Structural Characterization and Efficient Implementation Techniques for $A$-Stable High-Order Integration Methods

摘要

This paper presents structural characterization and performance enhancement strategies for the recently proposed $A$-stable and $L$-stable high-order integration methods based on the Obreshkov formula. It is demonstrated that although the Jacobian matrix in these methods has a bigger size than the Jacobian matrix in classical low-order methods, it enjoys a special structure that can be used to develop efficient factorization techniques. In addition, the paper proposes a method to reduce the number of Newton–Raphson iterations needed to converge in the large Jacobian domain.