### 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 language | English (US) |
---|---|

Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Applied Numerical Mathematics |

Volume | 27 |

Issue number | 1 |

State | Published - May 1998 |

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### 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.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 27, no. 1, pp. 1-12.

}

TY - JOUR

T1 - Construction of high order diagonally implicit multistage integration methods for ordinary differential equations

AU - Butcher, J. C.

AU - Jackiewicz, Zdzislaw

PY - 1998/5

Y1 - 1998/5

N2 - 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.

AB - 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.

KW - A-stability

KW - Fourier series method

KW - General linear method

KW - Least-squares minimization

KW - Ordinary differential equation

KW - Region of stability

KW - Runge-Kutta formula

UR - http://www.scopus.com/inward/record.url?scp=0032068326&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032068326&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032068326

VL - 27

SP - 1

EP - 12

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

IS - 1

ER -