Towards efficient implementation of singly-implicit methods

ABSTRACT

It has been observed that for problems of low dimension the transformations used in the implementation of singly-implicit Runge-Kutta methods consume an unreasonable share of the total computational costs. Two proposals for reducing these costs are presented here. The first makes use of an alternative transformation for which the combined operation counts of the transformations together with the iterations themselves are lower than for the standard implementation scheme for singly-implicit methods. The second proposal is to use a Runge-Kutta method for which the first row of the coefficient matrix is zero but which still possesses acceptable stability properties. It is hoped that by combining these two proposals increased efficiency in the implementation of Runge-Kutta methods for stiff problems can be achieved.

REFERENCES

• 1: BURRAGE, K. A special family of Runge-Kutta methods for solving stiff differential equations. BIT 18, 1 (1978), 22-24.
• 2: BURRAGE, K., BUTCHER, J. C., AND CHIPMAN, F. H. An implementation of singly-implicit Runge-Kutta methods. BIT 20, 3 (1980), 326-340.
• 3: BUTCHER, J.C. On the implementation of implicit Runge-Kutta methods. BIT 16, 3 (1976), 237-240.
• 4: J. C. Butcher, A Transformed implicit Runge-Kutta Method, Journal of the ACM (JACM), v.26 n.4, p.731-738, Oct. 1979 [doi>10.1145/322154.322163]
• 5: BUTCHER, J.C.A generalization of singly-implicit methods. BIT 21, 2 {1981), 175-189.
• 6: BUTCHER, J. C., BURRAGE, K., AND CHIPMAN. F.H. STRIDE--stable Runge-Kutta integrator for differential equations. Univ. of Auckland Computational Mathematics Rep. 20, 1979.
• 7: Patrick W. Gaffney, A Performance Evaluation of Some FORTRAN Subroutines for the Solution of Stiff Oscillatory Ordinary Differential Equations, ACM Transactions on Mathematical Software (TOMS), v.10 n.1, p.58-72, March 1984 [doi>10.1145/356068.356073]
• 8: WOLFBRANDT, A. A study of Rosenbrock processes with respectto order conditions and stiff stability. Dept. of Computer Science, Chalmers Univ. of Technology, GSteborg.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Initial value problems

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: One-step (single step) methods; Stiff equations
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Efficiency

General Terms:
Algorithms, Performance