Variable stepsize SDIMSIMs for ordinary differential equations

TL;DR

This paper introduces variable stepsize second derivative general linear methods directly on nonuniform grids, providing explicit examples up to order four and demonstrating their efficiency through numerical experiments.

Contribution

It presents a novel approach for variable stepsize SGLMs on nonuniform grids, expanding their applicability and confirming their theoretical convergence order.

Findings

Methods are efficient for nonstiff problems

Numerical experiments confirm theoretical order of convergence

Explicit examples up to order four are provided

Abstract

Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order $p$ and stage order $q=p$, some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.