TY - JOUR T1 - A-Stable and L-Stable Block Implicit One-Step Method AU - Bing Zhou JO - Journal of Computational Mathematics VL - 4 SP - 328 EP - 341 PY - 1985 DA - 1985/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9629.html KW - AB -

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 nonequidistant. 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 $km×km$ matrix during Newton-Raphson iterations, where $m$ is the number of differential equations. When k=4, for example, only a couple of $m×m$ matrices have to be inversed, but four values can be obtained at one time.