Piecewise analytical integration of chemical rate equations. II. Error analysis, implementation, and applications

摘要

The accuracy and stability of the piecewise analytical method of our previous work for the integration of chemical rate equations are discussed for piecewise constant and linear polynomial approximants. In both cases, the solutions are shown to have excellent stability properties, even for stiff systems of differential equations. Thus, the step size can be controlled by the truncation error alone. The piecewise constant method is shown to have a local truncation error varying as the cube of the step size, while the more accurate linear approximation has an error which is fifth power in the step size. These results are used to produce a successful algorithm for control of step size in a variable step, variable order method. This new method has several advantages over previous stiff integrators, such as Gearrsquo;s method: (1) The results are defined by analytical functions at all times, so that integrals, derivatives for other properties of the solutions may be obtained analytically. (2) The solutions are strictly nonhyphen;negative. (3) Considerably reduced computer storage is required. (4) Savings in computer time are projected for large calculations involving many species and reactions, and for ones with variable temperature.