Backward error analysis for conjugate symplectic methods

01/11/2022
by   Robert I McLachlan, et al.
0

The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/25/2020

Backward error analysis for variational discretisations of partial differential equations

In backward error analysis, an approximate solution to an equation is co...
research
07/05/2019

Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems based on the Wong--Zakai approximation

We investigate the stochastic modified equation which plays an important...
research
07/18/2022

Convergence acceleration for the BLUES function method

A detailed comparison is made between four different iterative procedure...
research
09/29/2020

Method of fundamental solutions for Neumann problems of the modified Helmholtz equation in disk domains

The method of the fundamental solutions (MFS) is used to construct an ap...
research
12/27/2020

Convergence, error analysis and longtime behavior of the Scalar Auxiliary Variable method for the nonlinear Schrödinger equation

We carry out the convergence analysis of the Scalar Auxiliary Variable (...
research
03/23/2022

A stochastic Hamiltonian formulation applied to dissipative particle dynamics

In this paper, a stochastic Hamiltonian formulation (SHF) is proposed an...
research
10/07/2019

Explicit and implicit error inhibiting schemes with post-processing

Efficient high order numerical methods for evolving the solution of an o...

Please sign up or login with your details

Forgot password? Click here to reset