1 Introduction
Abstract

This paper is concerned with the invariant discretization of differential equations admitting infinite-dimensional symmetry groups. By way of example, we first show that there are differential equations with infinite-dimensional symmetry groups that do not admit enough joint invariants preventing the construction of invariant finite difference approximations. To solve this shortage of joint invariants we propose to discretize the pseudo-group action. Computer simulations indicate that the numerical schemes constructed from the joint invariants of discretized pseudo-group can produce better numerical results than standard schemes.


Invariant Discretization of Partial Differential Equations Admitting Infinite-Dimensional Symmetry Groups

Raphaël Rebelo and Francis Valiquette

Centre de Recherche Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, H3C 3J7, Canada
Email: raph.rebelo@gmail.com Department of Mathematics, SUNY at New Paltz, New Paltz, NY 12561, USA
Email: valiquef@newpaltz.edu


Keywords: Infinite-dimensional Lie pseudo-groups, joint invariants, moving frames.

Mathematics subject classification (MSC2010): 58J70, 65N06


1 Introduction

For the last 20 years, a considerable amount of work has been invested into the problem of invariantly discretizing differential equations with symmetries. This effort is part of a larger program aiming to extend Lie’s theory of transformation groups to finite difference equations, [16]. With the emergence of physical models based on discrete spacetime, and in light of the importance of symmetry in our understanding of modern physics, the problem of invariantly discretizing differential equations is still of present interest. From a theoretical standpoint, working with invariant numerical schemes allows one to use standard Lie group techniques to find explicit solutions, [27], or compute conservation laws, [8]. From a more practical point of view, the motivation stems from the fact that invariant schemes have been shown to outperform standard numerical methods in a number of examples, [2, 6, 12, 26].

In general, to build an invariant numerical scheme one has to construct joint invariants (also known as finite difference invariants). These joint invariants are usually found using one of two methods. One can either use Lie’s method of infinitesimal generators which requires solving a system of linear partial differential equations, [7, 16], or the method of equivariant moving frames which requires solving a system of (nonlinear) algebraic equations, [12, 20]. Both approaches produce joint invariants which, in the coalescent limit, converge to differential invariants of the prolonged action. Thus far, the theory and applications found in the literature primarily deal with finite-dimensional Lie group actions and the case of infinite-dimensional Lie pseudo-groups as yet to be satisfactorily treated. Many partial differential equations in hydrodynamics or meteorology admit infinite-dimensional symmetry groups. The Navier–Stokes equation, [18], the Kadomtsev–Petviashvili equation, [5], and the Davey–Stewartson equations, [4], are classical examples of such equations. Linear or linearizable partial differential equations also form a large class of equations admitting infinite-dimensional symmetry groups.

To construct invariant numerical schemes of differential equations admitting symmetries, one of the main steps consists of finding joint invariants that approximate the differential invariants of the symmetry group. For finite-dimensional Lie group actions, this can always be done by considering the product action on sufficiently many points. Unfortunately, as the next example shows, the same is not true for infinite-dimensional Lie pseudo-group actions.

Example 1.1.

Let be a local diffeormorphism of . Throughout the paper we will use the infinite-dimensional pseudo-group

(1.1)

acting on , to illustrate the theory and constructions. The pseudo-group (1.1) was introduced by Lie, [15, p.373], in his study of second order partial differential equations integrable by the method of Darboux. It also appears in Vessiot’s work on group splitting and automorphic systems, [29], in Kumpera’s investigation of Lie’s theory of differential invariants based on Spencer’s cohomology, [13], and recently in [21, 22, 25] to illustrate a new theoretical foundation of moving frames.

The differential invariants of the pseudo-group action (1.1) can be found in [22]. One of these invariants is

(1.2)

With (1.2) it is possible to form the partial differential equation

(1.3)

which was used in [25] to illustrate the method of symmetry reduction of exterior differential systems.

By construction, Equation (1.3) is invariant under the pseudo-group111Equation (1.3) admits a larger symmetry group given by , , , with , . This pseudo-group is considered in Example 3.20. (1.1). To obtain an invariant discretization of (1.3), an invariant approximation of the differential invariant (1.2) must be found. To discretize the invariant (1.2), the multi-index is introduced to label sample points:

(1.4)

Following the general philosophy, [7, 12, 16, 20], the pseudo-group (1.1) induces the product action

(1.5)

on the discrete points (1.4). On an arbitrary finite set of points, we claim that the only joint invariants are

(1.6)

To see this, let be a finite subset of , and assume for . Since the components are generically distinct and is an arbitrary local diffeomorphism, the pseudo-group parameters

(1.7)

are independent. Hence, as shown in [10], the pseudo-group (1.5) shares the same invariants as its Lie completion

(1.8)

where for each different subscript , the functions are functionally independent local diffeomorphisms222It is customary to use the notation to denote the value of the function at the point , and this is the convention used in Sections 3, 4, and 5. In equation (1.8), the subscript attached to the diffeomorphism has a different meaning. Here, the subscript is used to denote different diffeomorphisms. Thus, the pseudo-group (1.5) is contained in the Lie completion (1.8). This particular use of the subscript only occurs in (1.8).. For the Lie completion (1.8), it is clear that (1.6) are the only admissible invariants. Hence, generically, we conclude that it is not possible to approximate the differential invariant (1.2) by joint invariants.

To construct additional joint invariants, invariant constraints on the independent variables need to be imposed to reduce the number of pseudo-group parameters (1.7). To reduce this number as much as possible, we assume that

(1.9)

Equation (1.9) is seen to be invariant under the product action (1.5) since

when (1.9) holds. Equation (1.9) implies that is independent of the index :

To cover (a region of) the -plane,

must hold. Since the variables are invariant under the product action (1.5) we can, for simplicity, set

(1.10)

where and are constants. To respect the product action (1.5) we cannot require the step size to be constant as this is not an invariant assumption of the pseudo-group action. Thus, in general, the mesh in the independent variables will be rectangular with variable step sizes in , see Figure 1.

Figure 1: Rectangular mesh.

Repeating the argument above, when (1.9) and (1.10) hold, the joint invariants of the product action (1.5) are

(1.11)

Introducing the dilation group

(1.12)

we see that the differential invariant (1.2) cannot be approximated by the joint invariants (1.11). Indeed, since the invariants are homogeneous of degree 0, any combination of the invariants (1.11) will converge to a differential invariant of homogeneous degree 0. On the other hand, the differential invariant (1.2) is homogeneous of degree under (1.12).

As it stands, it is not possible to construct joint invariants that approximate the differential invariant (1.2). To remedy the problem, one possibility is to reduce the size of the symmetry group by considering sub-pseudo-groups. For the diffeomorphism pseudo-group , since the largest non-trivial sub-pseudo-group is the special linear group , [19], this approach drastically changes the nature of the action as it transitions from an infinite-dimensional transformation group to a three-dimensional group of transformations. In this paper we are interested in preserving the infinite-dimensional nature of transformation groups and propound another suggestion. Taking the point of view that the notion of derivative is not defined in the discrete setting, we propose to discretize infinite-dimensional pseudo-group actions. In other words, derivatives are to be replaced by finite difference approximations. For the pseudo-group (1.1), instead of considering the product action (1.5), we suggest to work with the first order approximation

(1.13)

In Section 3, joint invariants of the pseudo-group action (1.13) are constructed and an invariant numerical scheme approximating (1.3) is obtained in Section 4.

To develop our ideas we opted to use the theory of equivariant moving frames, [20, 22], but our constructions can also be recast within Lie’s infinitesimal framework. In Section 2, the concept of an infinite-dimensional Lie pseudo-group is recalled and the equivariant moving frame construction is summarized. In Section 3, pseudo-group actions are discretized and the equivariant moving frame construction is adapted to those actions. Along with (1.1), the pseudo-group

(1.14)

with and , will stand as a second example to illustrate our constructions. We choose to work with the pseudo-groups (1.1) and (1.14) to keep our examples relatively simple. Furthermore, these pseudo-groups have been extensively used in [21, 22, 23, 24] to illustrate the (continuous) method of moving frames. With these well-documented examples, it allowed us to verify that our discrete constructions and computations did converge to their continuous counterparts.

Finally, in Section 5 an invariant numerical approximation of (1.3) is compared to a standard discretization of the equation. Our numerical tests show that the invariant scheme is more precise and stable than the standard scheme.

2 Lie Pseudo-groups and moving frames

For completeness, we begin by recalling the definition of a pseudo-group, [3, 13, 14, 21, 22, 28]. Let be an -dimensional manifold. By a local diffeomorphism of we mean a one-to-one map defined on open subsets , , with inverse .

Definition 2.1.

A collection of local diffeomorphisms of is a pseudo-group if

  • is closed under restriction: if is an open set and is in , then so is the restriction for all open .

  • Elements of can be pieced together: if are open subsets, , and is a local diffeomorphism with for all , then .

  • contains the identity diffeomorphism:   for all .

  • is closed under composition: if and are two diffeomorphisms belonging to , and , then .

  • is closed under inversion: if is in then so is .

Example 2.2.

One of the simplest pseudo-group is given by the collection of local diffeomorphisms of a manifold . All other pseudo-groups defined on are sub-pseudo-groups of .

For , let denote the bundle formed by the order jets of local diffeomorphisms of . Local coordinates on are given by , where are the source coordinates of the local diffeomorphism, , its target coordinates, and collects the derivatives of the target coordinates with respect to the source coordinates of order . For , the standard projection is denoted .

Definition 2.3.

A pseudo-group is called a Lie pseudo-group of order if, for all finite

  • forms a smooth embedded subbundle,

  • the projection is a fibration,

  • every local diffeomorphism satisfying belongs to ,

  • is obtained by prolongation.

In local coordinates, the subbundle is characterized by a system of order (formally integrable) partial differential equations

(2.1)

called the order determining system of the pseudo-group. A Lie pseudo-group is said to be of finite type if the solution space of (2.1) only involves a finite number of arbitrary constants. Lie pseudo-groups of finite type are thus isomorphic to local Lie group actions. On the other hand, a Lie pseudo-group is of infinite type if it involves arbitrary functions.

Remark 2.4.

Linearizing (2.1) at the identity jet yields the infinitesimal determining equations

(2.2)

for an infinitesimal generator

(2.3)

The vector field (2.3) is in the Lie algebra of infinitesimal generators of if its components are solution of (2.2). Given a differential equation with symmetry group , the infinitesimal determining system (2.2) is equivalent to the equations obtained by Lie’s standard algorithm for determining the symmetry algebra of the differential equation , [18].

Example 2.5.

The pseudo-group (1.1) is a Lie pseudo-group. The first order determining equations are

(2.4)

If

denotes a (local) vector fields in , the linearization of (2.4) at the identity jet yields the first order infinitesimal determining equations

The general solution to this system of equations is

where is an arbitrary smooth function.

Given a Lie pseudo-group acting on , we are now interested in the induced action on -dimensional submanifolds with . It is customary to introduce adapted coordinates

(2.5)

on so that, locally, a submanifold transverse to the vertical fibre is given as the graph of a function . For each integer , let denote the order submanifold jet bundle defined as the set of equivalence classes under the equivalence relation of order contact, [19]. For , let denote the canonical projection. In the adapted system of coordinates , coordinates on are given by

(2.6)

where denotes the collection of derivatives of order .

Alternatively, when no distinction between dependent and independent variables is made, a submanifold can be locally parameterized by variables so that

In the numerical analysis community, the variables are called computational variables, [9]. We let denote the order jet space of submanifolds parametrized by computational variables. Local coordinates on are given by

(2.7)

with , , and . The transition between the jet coordinates (2.6) and (2.7) is given by the chain rule. Provided

(2.8)

successive application of the implicit total differential operators

(2.9)

to the dependent variables will give the coordinate expressions for the derivatives of in terms of the derivatives of and :

(2.10)

Given a Lie pseudo-group acting on , the action is prolonged to the computational variables by requiring that they remain unchanged:

By abuse of notation we still use to denote the extended action on .

The complete theory of moving frames for infinite-dimensional Lie pseudo-groups can be found in [22]. For reasons that will become more apparent in the next section we recall the main constructions over the jet bundle rather than . Using (2.10) one can translate the constructions from to . Let

(2.11)

denote the order lifted bundle. Local coordinates on are given by , where the base coordinates are the submanifold jet coordinates and the fibre coordinates are the pseudo-group parameters where . A local diffeomorphism acts on by right multiplication:

(2.12)

where defined. The second component of (2.12) corresponds to the usual right multiplication of the pseudo-group onto , [22]. The first component is the prolonged action of the pseudo-group onto the jet space . Coordinate expressions for the prolonged action are obtained by differentiating the target coordinates with respect to the computational variables :

(2.13)

where . The expressions (2.13) are invariant under the lifted action (2.12) and these functions are called lifted invariants.

Definition 2.6.

A (right) moving frame of order is a -equivariant section of the lifted bundle .

In local coordinates, the notation

is used to denote an order right moving frame. Right equivariance means that for

where defined.

Definition 2.7.

Let

denote the isotropy subgroup of . The pseudo-group is said to act freely at if . The pseudo-group is said to act freely at order if it acts freely on an open subset , called the set of regular -jets.

Theorem 2.8.

Suppose acts freely on , with its orbits forming a regular foliation. Then an order moving frame exists in a neighbourhood of .

Once a pseudo-group acts freely, a result known as the persistence of freeness, [23, 24], guarantees that the action remains free under prolongation.

Theorem 2.9.

If a Lie pseudo-group acts freely at , then it acts freely at any , , with .

Remark 2.10.

Theorems 2.8 and 2.9 also hold when the pseudo-group action is locally free, meaning that the isotropy group is a discrete subgroup of .

An order moving frame is constructed through a normalization procedure based on the choice of a cross-section to the pseudo-group orbits. The associated (locally defined) right moving frame section is uniquely characterized by the condition that . In coordinates, assuming that

(2.14)

is a coordinate cross-section, the moving frame is obtained by solving the normalization equations

for the pseudo-group parameters . As one increases the order from to , a new cross-section must be selected. These cross-sections are required to be compatible meaning that for all . This in turn, implies the compatibility of the moving frames: , where is the standard projection.

Proposition 2.11.

Let be an order right moving frame. The normalized invariants

form a complete set of differential invariants of order .

Example 2.12.

In this example we construct a moving frame for the pseudo-group (1.1). The computations for graphs of functions appear in [22]. In preparation for the next section we revisit the calculations using the computational variables so that , and . To simplify the computations let

(2.15a)
where and are constants, and assume that
(2.15b)

In other words, is a function of the computational variable . We note that the constraints (2.15) are invariant under the pseudo-group action (1.1). For the variable, this is straightforward as it is an invariant. The invariance of (2.15b) follows from the chain rule:

The non-degeneracy condition (2.8) requires the invariant constraint to be satisfied.

Up to order 2, the prolonged action is

(2.16)

A cross-section to the prolonged action (2.16) and its prolongation is given by

(2.17)

Solving the normalization equations

for the pseudo-group parameters we obtain the right moving frame

In general,

(2.18)

Substituting the pseudo-group normalizations (2.18) into the prolonged action (2.16) yields the normalized differential invariants

(2.19)
Remark 2.13.

To transition between the expressions obtained in Example 2.12 and those appearing in [22], it suffices to use the chain rule. When (2.15) holds,

so that

(2.20)

The prolonged action on , , , can then be obtained by substituting (2.16) into (2.20). For example,

In the jet variables a cross-section is given by, [22],

(2.21)

and the corresponding moving frame is

(2.22)

Expressing in terms of the derivatives , using (2.20), one sees that (2.21) and (2.22) are equivalent to (2.17) and (2.18) in the computational variable framework. In the following, the cross-sections (2.17) and (2.21) (and the corresponding moving frames (2.18) and (2.22)) are said to be equivalent.

Not all cross-sections are equivalent. For example, instead of using the cross-section (2.17), it is also possible to choose the (non-minimal) cross-section

(2.23)

Since (2.21) is not related to (2.23) by the substitutions (2.20), the cross-section (2.23) is said to be inequivalent to (2.21).

Definition 2.14.

Let be a Lie pseudo-group acting on and . A cross-section is said to be equivalent with the cross-section if the defining equation (2.14) of are obtained from those of by expressing the submanifold jet in terms of using the relations (2.10).

3 Discrete pseudo-groups and moving frames

Let denote the -fold Cartesian product of a manifold . Discrete points in are labelled using the multi-index notation

(3.1)

The multi-index notation (3.1) can be related to the continuous theory of Section 2 in the following way. The multi-index can be thought as sampling the computational variables on a unit hypercube grid. Thus, the notation can be understood as sampling a submanifold parameterized by at the integer points .

To mimic the continuous theory of moving frames in the finite difference setting, a discrete counterpart to the submanifold jet space is introduced.

Definition 3.1.

Let be a manifold with local coordinate system . The -fold joint product of is a subset of the -fold Cartesian product given by

Definition 3.2.

The order forward discrete jet at the multi-index is the point

(3.2)

where with

and is a non-negative multi-index of order .

In dimension 2, when , Figure 2 shows the multi-indices contained in a forward discrete jet or order . In general, the multi-indices included in are those contained in the interior and boundary of the right isosceles triangle with vertices at , and .

(a)
(b)
(c)
Figure 2: Multi-indices occurring in for .
Definition 3.3.

The order forward joint jet space is the collection of forward discrete jets (3.2):

For , will denote the projection obtained by truncating , , to

Let us explain how can be understood as a discrete representation of the submanifold jet space . For this, let be the element of the standard orthonormal basis of , and let

denote the forward shift operator in the component. Then, on a unit hypercube grid in the computational variables, the derivative operators can be approximated by the forward difference

where is the identity map. Then, for a non-negative multi-index ,

(3.3)

is a forward difference approximation of the derivative at the point . Making the change of variables , we have that

is a finite difference approximation of the submanifold jet at the point on a unit hypercube grid. In this sense, can be thought as a discrete counterpart to the submanifold jet in the computational variable formalism, [9].

Remark 3.4.

In (3.3) and elsewhere, the usual derivative notation is supplemented by a superscript to denote (forward) discrete derivatives. The superscript indicates where the derivative is evaluated.

Remark 3.5.

It is also possible to introduce a backward discrete jet space by introducing the backward differences

For numerical purposes, it might be preferable to consider symmetric discrete jets, but to simplify the exposition we restrict ourself to forward differences. All constructions can be adapted to these alternative settings.

Now, assume that the discrete counterpart of the non-degeneracy condition (2.8) holds. Namely,

(3.4)

Then, discrete approximations of the derivatives can be obtained as follows:

  1. compute the expressions (2.10),

  2. replace the derivatives by the difference operators .

Since the independent variables do not have to form a rectangular grid, the finite difference approximations will hold on any admissible mesh. Having these expressions will be important as below a Lie pseudo-group will act on and the expressions for need to hold on general meshes, [7, 16, 26].

Using the approximations , a finite difference approximation of the jet space is given by

Example 3.6.

To illustrate the above discussion, we consider the case of two independent variables