Abstract
This paper is concerned with the invariant discretization of differential equations admitting infinitedimensional symmetry groups. By way of example, we first show that there are differential equations with infinitedimensional 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 pseudogroup action. Computer simulations indicate that the numerical schemes constructed from the joint invariants of discretized pseudogroup can produce better numerical results than standard schemes.
Invariant Discretization of Partial Differential Equations Admitting InfiniteDimensional Symmetry Groups
Raphaël Rebelo and Francis Valiquette
Centre de Recherche Mathématiques, Université de Montréal, C.P. 6128, Succ. CentreVille, 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: Infinitedimensional Lie pseudogroups, 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 finitedimensional Lie group actions and the case of infinitedimensional Lie pseudogroups as yet to be satisfactorily treated. Many partial differential equations in hydrodynamics or meteorology admit infinitedimensional 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 infinitedimensional 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 finitedimensional 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 infinitedimensional Lie pseudogroup actions.
Example 1.1.
Let be a local diffeormorphism of . Throughout the paper we will use the infinitedimensional pseudogroup
(1.1) 
acting on , to illustrate the theory and constructions. The pseudogroup (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 pseudogroup 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 pseudogroup^{1}^{1}1Equation (1.3) admits a larger symmetry group given by , , , with , . This pseudogroup 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 multiindex is introduced to label sample points:
(1.4) 
Following the general philosophy, [7, 12, 16, 20], the pseudogroup (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 pseudogroup parameters
(1.7) 
are independent. Hence, as shown in [10], the pseudogroup (1.5) shares the same invariants as its Lie completion
(1.8) 
where for each different subscript , the functions are functionally independent local diffeomorphisms^{2}^{2}2It 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 pseudogroup (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 pseudogroup 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 pseudogroup action. Thus, in general, the mesh in the independent variables will be rectangular with variable step sizes in , see Figure 1.
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 subpseudogroups. For the diffeomorphism pseudogroup , since the largest nontrivial subpseudogroup is the special linear group , [19], this approach drastically changes the nature of the action as it transitions from an infinitedimensional transformation group to a threedimensional group of transformations. In this paper we are interested in preserving the infinitedimensional 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 infinitedimensional pseudogroup actions. In other words, derivatives are to be replaced by finite difference approximations. For the pseudogroup (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 pseudogroup 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 infinitedimensional Lie pseudogroup is recalled and the equivariant moving frame construction is summarized. In Section 3, pseudogroup actions are discretized and the equivariant moving frame construction is adapted to those actions. Along with (1.1), the pseudogroup
(1.14) 
with and , will stand as a second example to illustrate our constructions. We choose to work with the pseudogroups (1.1) and (1.14) to keep our examples relatively simple. Furthermore, these pseudogroups have been extensively used in [21, 22, 23, 24] to illustrate the (continuous) method of moving frames. With these welldocumented examples, it allowed us to verify that our discrete constructions and computations did converge to their continuous counterparts.
2 Lie Pseudogroups and moving frames
For completeness, we begin by recalling the definition of a pseudogroup, [3, 13, 14, 21, 22, 28]. Let be an dimensional manifold. By a local diffeomorphism of we mean a onetoone map defined on open subsets , , with inverse .
Definition 2.1.
A collection of local diffeomorphisms of is a pseudogroup 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 pseudogroup is given by the collection of local diffeomorphisms of a manifold . All other pseudogroups defined on are subpseudogroups 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 pseudogroup is called a Lie pseudogroup 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 pseudogroup. A Lie pseudogroup is said to be of finite type if the solution space of (2.1) only involves a finite number of arbitrary constants. Lie pseudogroups of finite type are thus isomorphic to local Lie group actions. On the other hand, a Lie pseudogroup 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 pseudogroup (1.1) is a Lie pseudogroup. 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 pseudogroup 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 pseudogroup 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 infinitedimensional Lie pseudogroups 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 pseudogroup 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 pseudogroup onto , [22]. The first component is the prolonged action of the pseudogroup 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 pseudogroup is said to act freely at if . The pseudogroup 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 pseudogroup 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 pseudogroup acts freely at , then it acts freely at any , , with .
Remark 2.10.
An order moving frame is constructed through a normalization procedure based on the choice of a crosssection to the pseudogroup 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 crosssection, the moving frame is obtained by solving the normalization equations
for the pseudogroup parameters . As one increases the order from to , a new crosssection must be selected. These crosssections 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 pseudogroup (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 pseudogroup 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 nondegeneracy condition (2.8) requires the invariant constraint to be satisfied.
Up to order 2, the prolonged action is
(2.16) 
A crosssection to the prolonged action (2.16) and its prolongation is given by
(2.17) 
Solving the normalization equations
for the pseudogroup parameters we obtain the right moving frame
In general,
(2.18) 
Substituting the pseudogroup 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 crosssection 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 crosssections (2.17) and (2.21) (and the corresponding moving frames (2.18) and (2.22)) are said to be equivalent.
3 Discrete pseudogroups and moving frames
Let denote the fold Cartesian product of a manifold . Discrete points in are labelled using the multiindex notation
(3.1) 
The multiindex notation (3.1) can be related to the continuous theory of Section 2 in the following way. The multiindex 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 multiindex is the point
(3.2) 
where with
and is a nonnegative multiindex of order .
In dimension 2, when , Figure 2 shows the multiindices contained in a forward discrete jet or order . In general, the multiindices included in are those contained in the interior and boundary of the right isosceles triangle with vertices at , and .
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 nonnegative multiindex ,
(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 nondegeneracy condition (2.8) holds. Namely,
(3.4) 
Then, discrete approximations of the derivatives can be obtained as follows:

compute the expressions (2.10),

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 pseudogroup 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