Partitioned AVF methods

TL;DR

This paper introduces a partitioned AVF method that preserves energy in Hamiltonian systems while reducing computational cost through variable partitioning and semi-implicit schemes, with enhanced accuracy and additional conservation properties.

Contribution

It proposes a novel partitioned AVF approach that improves efficiency, accuracy, and conservation properties for nonlinear Hamiltonian systems.

Findings

The partitioned AVF methods are computationally cheaper than the classic AVF.

The schemes preserve energy and, for specific problems, additional conserved quantities.

Numerical tests confirm the theoretical advantages of the proposed methods.

Abstract

The classic second-order average vector field (AVF) method can exactly preserve the energy for Hamiltonian ordinary differential equations and partial differential equations. However, the AVF method inevitably leads to fully-implicit nonlinear algebraic equations for general nonlinear systems. To address this drawback and maintain the desired energy-preserving property, a first-order partitioned AVF method is proposed which first divides the variables into groups and then applies the AVF method step by step. In conjunction with its adjoint method we present the partitioned AVF composition method and plus method respectively to improve its accuracy to second order. Concrete schemes for two classic model equations are constructed with semi-implicit, linear-implicit properties that make considerable lower cost than the original AVF method. Furthermore, additional conservative property can be generated besides the conventional energy preservation for specific problems. Numerical verification of these schemes further conforms our results.