Abstract
An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed in this fashion have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially available approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Five illustrative examples are presented.
Highlights
We consider the initial value problem for linear or nonlinear ordinary differential equations
We introduce here a broadly applicable inverse method that constructs a neighbor of a given numerical approximate solution; the neighboring problem does exactly satisfy the original differential equations and serves as an excellent benchmark problem
When we use the IMSL (1989) subroutines DIVPRK and DIVPBS as solvers, we show the utility of this methodology for two celestial mechanics problems (Krogh, 1973) that have been used as test problems several times in the literature
Summary
We consider the initial value problem for linear or nonlinear ordinary differential equations. Whenever we know the true solutions of a test problem, we can investigate the relationship between the true, or global error and the tuning parameters of a given code (e .g., step size, local error tolerance, order, etc.). We introduce here a broadly applicable inverse method that constructs a neighbor of a given numerical approximate solution; the neighboring problem does exactly satisfy the original differential equations (with a known, small forcing function) and serves as an excellent benchmark problem. It is certainly true that there are open questions on this issue needing further investigation; by constructing a family of neighboring benchmark problems, it is usually possible to judge the size of the neighborhood in which the convergence and accuracy properties are relatively invariant with respect to the perturbation. We consider a typical stiff problem and discuss some limitations and restrictions of this methodology
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