On some highorder accurate fully discrete Galerkin methods for the Kortewegde Vries equation
Authors:
Vassilios A. Dougalis and Ohannes A. Karakashian
Journal:
Math. Comp. 45 (1985), 329345
MSC:
Primary 65M60
MathSciNet review:
804927
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Abstract: We construct and analyze fully discrete Galerkin (finiteelement) methods of high order of accuracy for the numerical solution of the periodic initialvalue problem for the Kortewegde Vries equation. The methods are based on a standard space discretization using smooth periodic splines on a uniform mesh. For the time stepping, we use two schemes of third (resp. fourth) order of accuracy which are modifications of wellknown, diagonally implicit RungeKutta methods and require the solution of two (resp. three) nonlinear systems of equations at each time step. These systems are solved approximately by Newton's method. Provided the initial iterates are chosen in a specific, accurate way, we show that only one Newton iteration per system is needed to preserve the stability and order of accuracy of the scheme. Under certain mild restrictions on the space mesh length and the time step we prove error estimates of optimal rate of convergence for both schemes.
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 R. Alexander, "Diagonally implicit RungeKutta methods for stiff O.D.E.'s," SIAM J. Numer. Anal., v. 14, 1976, pp. 10061021. MR 0458890 (56:17089)
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DOI:
http://dx.doi.org/10.1090/S00255718198508049278
PII:
S 00255718(1985)08049278
Article copyright:
© Copyright 1985
American Mathematical Society
