Stable variable step stiff methods for ordinary differential equations

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1983
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Rockswold, Gary
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Mathematics
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The (alpha)-type variable step variable formula method was first shown to be stable by Zlatev in 1978 and more generally in 1981. These methods are studied for their use in the numerical solution of stiff ordinary differential equations with initial conditions;Two parameter families of these (alpha)-type formulas for orders 2-5 are analyzed. Theorems characterizing A(,0)-stability and A-stability are proved. It is shown that no order 5 A(,0)-stable member exists. A((theta))-stability is also discussed;A generalization of the above types of stability is defined for the variable coefficient methods with a variable grid spacing. Theorems for orders 2-3 are proved which provide bounds on the parameters and allowable step size changing sequences in order to increase generalized A((theta))-stability is discussed. A numerical solution for the order 4 case is given. Selection of these formulas is done and comparisons with the variable coefficient backward differentiation formulas are made;A general purpose computer code which uses the (alpha)-type variable step variable formula method is given. Numerical testing on a set of test problems is performed and comparisons are made to three different computer codes.

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Sat Jan 01 00:00:00 UTC 1983