Discrete conservation laws for finite element discretisations of multisymplectic PDEs

https://doi.org/10.1016/j.jcp.2021.110520Get rights and content

Highlights

  • Arbitrary order space-time finite element discretisations for multisymplectic PDEs.

  • Local momentum and/or energy conservation laws.

  • Existence uniqueness proofs of the schemes for class of nonlinear wave equations.

  • Extensive numerical experiments with wave and nonlinear Schrodinger equations.

Abstract

In this work we propose a new, arbitrary order space-time finite element discretisation for Hamiltonian PDEs in multisymplectic formulation. We show that the new method which is obtained by using both continuous and discontinuous discretisations in space, admits a local and global conservation law of energy. We also show existence and uniqueness of solutions of the discrete equations. Further, we illustrate the error behaviour and the conservation properties of the proposed discretisation in extensive numerical experiments on the linear and nonlinear wave equation and the nonlinear Schrödinger equation.

Introduction

Finite element discretisations of space-time variational problems have seen a revival of interest in the recent literature [1], [2], [3], [4], with their origin going back to the work of [5], [6]. The focus of the present work is the structure-preserving discretisation of variational problems using finite element methods. The point of departure is the variational space-time formulation of PDE problems arising as the Euler-Lagrange equations of a space-time action functional. Formally via the Legendre transform one obtains the Hamiltonian formulation of these partial differential equations [7]. However a space-time analogue of the Legendre transform gives rise to the so called multisymplectic formulation of these PDEs as originally proposed by Bridges [8], and intrinsically generalised in [9], [10].

There are two main proposed discretisation approaches to the multisymplectic formulation of Hamiltonian PDEs. The first is inspired by a technique proposed by Veselov to discretise Hamiltonian ODEs and consists of discretising the Lagrangian density to yield a discrete analogue of the variational principle, and then extremising to obtain discrete Euler Lagrange equations, [10]. The second is obtained by first extremising the variational principle, writing the multisymplectic PDEs in the strong multisymplectic formulation, and then discretising these equations with structure preserving approximations, [11]. The first approach leads automatically to the conservation of a discrete multisymplectic conservation law and the corresponding numerical methods typically show as a side effect good local conservation of a modified energy and momentum in numerical experiments. The second approach is more flexible, while discretisations with similar properties can be obtained using appropriate multisymplectic schemes [11], this formulation can be also easily adopted to obtain methods satisfying local conservation laws of energy or momentum [12]. In the sequel we shall follow the latter approach. We note for Hamiltonian ODEs that symplectic schemes are often preferred over conservative schemes as they allow for backward error analysis, and the resulting schemes will preserve a modified conserved quantity [13]. A backward error analysis for multisymplectic schemes in the PDE setting is not yet fully developed, [14], [15]. On the other hand, being able to bound the approximation by the energy often proves to be an invaluable property for convergence studies of both geometric numerical integrators [16], [17], [18] and finite element schemes via energy arguments [19].

Most of the proposed discretisations of multisymplectic PDEs are proposed in the framework of finite differences with restrictions to rectangular domains, are not easy to use on irregular domains, and are often of low order with some exceptions [20], [21], [22], [23], [24], [25]. Due to the nature of their formulation, finite element methods may overcome these issues. It has also been remarked [26] that some of the proposed multisymplectic discretisations might not be well defined locally and or globally or not have solutions/unique solutions, [27], [24].

Standard finite element methods, when arising from boundary value problems, do not historically lead to discretisations which may be interpreted locally. Indeed, we may on a case-by-case basis move out of the finite element framework through the assembling of the associated algebraic system which may then be interpreted locally similarly to a spatial finite difference discretisation, see for example [28]. The recent work [22] utilises the hybridisable discontinuous Galerkin framework [29], in which the solution inside an element and on the element boundaries are considered independently and coupled to ensure global communication of the solution. Fortuitously, standard finite element methods also fit within this framework [30], and as such global solution properties may be naturally restricted to the patch surrounding a single element.

Another novel approach which allows for the incorporation of spatially local conservation laws are local discontinuous Galerkin (ldG) methods [31], [32], [33], in which a given PDE is reduced to a first order system through the incorporation of auxiliary variables. The resulting discretisation is comprised of discontinuous approximations of first derivatives utilising either upwind or downwind flux contributions, which allow for the development of schemes which preserve conservation laws for PDEs with even order spatial derivatives [34], [35]. A crucial implementational benefit of these methods is that due to the discontinuous nature of the approximation the discrete system may be rewritten as a single equation, and as such the resulting algorithms are of a competitive complexity. In this work, we shall consider similar spatial discretisations, however, due to the general multisymplectic framework considered we choose an average spatial flux, as opposed to the upwind/downwind considered in the ldG setting. Our choice here is similar to that made in [36]. This average spatial flux choice allows us to utilise a discrete integration by parts which is fundamental to the preservation of conservation laws in a general setting. If we restrict ourselves to consider multisymplectic PDEs with even order spatial derivatives, appropriate modifications of the spatial derivative may be designed to preserve local conservation laws for this class of multisymplectic PDEs with upwind/downwind fluxes, which often yields an optimal rate of convergence in the resulting numerical scheme.

Temporally, finite element methods are a competitive, well studied, class of methods [37], [38], [39] dating back to 1969 [40]. Further to this, the conforming approximation is well known to preserve energy for Hamiltonian problems [41], [42], [43], [44]. For space-time finite element approximations the temporal are typically nonconforming, with the standard choice of a discontinuous upwind flux introducing artificial diffusion [45] which facilitates stability, even when the space-time algorithm is adapted. Here we shall focus our attention on the temporally conforming approximation, with a view to extend this work to incorporate a new form of conservative space-time adaptive algorithm with inherent stability building on ideas developed in [44].

The outline of the paper is as follows, in Section 2 we briefly review the main features of multisymplectic PDEs including examples; in Section 3 we present the continuous space-time finite element discretisation and prove existence and uniqueness of solutions of the discrete equations at the end of the section; in Section 4 we extend the method to the spatially discontinuous case; in Section 5 we conduct numerical experiments; and in Section 6 we conclude.

Section snippets

Multisymplectic PDEs

Let u=u(t,x) where t[0,T] and xS1=[0,1) periodic and consider the space-time variational problem0=δ[0,T]×S1L(u,ut,ux)dxdt, where L(u,ut,ux) is some given Lagrangian density function. Multisymplectic PDEs may arise naturally from such problems through the following methodology. Introducing the auxiliary variablesv:=Lut,w:=Lux, and assuming that ut=ut(v), ux=ux(w) are invertible functions, we can define the Hamiltonian densityS(u,v,w):=vut+wuxL(u,ut(v),ux(w)). We may then express the

Continuous space-time finite element approximations

We shall now focus on the design of a conforming space-time finite element method which is constructed through the tensor product of spatial and temporal finite element discretisations. This decoupling of the spatial and temporal discretisations allows us to combine a conservative spatial method with a temporally conservative method, which for the study of Hamiltonian ODEs and Hamiltonian PDEs are fundamentally different, see [44].

Before defining our finite element space we must give

Spatially discontinuous finite element approximation

We shall now focus our attention on a spatially discontinuous approximation. Such an approximation allows us to concisely localise our conservation laws spatially, in addition to the benefits discussed in Remark 3.5.

As we consider discontinuous function spaces we require the following additional definitions to concisely describe the method about points with multiple values.

Definition 4.1 Jumps and averages

Due to the discontinuous nature of the finite element space finite element functions are permitted to be multi-valued at

Numerical experiments

In this section we illustrate the numerical performance of the numerical methods designed in Section 3 and Section 4 for select examples through summarising extensive numerical experiments. The brunt of the computational work here has been conducted by Firedrake [52], [53] and utilises PETsc solvers [54]. Throughout we shall use direct linear solvers and set our nonlinear solver tolerance to be 1012. We employ a Gauss quadrature which is either of high enough order to be able to integrate

Conclusion

We introduced space-time finite element approximations which preserve geometric structure, namely the energy conservation law, of an arbitrary parabolic multisymplectic PDE. Furthermore we proved existence and uniqueness of this approximation for the wave equation, followed by an extensive numerical study showing good long term behaviour of the solution and convergence of the simulations. Experimentally, the approximations converged optimally in time, with sub-optimal spatial behaviour for both

CRediT authorship contribution statement

Elena Celledoni: Conceptualization, Methodology. James Jackaman: Conceptualization, Methodology, Software.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work has been partially supported through the IMA small grant SG2018/180, the Canadian Research Chairs and NSERC Discovery grant programs (J.J.), H2020-MSCA-RISE-691070 CHiPS and the Simons Foundation (E.C.). Both authors would additionally like to acknowledge the support of the Isaac Newton Institute for Mathematical Sciences, Cambridge through the EPSRC grant EP/K032208/1.

The authors also gratefully acknowledge the suggestions of Dr Colin Cotter which proved invaluable for the space-time

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