A-Stable And L-Stable Block Implicit One-Step Method
Bing Zhou 11 Shanghai Teachers University, China
Received 1984-11-24 Revised Online 2006-11-24
A class of methods for solving the initial problem for ordinary differential equations are studied. We develop k-block implicit one step methods whose nodes in a block are noneqidistant. When the components of the node vector are related to the zeros of Jacobi's orthogonal polynomials, we can derive a subclass of formulas which are A or L-stable. The order can be arbitrarily high with A-or L-stability. We suggest a modified algorithm which avoids the inversion of a kmXkm matrix during Newton-Raphson iterations, where m is the number of differential equations. when k=4, for example, only a couple of mXm matrices have to be inversed, but four values can be obtained at one time.