Does error control suppress spuriosity?

In the numerical solution of initial value ordinary differential equations, to what extent does local error control confer global properties? This work concentrates on global steady states or fixed points. It is shown that, for systems of equations, spurious fixed points generally cease to exist when local error control is used. For scalar problems, on the other hand, locally adaptive algorithms generally avoid spurious fixed points by an indirect method---the stepsize selection process causes spurious fixed points to be unstable. However, problem classes exist where, for arbitrarily small tolerances, stable spurious fixed points persist with significant basins of attraction. A technique is derived for generating such examples

Linear equations in general purpose codes for stiff ODEs

It is noted that it is possible to improve significantly the handling of linear problems in a general-purpose code with very little trouble to the user or change to the code. In such situations analytical evaluation of the Jacobian is a lot cheaper than numerical differencing. A slight change in the point at which the Jacobian is evaluated results in a more accurate Jacobian in linear problems. (RWR

Efficient use of implicit formulas with predictor-corrector error estimate

AbstractWith current implementations of implicit formulas for the solution of ordinary differential equations, one first solves the algebraic equations of the formula and then tests to see if the local error is acceptable. When the local error is estimated by comparing predicted and final corrected value, an extremely cheap necessary condition on the first correction is developed which avoids the expense of solving the algebraic equations only to reject the step. This condition and another provide early prediction of all modes of failure to take a step with an implicit formula

Numerical solution of ordinary differential equations

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The Matlab ODE Suite

This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in Matlab