Ill-conditioned matrices and the integration of stiff ODEs

Abstract Popular methods for the integration of a stiff initial-value problem for a system of ordinary differential equations (ODEs) require the solution of systems of linear equations. It is shown that the matrices are very ill-conditioned. Implicit linear multistep methods (LMMs) can be evaluated accurately by iteration, even when the matrices are very ill-conditioned. Although semi-implicit methods do not involve iteration, it is observed that codes based on these methods cope with ill-conditioned matrices about as well as codes based on LMMs. An explanation is provided for this fact.

[1]  P. Deuflhard,et al.  A semi-implicit mid-point rule for stiff systems of ordinary differential equations , 1983 .

[2]  A. C. Hindmarsh,et al.  Applications of EPISODE: an experimental package for the integration of systems of ordinary differential equations. [In FORTRAN for CDC 7600] , 1975 .

[3]  M. K. Gordon,et al.  Typical problems for stiff differential equations , 1975, SGNM.

[4]  R. Skeel.  Iterative refinement implies numerical stability for Gaussian elimination , 1980 .

[5]  T. E. Hull,et al.  Comparing numerical methods for stiff systems of O.D.E:s , 1975 .

[6]  A. Sluis.  Condition numbers and equilibration of matrices , 1969 .

[7]  Lawrence F. Shampine,et al.  Control of step size and order in extrapolation codes , 1987 .

[8]  C. W. Gear,et al.  Numerical initial value problem~ in ordinary differential eqttations , 1971 .

[9]  A. Sluis.  Stability of solutions of linear algebraic systems , 1970 .

[10]  Peter Deuflhard,et al.  Recent progress in extrapolation methods for ordinary differential equations , 1985 .

[11]  Lawrence F. Shampine,et al.  Implementation of Rosenbrock Methods , 1982, TOMS.

[12]  L. Shampine.  Implementation of Implicit Formulas for the Solution of ODEs , 1980 .

[13]  Lawrence F. Shampine,et al.  Evaluation of a Test Set for Stiff ODE Solvers , 1981, TOMS.

[14]  Robert D. Skeel,et al.  Scaling for Numerical Stability in Gaussian Elimination , 1979, JACM.

[15]  T. E. Hull,et al.  Comparing Numerical Methods for the Solution of Stiff Systems of ODEs Arising in Chemistry , 1976 .

[16]  L. Shampine.  Practical improvements to SIMPR codes for stiff ODEs , 1989, Conference on Numerical Ordinary Differential Equations.