Partitioned AVF 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.

23 pages, 20 figures

References in corpus (5)

• Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method
• Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
• On the Existence of Energy-Preserving Symplectic Integrators Based upon Gauss Collocation Formulae
• Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
• Classifying orbits in the classical Henon-Heiles Hamiltonian system

Cited by in corpus (8)

• A Roadmap for Discretely Energy-Stable Schemes for Dissipative Systems Based on a Generalized Auxiliary Variable with Guaranteed Positivity
• A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation
• Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs
• Structure-preserving algorithms for multi-dimensional fractional Klein-Gordon-Schrödinger equation
• Linearly implicit local and global energy-preserving methods for PDEs with a cubic Hamiltonian
• Two novel classes of arbitrary high-order structure-preserving algorithms for canonical Hamiltonian systems
• Mass- and energy-preserving exponential Runge-Kutta methods for the nonlinear Schrödinger equation
• Efficient energy-preserving exponential integrators for multi-components Hamiltonian systems