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

DOI:
https://doi.org/10.1090/S0025-5718-1972-0315899-8

MathSciNet review:
0315899

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Abstract | References | Similar Articles | Additional Information

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 is constructed and the arcs are joined together continuously, but not smoothly, to form a piecewise polynomial of degree and class . If the -point quadrature rule used for the innerproducts is of order , then the Galerkin method is of order at the mesh points. In between the mesh points, the th derivatives have accuracy of order , for and for .

**[1]**Owe Axelsson,*A class of 𝐴-stable methods*, Nordisk Tidskr. Informationsbehandling (BIT)**9**(1969), 185–199. MR**0255059****[2]**J. C. Butcher,*Implicit Runge-Kutta processes*, Math. Comp.**18**(1964), 50–64. MR**0159424**, https://doi.org/10.1090/S0025-5718-1964-0159424-9**[3]**J. C. Butcher,*Integration processes based on Radau quadrature formulas*, Math. Comp.**18**(1964), 233–244. MR**0165693**, https://doi.org/10.1090/S0025-5718-1964-0165693-1**[4]**F. H. Chipman,*Numerical Solution of Initial Value Problems Using -stable RungeKutta Processes*, Ph.D. Thesis, Univ. of Waterloo, Waterloo, Ontario, 1971.**[5]**G. J. Cooper,*Interpolation and quadrature methods for ordinary differential equations*, Math. Comp.**22**(1968), 69–76. MR**0224289**, https://doi.org/10.1090/S0025-5718-1968-0224289-7**[6]**Germund G. Dahlquist,*A special stability problem for linear multistep methods*, Nordisk Tidskr. Informations-Behandling**3**(1963), 27–43. MR**0170477****[7]**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**0239762****[8]**B. L. Ehle,*On Padé Approximations to the Exponential Function and -Stable Methods for the Numerical Solution of Initial Value Problems*, Ph.D. Thesis, Univ. of Waterloo, Waterloo, Ontario, 1969.**[9]**Preston C. Hammer and Jack W. Hollingsworth,*Trapezoidal methods of approximating solutions of differential equations*, Math. Tables Aids Comput.**9**(1955), 92–96. MR**0072547**, https://doi.org/10.1090/S0025-5718-1955-0072547-2**[10]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[11]**F. B. Hildebrand,*Introduction to numerical analysis*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. MR**0075670****[12]**Bernie L. Hulme,*One-step piecewise polynomial Galerkin methods for initial value problems*, Math. Comp.**26**(1972), 415–426. MR**0321301**, https://doi.org/10.1090/S0025-5718-1972-0321301-2**[13]**K. Wright,*Some relationships between implicit Runge-Kutta, collocation Lanczos 𝜏 methods, and their stability properties*, Nordisk Tidskr. Informationsbehandling (BIT)**10**(1970), 217–227. MR**0266439**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0315899-8

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

Article copyright:
© Copyright 1972
American Mathematical Society