One-step piecewise polynomial Galerkin methods for initial value problems

Author:
Bernie L. Hulme

Journal:
Math. Comp. **26** (1972), 415-426

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1972-0321301-2

MathSciNet review:
0321301

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Abstract | References | Similar Articles | Additional Information

Abstract: A new approach to the numerical solution of systems of first-order ordinary differential equations is given by finding local Galerkin approximations on each subinterval of a given mesh of size . One step at a time, a piecewise polynomial, of degree and class , is constructed, which yields an approximation of order at the mesh points and between mesh points. In addition, the th derivatives of the approximation on each subinterval have errors of order . The methods are related to collocation schemes and to implicit Runge-Kutta schemes based on Gauss-Legendre quadrature, from which it follows that the Galerkin methods are -stable.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0321301-2

Keywords:
Galerkin method,
initial value problems,
ordinary differential equations,
piecewise polynomials,
Gauss-Legendre quadrature,
collocation methods,
implicit Runge-Kutta methods,
-stable

Article copyright:
© Copyright 1972
American Mathematical Society