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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Discrete Galerkin and related one-step methods for ordinary differential equations

Author: Bernie L. Hulme
Journal: Math. Comp. 26 (1972), 881-891
MSC: Primary 65L05
MathSciNet review: 0315899
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Abstract: New techniques for numerically solving systems of first-order ordinary differential equations are obtained by finding local Galerkin approximations on each subinterval of a given mesh. Different classes of methods correspond to different quadrature rules used to evaluate the innerproducts involved. At each step, a polynomial of degree $ n$ is constructed and the arcs are joined together continuously, but not smoothly, to form a piecewise polynomial of degree $ n$ and class $ {C^0}$. If the $ n$-point quadrature rule used for the innerproducts is of order $ \nu + 1,\nu \geqq n$, then the Galerkin method is of order $ \nu $ at the mesh points. In between the mesh points, the $ j$th derivatives have accuracy of order $ O({h^{\min (\nu ,n + 1)}})$, for $ j = 0$ and $ O({h^{n - j + 1}})$ for $ 1 \leqq j \leqq n$.

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Keywords: Discrete Galerkin methods, initial value problems, ordinary differential equations, piecewise polynomials, collocation, quadrature, implicit Runge-Kutta methods, $ A$-stable
Article copyright: © Copyright 1972 American Mathematical Society