Equilibrium states of Runge Kutta schemes

ABSTRACT

Understanding the behavior of Runge-Kutta codes when stability considerations restrict the stepsize provides useful information for stiffness detection and other implementation details. Analysis of equilibrium states on test problems is presented which provides predictions and insights into this behavior. The implications for global error are also discussed.

REFERENCES

• 1: HALL, G. AND SULEIMAN, M.B. A single code for the solution of stiff and non-stiff ODE's. SIAM J. Sci. Stat. Comput. (to be published).
• 2: KROGH, F.T. Notes on partitioning in the solution of stiff equations. Computing Memo 488, Section 366, California Institute of Technology, Jet Propulsion Laboratory, Pasadena, Calif., 1982.
• 3: LINDBERG, B. Characterisation of optimal stepsize sequences for methods for stiff differential equations. SIAM J. Nurner. Anal. 14, 5 (1977), 859-887.
• 4: PETZOLD, L. Automatic selection of methods for solving stiff and nonstiff systems of ODE's. SIAM J. Sci. Stat. Comput. 4, 1 (1983), 136-148.
• 5: L. F. Shampine, Stiffness and Nonstiff Differential Equation Solvers, II: Detecting Stiffness with Runge-Kutta Methods, ACM Transactions on Mathematical Software (TOMS), v.3 n.1, p.44-53, March 1977 [doi>10.1145/355719.355722]
• 6: WILKINSON, J. C. Some topics in stability for numerical methods for ordinary differential equations. M.Sc. Thesis, Department of Mathematics, Victoria University of Manchester. (1979).

INDEX TERMS

Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Initial value problems; One-step (single step) methods
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis

General Terms:
Algorithms

Keywords:
Runge-Kutta methods, stiff systems