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A new construction of modified equations for variational integrators

Marcel Oliver, Sergiy Vasylkevych

TL;DR

The paper addresses the problem that traditional backward error analysis for variational integrators applied to PDEs induces unphysically fast modes via a Hamiltonian modification. It proposes a degenerate variational asymptotics approach that uses a near identity change in the variational principle to derive all orders of modified Lagrangians while keeping the dynamics on the original energy space. The authors derive a next order modified equation for the implicit midpoint rule applied to the semilinear wave equation, prove local well posedness, and extend the method to all orders for linear and nonlinear cases, with favorable frequency content and energy behavior observed numerically. This variational framework potentially enables robust long-time accuracy and opens avenues toward exponential backward error estimates for PDE integrators.

Abstract

The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified equations with increasingly high frequencies which increase the regularity requirements on the analysis. In this paper, we consider the next order modified equations for the implicit midpoint rule applied to the semilinear wave equation to give a proof-of-concept of a new construction which works directly with the variational principle. We show that a carefully chosen change of coordinates yields a modified system which inherits its analytical properties from the original wave equation. Our method systematically exploits additional degrees of freedom by modifying the symplectic structure and the Hamiltonian together.

A new construction of modified equations for variational integrators

TL;DR

The paper addresses the problem that traditional backward error analysis for variational integrators applied to PDEs induces unphysically fast modes via a Hamiltonian modification. It proposes a degenerate variational asymptotics approach that uses a near identity change in the variational principle to derive all orders of modified Lagrangians while keeping the dynamics on the original energy space. The authors derive a next order modified equation for the implicit midpoint rule applied to the semilinear wave equation, prove local well posedness, and extend the method to all orders for linear and nonlinear cases, with favorable frequency content and energy behavior observed numerically. This variational framework potentially enables robust long-time accuracy and opens avenues toward exponential backward error estimates for PDE integrators.

Abstract

The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified equations with increasingly high frequencies which increase the regularity requirements on the analysis. In this paper, we consider the next order modified equations for the implicit midpoint rule applied to the semilinear wave equation to give a proof-of-concept of a new construction which works directly with the variational principle. We show that a carefully chosen change of coordinates yields a modified system which inherits its analytical properties from the original wave equation. Our method systematically exploits additional degrees of freedom by modifying the symplectic structure and the Hamiltonian together.
Paper Structure (12 sections, 4 theorems, 126 equations, 2 figures)

This paper contains 12 sections, 4 theorems, 126 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The semilinear wave equation
  3. Variational integrators
  4. Backward error analysis on the Hamiltonian side
  5. Backward error analysis on the Lagrangian side
  6. Method of degenerate variational asymptotics
  7. Initialization
  8. Numerical Experiments
  9. Well-posedness of the new modified system
  10. All-order modified equations: the linear case
  11. High-order modified equations: the nonlinear case
  12. Conclusions

Key Result

Lemma 1

For every $C>0$ there exists $h_*>0$ such that for every $h \in (0,h_*]$, $u \in H^1(S^1)$ with $\lVert u \rVert_{H^1}^{} \leq C$, and $z \in L^2(S^1)$, the equation $K(u)v=z$ has a unique solution $v \in H^2(S^1)$ and there exists a constant $c=c(C)>0$ such that and Moreover, for fixed $z \in L^2(S^1)$ and under the above bounds on $u$ and $h$, the mapping $u \mapsto K(u)^{-1}z$ is uniformly Li

Figures (2)

  • Figure 1: Scaling of the error at final time $T=0.5$.
  • Figure 2: Approximate energy preservation of the implicit midpoint rule and modified equations. The Hamiltonian modified equation has a numerical blowup due to a CFL violation while the variational modified equations is stably solved under identical conditions using a fourth order explicit Runge--Kutta scheme with a fixed step size $\Delta t=0.025<h=0.037$.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Corollary 4
  • proof