Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs

11/17/2020
by   Yonghui Bo, et al.
0

In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such novel strategy is based on the newly developed exponential scalar variable (ESAV) approach that can remove the bounded-from-blew restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products, which make it more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize the symplectic Runge-Kutta method for both solution variables and the auxiliary variable, where the values of internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms are of constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes.

READ FULL TEXT

page 15

page 16

research
02/28/2021

High-order linearly implicit structure-preserving exponential integrators for the nonlinear Schrödinger equation

A novel class of high-order linearly implicit energy-preserving exponent...
research
12/18/2019

The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing

In this paper, we consider an exponential scalar auxiliary variable (E-S...
research
07/15/2020

Mass- and energy-preserving exponential Runge-Kutta methods for the nonlinear Schrödinger equation

In this paper, a family of arbitrarily high-order structure-preserving e...
research
06/08/2020

Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow

Reductions of the self-consistent mean field theory model of amphiphilic...
research
04/23/2020

Two novel classes of energy-preserving numerical approximations for the sine-Gordon equation with Neumann boundary conditions

We develop two novel classes of energy-preserving algorithms for the sin...
research
07/12/2021

Extending nonstandard finite difference schemes rules to systems of nonlinear ODEs with constant coefficients

In this paper, we present a reformulation of Mickens' rules for nonstand...
research
06/29/2022

MARS : a Method for the Adaptive Removal of Stiffness in PDEs

The E(xplicit)I(implicit)N(null) method was developed recently to remove...

Please sign up or login with your details

Forgot password? Click here to reset