TY - JOUR T1 - Nonlinear Implicit One-Step Schemes for Solving Initial Value Problems for Ordinary Differential Equation with Steep Gradients AU - Sun , Jia-Chang AU - Jackson , Ken JO - Journal of Computational Mathematics VL - 3 SP - 264 EP - 281 PY - 1983 DA - 1983/01 SN - 1 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9703.html KW - AB -

A general theory for nonlinear implicit one-step schemes for solving initial value problems for ordinary differential equations is presented in this paper. The general expansion of "symmetric" implicit one-step schemes having second-order is derived and stability and convergence are studied. As examples, some geometric schemes are given.
Based on previous work of the first author on a generalization of means, a fourth-order nonlinear implicit one-step scheme is presented for solving equations with steep gradients. Also, a hybrid method based on the GMS and a fourth-order linear scheme is discussed. Some numerical results are given.