Partitioned AVF methods
For researchers in numerical methods for Hamiltonian systems, this work offers more efficient energy-preserving schemes, though the improvements are incremental.
The paper proposes a first-order partitioned AVF method to reduce the computational cost of the classic second-order AVF method while preserving energy, and extends it to second-order accuracy via composition and plus methods, achieving semi-implicit and linear-implicit schemes with lower cost.
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.