TY - JOUR
T1 - Construction of high order diagonally implicit multistage integration methods for ordinary differential equations
AU - Butcher, J. C.
AU - Jackiewicz, Zdzislaw
N1 - Funding Information:
The work of this author was assisted by the Marsden Fund of New Zealand. 2 The work of this author was partially supported by the National Science Foundation under grant NSF DMS-9208048.
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
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U2 - 10.1016/S0168-9274(97)00109-8
DO - 10.1016/S0168-9274(97)00109-8
M3 - Article
AN - SCOPUS:0032068326
SN - 0168-9274
VL - 27
SP - 1
EP - 12
JO - Applied Numerical Mathematics
JF - Applied Numerical Mathematics
IS - 1
ER -