time Dependent Problems and Difference Methods, Wiley, New York, 1995.
B. Lindberg, Error estimation and iterative improvement for discretization algorithms, BIT, 20 (1980), pp. 486–500.
V. Pereyra, Accelerating the convergence of discretizations algorithms, SIAM J. Numer. Anal., 4 (1967), pp. 508–532.
V. Pereyra, Iterated deferred corrections for nonlinear operator equations, Num er. Math., 10 (1967), pp. 316–323.
V. Pereyra, Iterated deferred corrections for nonlinear operator equations, Num er. Math., 11 (1968), pp. 111–125.
V. Pereyra, Highly accurate numerical solution of quasilinear elliptic boundary-value problems in n dimensions, Math. Comp., 11 (1970), pp. 771–783.
V. Pereyra, W. Proskurowski, and O. Widlund, High order fast Laplace solvers for the Dirichlet problem on general regions, Math. Comp., 31 (1977), pp. 1–16.
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Gustafsson, B., Kress, W. Deferred Correction Methods for Initial Value Problems. BIT Numerical Mathematics 41, 986–995 (2001). https://doi.org/10.1023/A:1021937227950
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DOI: https://doi.org/10.1023/A:1021937227950