Partitioned averaged vector field methods

Highlights: • We propose a PAVF method to improve the computational efficiency of the conventional AVF method. • We provide a general partition strategy to construct the PAVF method. • The constructed PAVF method is semi-linear or linearly implicit even for nonlinear problems. • High-order schemes can be obtained by the composition or plus in conjunction of the adjoint method. • Concrete numerical schemes are constructed for two Hamiltonian systems, including ODE and PDE cases. The classic second-order averaged vector field (AVF) method can exactly preserve the energy for Hamiltonian systems. 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.