Optimal Second Derivative Methods for Stiff Systems
In implementing numerical methods for stiff ordinary differential equations several difficulties arise which are not usually encountered with methods designed for non-stiff equations. These difficulties include the choice of the basic formula, the choice of the iteration scheme to solve the implicit set of equations associated with each step, and the choice of a valid error estimate and step control strategy. We will introduce a class of second derivative multistep formulas suitable for stiff equations and show how they have been implemented in an efficient variable-order method.
We will also discuss the special difficulty associated with large systems and discuss how, in this case, one must take advantage of a sparse Jacobian. We will consider two modified versions of our method which are suitable for the solution of large systems. One modification involves the introduction of complex arithmetic and the second involves the introduction of a different class of basic second derivative formulas. Numerical results will be given for the original method and each of the modified versions.
KeywordsIteration Scheme Derivative Method Stiff System Complex Arithmetic Derivative Formula
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- 1972 Enright, W. H., “Studies in the numerical solution of stiff ordinary differential equations,” PhD Thesis, University of Toronto, Computer Science Report 42.Google Scholar