Construction of high order diagonally implicit multistage integration methods for ordinary differential equations

J. C. Butcher, Zdzislaw Jackiewicz

Research output: Contribution to journalArticle

43 Citations (Scopus)

Abstract

The identification of high order diagonally implicit multistage integration methods with appropriate stability properties requires the solution of high dimensional nonlinear equation systems. The approach to the solution of these equations, and hence the construction of suitable methods, that we will describe in this paper, is based on computation of the coefficients of the stability polynomial by a variant of the Fourier series method and solving the resulting systems of polynomial equations by least squares minimization. Examples of explicit and implicit methods of order 5 and 6 are given which are appropriate for nonstiff or stiff differential systems in a sequential computing environment. The coefficients of these methods were obtained numerically with the aid of lmdif. f and lmder. f from MINPACK. These programs minimize the sum of the squares of nonlinear functions by a modification of the Levenberg-Marquardt algorithm. The derived explicit and implicit methods have the same stability properties as explicit Runge-Kutta and SDIRK methods, respectively, of the same order.

Original languageEnglish (US)
Pages (from-to)1-12
Number of pages12
JournalApplied Numerical Mathematics
Volume27
Issue number1
StatePublished - May 1998

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Ordinary differential equations
Ordinary differential equation
Higher Order
Polynomials
Explicit Methods
Implicit Method
Fourier series
Nonlinear equations
Levenberg-Marquardt Algorithm
Stiff Systems
Polynomial equation
Runge-Kutta
Coefficient
Nonlinear Function
Differential System
Least Squares
Nonlinear Equations
High-dimensional
Minimise
Polynomial

Keywords

  • A-stability
  • Fourier series method
  • General linear method
  • Least-squares minimization
  • Ordinary differential equation
  • Region of stability
  • Runge-Kutta formula

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Modeling and Simulation

Cite this

Construction of high order diagonally implicit multistage integration methods for ordinary differential equations. / Butcher, J. C.; Jackiewicz, Zdzislaw.

In: Applied Numerical Mathematics, Vol. 27, No. 1, 05.1998, p. 1-12.

Research output: Contribution to journalArticle

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