1 Introduction
In recent years, much attention has been given to the design and analysis of numerical methods for differential equations that can capture geometric properties of the exact flow. The increased interest in this subject can mainly be attributed to the superior qualitative behaviour over long time integration of such structurepreserving methods, see [1, 2, 3]. A popular class of structurepreserving methods are energypreserving methods. In particular, the energy preservation property has been found to be crucial in the proof of stability for several of these numerical methods, see e.g [4].
Energypreserving methods are well studied for finitedimensional Hamiltonian systems [5, 6, 7, 8]
. It is also highly conceivable that the ideas behind the finitedimensional setting can be extended to the infinitedimensional Hamiltonian systems or Hamiltonian partial differential equations (PDEs)
[9]. There are two popular ways to construct energypreserving methods for Hamiltonian PDEs. One approach is to semidiscretize the PDE in space so that one obtains a system of Hamiltonian ordinary differential equations (ODEs), and then apply an energypreserving method to this semidiscrete system, see for example
[8]. In this way, it is straightforward to generalise the energypreserving methods for finitedimensional Hamiltonian systems to Hamiltonian PDEs. However, such methods conserve only a global energy that relies on a proper boundary condition, such as a periodic boundary condition. If this is not present, the energypreserving property will be destroyed. The other approach is based on a reformulation of the Hamiltonian PDE into a multisymplectic form, which provides the PDE with three local conservation laws: the multisymplectic conservation law, the energy conservation and the momentum conservation law [10, 11, 12]. Then one may consider methods that preserve the local conservation laws, see for example [13]. These locally defined properties are not dependent on the choice of boundary conditions, giving the methods that preserve local energy an advantage over methods that preserve a global energy, especially since local conservation laws will always lead to global conservation laws whenever periodic boundary conditions are considered. The concept of a multisymplectic structure for PDEs was introduced by Bridges in [10, 11], see also [14] for a framework based on a Lagrangian formulation of the Cartan form. Local energypreserving methods were first studied in [15], and have garnered much interest recently, see for example [13, 16, 17].Most of the local energypreserving methods proposed so far are fully implicit methods, for which a nonlinear system must be solved at each time step. This is normally done by using an iterative solver where a linear system is solved at each iteration, which can lead to computationally expensive procedures, especially since the number of iterations needed in general increases with the size of the system. A fully explicit method on the other hand, may oversimplify the problem and often has inferior stability properties, so that a strong restriction on the grid ratio is needed. A good alternative may therefore be to develop linearly implicit schemes, where the solution at the next time step is found by solving only one linear system.
One example of linearly implicit methods for Hamiltonian ODEs is Kahan’s method, which was designed for solving quadratic ODEs [18] and whose geometric properties have been studied in a series of papers by Celledoni et al. [19, 20, 21]. For Hamiltonian PDEs, Matsuo and Furihata proposed the idea of using multiple points to discretize the variational derivative and thus design linearly implicit energypreserving schemes [22]. Dahlby and Owren generalised this concept and developed a framework for deriving linearly implicit energypreserving multistep methods for Hamiltonian PDEs with polynomial invariants [23]. A comparison of this approach and Kahan’s method applied to PDEs is given in [24]
. Recently, more work has been put into developing linearly implicit energypreserving schemes for Hamiltonian PDEs, e.g. the partitioned averaged vector field (PAVF) method
[25] and schemes based on the invariant energy quadratization (IEQ) approach [26] or the multiple scalar auxiliary variables (MSAV) approach [27]. However, little attention has been given to linearly implicit local energypreserving methods. To the best of the authors’ knowledge, the only existing method is one based on the IEQ approach, specific for the sineGordon equation [28]. In this paper, we use Kahan’s method to construct a linearly implicit method that preserves a discrete approximation to the local energy for multisymplectic PDEs with a cubic energy function.The rest of this paper is organized as follows. First, we give an overview of Kahan’s method and formulate it by using a polarised energy function. A brief introduction to multisymplectic PDEs and their conservation laws are presented in Section 3. In Section 4, new linearly implicit local and global energypreserving schemes are presented. Numerical examples for the Korteweg–de Vries (KdV) and Zakharov–Kuznetsov equations are given in Section 5, before we end the paper with some concluding remarks.
2 Kahan’s method
Consider an ODE system
(2.1) 
where is an valued quadratic form, is a symmetric constant matrix, and is a constant vector. Kahan’s method is then given by
where
is the symmetric bilinear form obtained by polarisation of the quadratic form [19]. Polarisation, which maps a homogeneous polynomial function to a symmetric multilinear form in more variables, was used to generalise Kahan’s method to higher degree polynomial vector fields in [29].
Suppose we restrict the problem (2.1) to be a Hamiltonian system on a Poisson vector space with a constant Poisson structure:
(2.2) 
where
is a constant skewsymmetric matrix, and
is a cubic polynomial function. We first consider the Hamiltonian to be homogeneous. Then, following the result in Proposition of [29], Kahan’s method can be reformulated as(2.3) 
where is a symmetric
tensor satisfying
. Consider the tensor , where , with being the Hessian of ; then we can rewrite Kahan’s method (2.3) as(2.4) 
where denotes the partial derivative with respect to the first argument of .
Consider then the cases where the Hamiltonian in problem (2.2) is nonhomogeneous, i.e. of the general form
(2.5) 
where is the linear part of and thus a symmetric matrix whose elements are homogeneous linear polynomials, is the constant part of and thus a symmetric constant matrix, is a constant vector and is a constant scalar. We follow the technique in [19], adding one variable to to get , extending to by adding a zero initial row and a zero initial column, considering a homogeneous function based on the nonhomogeneous Hamiltonian such that , and finally solving instead of (2.2) the equivalent, homogeneous cubic Hamiltonian problem
with . In this way we can still get the reformulation of Kahan’s method as (2.4) with
(2.6) 
Remark 1.
The valued function in (2.6) has the following properties:

is symmetric^{1}^{1}1Denote the elements in by , where , , are scalars and is the th element of . We have that satisfies since , which is unchanged under any permutation of . This provides the symmetry of . w.r.t. , and ,

,

is symmetric w.r.t. and .
In this paper, we will use the form of Kahan’s method in (2.4) to prove the energy preservation of the proposed methods.
3 Conservation laws for multisymplectic PDEs
Many PDEs, including all onedimensional Hamiltonian PDEs, can be written on the multisymplectic form
(3.1) 
where , are two constant skewsymmetric matrices and is a scalarvalued function. Following the results about multisymplectic structure in [10], it can be shown that multisymplectic PDEs satisfy the following local conservation laws [30]: the multisymplectic conservation law
the local energy conservation law (LECL)
(3.2) 
and the local momentum conservation law (LMCL)
where and satisfy
Decomposition of the matrices is done to make deduction of the conservations laws for energy and momentum more efficient [12, Section 12.3.1].
The multisymplectic form (3.1) can also be generalised to problems in higher dimensional spaces. Consider spatial dimensions; based on the work by Bridges [10], a multisymplectic PDE can then be written as
(3.3) 
where , are constant skewsymmetric matrices and is a smooth functional. Equation (3.3) has the following local energy conservation law:
(3.4) 
where , , and are splittings of satisfying .
Say we have (3.3) defined on the spatial domain with periodic boundary conditions. Integrating over the domain on both sides of the equation (3.4) and using the periodic boundary condition then leads to the global energy conservation law for the multisymplectic PDEs,
(3.5) 
where .
Example 1.
Korteweg–de Vries equation. Consider the KdV equation for modeling shallow water waves,
(3.6) 
where . Introducing the potential , momenta and the variable by the covariant Legendre transform from the Lagrangian, we obtain
(3.7) 
from which we find the multisymplectic formulation (3.1) for the KdV equation with , the Hamiltonian , and
As for the conservation laws, there are many choices of and , for example , or and being the upper triangular parts of and , respectively.
Example 2.
Zakharov–Kuznetsov equation. Zakharov and Kuznetsov introduced in [31] a (2+1)dimensional generalisation of the KdV equation which includes weak transverse variation,
(3.8) 
A multisymplectification of this leads to a system (3.3) for two spatial dimensions,
(3.9) 
Following [9], we have that (3.8) is equivalent to a system of firstorder PDEs,
(3.10) 
which is (3.9) with , the Hamiltonian , and the skewsymmetric matrices
4 New linearly implicit energypreserving schemes
In [16], Gong, Cai and Wang present a scheme that preserves the local energy conservation law (3.4) of a onedimensional multisymplectic PDE, obtained by applying the midpoint rule in space and the averaged vetor field (AVF) method in time. They also present schemes that preserve the global energy, but not (3.4), obtained by considering spatial discretizations that preserve the skewsymmetric property of the difference operator . We build on their work by considering Kahan’s method for the discretization in time, ensuring linearly implicit schemes and also energy preservation.
To introduce our new schemes, we begin with some basic difference operators:
The operators satisfy the following properties [13]:

All the operators commute with each other, e.g.
.

They satisfy the discrete Leibniz rule
Specifically,
One can obtain a series of similar commutative equations and discrete Leibniz rules that are not presented here, but which are also crucial in the proofs of the preservation properties of the schemes to be introduced in the remainder of this section.
4.1 A local energypreserving scheme for multisymplectic PDEs
In this section, we apply the midpoint rule in space and Kahan’s method in time to construct a local energypreserving method for multisymplectic PDEs. Introducing the concept by first considering the onedimensional system (3.1), we apply the midpoint rule in space to get
Then applying Kahan’s method gives us the linearly implicit local energypreserving (LILEP) scheme
(4.1) 
Here we consider of the form , as in (2.5), and accordingly of the form (2.6).
Theorem 1.
Proof.
Taking the inner product with on both sides of (4.1) and using the skewsymmetry of matrix , we have
(4.4) 
Taking the inner product with on both sides of (4.1), we get
(4.5) 
Taking the inner product with on both sides of the scheme (4.1) for the next time step, we get
(4.6) 
Adding equations (4.4), (4.5) and (4.6) and using the skewsymmetry of matrix , we obtain
(4.7) 
On the other hand, using the aforementioned commutative laws and discrete Leibniz rules for the operators, we can deduce
(4.8) 
Using the above relations (4.8), the fact that and the result (4.7), we obtain
∎
Corollary 1.
The polarised global energy may be considered as a function of the solution in time step only, similarly to the modified Hamiltonian defined in Proposition 3 of [19].
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