Discrete Galerkin and related one-step methods for ordinary differential equations
HTML articles powered by AMS MathViewer
- by Bernie L. Hulme PDF
- Math. Comp. 26 (1972), 881-891 Request permission
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 $1 \leqq j \leqq n$.References
- Owe Axelsson, A class of $A$-stable methods, Nordisk Tidskr. Informationsbehandling (BIT) 9 (1969), 185–199. MR 255059, DOI 10.1007/bf01946812
- J. C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50–64. MR 159424, DOI 10.1090/S0025-5718-1964-0159424-9
- J. C. Butcher, Integration processes based on Radau quadrature formulas, Math. Comp. 18 (1964), 233–244. MR 165693, DOI 10.1090/S0025-5718-1964-0165693-1 F. H. Chipman, Numerical Solution of Initial Value Problems Using $A$-stable RungeKutta Processes, Ph.D. Thesis, Univ. of Waterloo, Waterloo, Ontario, 1971.
- G. J. Cooper, Interpolation and quadrature methods for ordinary differential equations, Math. Comp. 22 (1968), 69–76. MR 224289, DOI 10.1090/S0025-5718-1968-0224289-7
- Germund G. Dahlquist, A special stability problem for linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 3 (1963), 27–43. MR 170477, DOI 10.1007/bf01963532
- Byron L. Ehle, High order $A$-stable methods for the numerical solution of systems of D.E.’s, Nordisk Tidskr. Informationsbehandling (BIT) 8 (1968), 276–278. MR 239762, DOI 10.1007/bf01933437 B. L. Ehle, On Padé Approximations to the Exponential Function and $A$-Stable Methods for the Numerical Solution of Initial Value Problems, Ph.D. Thesis, Univ. of Waterloo, Waterloo, Ontario, 1969.
- Preston C. Hammer and Jack W. Hollingsworth, Trapezoidal methods of approximating solutions of differential equations, Math. Tables Aids Comput. 9 (1955), 92–96. MR 72547, DOI 10.1090/S0025-5718-1955-0072547-2
- Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
- F. B. Hildebrand, Introduction to numerical analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. MR 0075670
- Bernie L. Hulme, One-step piecewise polynomial Galerkin methods for initial value problems, Math. Comp. 26 (1972), 415–426. MR 321301, DOI 10.1090/S0025-5718-1972-0321301-2
- K. Wright, Some relationships between implicit Runge-Kutta, collocation Lanczos $\tau$ methods, and their stability properties, Nordisk Tidskr. Informationsbehandling (BIT) 10 (1970), 217–227. MR 266439, DOI 10.1007/bf01936868
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Math. Comp. 26 (1972), 881-891
- MSC: Primary 65L05
- DOI: https://doi.org/10.1090/S0025-5718-1972-0315899-8
- MathSciNet review: 0315899