Ill-conditioned matrices and the integration of stiff ODEs
The article is concerned with the numerical integration of stiff initial value problems for systems of ordinary differential equations. It is shown that the matrices occurring in the linear algebraic systems, which have to be solved when implicit linear multistep methods are applied, are ``almost always'' very ill-conditioned. Nevertheless, multistep methods can be evaluated accurately by iteration, since (resp. if) each iteration step improves the solution by one or a few accurate digits. Even semi- implicit methods perform well although they do not involve iteration. This is explained by the fact that the right-hand sides occurring in the linear algebraic systems behave in a way which makes the quotient of the relative errors of solution and right-hand side much less than the matrix condition.
systems of ordinary differential equations
scientific article; zbMATH DE number 540485
scientific article