Acta Polytechnica

In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing a partial differential equation on an arbitrary lattice. An open problem is the meaning of a lattice which does not satisfy the Clairaut– Schwarz–Young theorem. To analyze it we apply the procedure on a simple example, the potential Burgers equation with two different lattices, an orthogonal lattice which is invariant under the symmetries of the equation and satisfies the commutativity of the partial difference operators and an exponential lattice which is not invariant and does not satisfy the Clairaut–Schwarz–Young theorem. A discussion on the numerical results is presented showing the different behavior of both schemes for two different exact solutions and their numerical approximations.


imple example
the potential Burgers equation.

Introduction

The construction of difference equations, written as invariants of the continuous group of symmetries of differential equations, is part of a project to apply symmetry group to the numerical solution of differential equations [2, 3, 6-9, 11-13, 15-17, 20-24].This project has accomplished a significative advance since its first introduction last century.In particular, the construction of invariant schemes has proven to be a very fruitful approach in the construction of numerical schemes for ordinary differential equations [2,3], in cases when the usual approaches present serious problems of convergence and accuracy as when the behavior of the solutions is studied in the neighborhood of a singularity.In this problem a deeper understanding of the mathematics involved in its relation with numerics can provide results important for their applicat ons to problems in Physics and Mathematics.

The explicit construction of schemes invariant under the group (or a subgroup) of symmetries of a differential equation might be a favorable base for the computation of numerical solutions of differential equations providing better results than those obtained via the standard numerical techniques like the Runge-Kutta or other suited methods.The usual procedure in this framework is to compute the symmetry group of the differential equation and then compute the action of the group on an invariant lattice and invariant difference equation.However since the continuous (differential) and discrete (difference) calculus presents substantial differences, a special care has to be taken to assure the consistency of the approach.For example, the Schwarz theorem on the equality of the cross derivatives, which is satisfied in the continuous case under some mild conditions on the functions, is not valid in the discrete case for a general lattice.

Recently it has been shown [14] that the discrete Schwarz theorem, equality of the cross differences, imposes strong restrictions on the lattice.Moreover, the construction of the discrete invariant scheme starting from the discrete group invariants is not at all obvious as it is obtained by a proper combination of the continuous l mits of the various discrete invariants.

A different but equally important approach is the structure-preserving discretization introduced by Marsden et.al. [5,19]: symplectic integrators for Hamiltonian systems, symmetric integrators for reversible systems, methods preserving first integrals and numerical methods on manifolds, including Lie group methods and integrators for constrained mechanical systems, and methods for pro lems with highly oscillatory solutions.

This article is a continuation of our work on the construction of partial difference schemes [14].In the previous work we concentrated on the Schwarz theorem.Here we introduce by a one to one correspondence a new set of discrete coordinates for partial difference equations, whose continuos limit provide the set of partial derivatives, to describe partial difference equations on the lattice and, as an example, we apply our results to the invariant discretization of the Burgers equation.In terms of these coordinate systems we can write immediately the discrete counterpart of any continuous invariant, so we can discretize in a non trivial straightforward way any partial differential equation which is described in terms of invariants f a given Lie algebra of symmetries.

In Section 2 we study stencils for scalar partial differential equations and show the constraints on the group transformations due to the Schwarz theorem, while in Section 3 we describe a set of discrete invariants for the Burgers equation whose natural continuous limit is the viscous Burgers equation.Some concluding

emarks are presente
in Section 4.


Stencils for P∆E's

Let us present the case of O∆E's of order K for one dependent variable u n (x n ) and one independent variable x n .A natural stencil is given by the points {x n+k−1 , u n+k−1 , 1 ≤ k ≤ K + 1} for some fixed n.An alternative set of coordinates o 1], T x u n = u n+1 , k ∈ Z + ( → 0, p (k) n+k → d k u(x) dx k(3)
If we transform the standa +K k=n (ξ k ∂ x k + r field
X = ξ n ∂ xn + φ n ∂ un(5)
t k + K k=1 φ (k) n+k ∂ p (k) n+k(6)
The general formulas for th (k)n+k = D x φ (k−1) n+k−1 − p (k) n+k D x ξ n k = 1, 2, • • • , K.
It is worthwhile to notice that both the higher order discrete derivatives and their corresponding prolongations are written in terms of u n and (ξ n , φ n ) applying onto them a difference operator at difference from the approach considered in [24].In this way this approach is easily extendible to the case of partial difference equations.

In the case of partial difference equations, we consider here only the case of one dependent variable and two independent variables as will be the example we will consider later.Moreover, as we will consider a nonlinear P∆E of second order we limit ourselves to a stencil of six points (n, m), (n+1, m), (n, m+1), (n+2, m), (n, m+2), (n+1, m+1), the minimum number of points necessary to get all partial second derivatives at the first order approximation.As we consider the variables x, y and u(x, y) in all points, we have a total set of 18 data, 12 related to the independent variables and 6 to the dependent one.The extension to more variables and higher order equations requires just more calculations but it is straightforward.Having 12 data for the independent variables we can construct from them 10 differences on a ste h y 1,0 = y 1,1 − y 1,0 ,(8)
where for convenience of notation here and in the following, whenever it may not create misunderstanding, we have indicated just the distance from the values n, m of the indices.From the values of the dependent variables in the 6 points we ca 2 u 0,0 , D x D y u 0,0 ,(9)
where the operators D x and D y , intr 0,0 m = m = f n,m+1 , ∆ m = T m − 1.
We cannot define D y D x u 0,0 in a way independent from the 6 quantities ( 9), since it can be written in term of ( 8), (9).F 0,1 σ y 0,0 − σ x 0,0 σ y 0,0
A simpler way to get a unique correspondence between the two different sets of coordinates of the stencils (that is, using D x D y u 0,0 or D y D x u 0,0 ) is to require the validity of the Clairaut-Schwarz-Young theorem i.e.D x D y u 0,0 = D y D x u 0,0 and this is obtained when the following constraint ,m = h x n,m+1 ≡ h x n ,(12)σ y n,m = σ y n,m+1 ≡ σ y n , h y n,m = h y n+1,m ≡ h y m .
Using the operators D x and D y introduced in [14] we can transform the standard "discrete" prolongation [17] pr Xn,m = Xn,m + Xn+1,m + Xn+2,m + Xn,m+1 + Xn,m+2 + Xn+1,m n,m + φ n,m ∂ un,m(14)
to the new set of variables (8,9).In such a case we get pr Xn,m = Xn,m +
(i,j)=0,1 η (x) n+i,m+j ∂ h x n+i,m+j + χ (x) n+i,m+j ∂ σ x n+i,m+j + η (y) n+i,m+j ∂ h y n+i,m+j +(15) +χ (y) n+i,m+j ∂ σ y n+i,m+j + φ (1,x) n,m ∂ l generator ( 15) onto (12) we get that both functions ξ n,m (x n,m , y n,m , u n,m ) and τ n,m (x n,m , y n,m , u n,m ) must satisfy the equation
(1,x) n,m , φ (1,y) n,m , φ (2,xx) n,m , φξ n,m+1 − ξ n,m − ξ n+1,m+1 + ξ n+1,m = 0,(17)τ n,m+1 − τ n,m − τ n+1,m+1 + τ n+1,m = 0,
i.e. a discrete wave equation.This is the consequence on the symmetry coefficients of the Clairaut-Schwarz-Young theorem.Equation ( 17) for ξ n,m and τ n,m are to be added to the determining equations for the lattice.They are additional conditions for the symmetries of lattice equations when we assume the validity of the Clairaut-Schwarz-Young theorem.In this case, if, for example, the difference equation for u n,m involves second order shifts like u n+2,m or u n,m+2 so that u n,m , u n,m+1 and u n+1,m are independent variables, then from (17) we get ξ n,m (x n,m , y n,m , u n,m ) = ξ n,m (x n,m , y n,m ).

If the lattice equations in our scheme involve two sites shifted points like x n+2,m , y n+2,m or x n,m+2 , y n,m+2 so that x n,m , x n,m+1 , x n+1,m , y n,m , y n,m+1 and y n+1,m are independent variables then ξ n,m (x n,m , y n,m , u n,m ) = ξ n,m .In this case the general solution of ( 17) is given by ξ n,m = f x n + g x m and τ n,m = f y n + g y m .It is worthwhile, in view of the application to be carried out in next Section, to compute the lowest order discrete derivatives of monomi

s in x and y:
D x x n,m = 1, D x y n,m =
0, D y x n,m = 0, D y m = 2x n,m + h y n,m (h x n,m ) 2 − σ y n,m (σ x n,m ) 2 h y n,m h x n,m − σ y n,m σ x n,m = 2x n,m + ∆ x xx , D x x n,m y n,m = y n,m + h y n,m σ y n,m (h x n,m − σ x n,m ) h y n,m h x n,m − σ y n,m σ x n,m = y n,m + ∆ x xy , D x y 2 n,m = − h y n,m σ y n,m (h y n,m − σ y n,m ) h y n,m h x n,m − σ y n,m σ x n,m = ∆ x yy , D y x 2 n,m = − h x n,m σ x n,m (h x n,m − σ x n,m ) h y n,m h x n,m − σ y n,m σ x n,m = ∆ y xx , D y x n,m y n,m = x n,m + h x n,m σ x n,m (h y n,m − σ y n,m ) h y n,m h x n,m − σ y n,m σ x n,m = x n,m + ∆ y xy , D y y 2 n,m = 2y n,m + h x n,m (h y n,m ) 2 − σ x n,m (σ y n,m ) 2 h y n,m h x n,m − σ y n,m σ x n,m = 2y n,m + ∆ y yy .
where the quantities ∆ x xx , ∆ x xy , ∆ x yy , ∆ y xx , ∆ y xy and ∆ y yy are quantities which go to zero in the continuous limit when the h and the σ go to zero and are explicitly defined in (18).


Example: the potential Burgers equation

The Burgers equation
u t = νu xx + uu x ,(19)
a very well known partial differential equation, appears as a simplification of the Navier -Stokes equation and has been studied from many, if not all, points of view.It was proposed as a model for a viscous fluid, with a viscosity parameter ν.When the viscosity parameter ν is set equal to zero, the Burgers equation degenerates into a qu silinear first order equation which is the prototype of a class of equations with nonlinear phenomena such as shock waves.In fact, the limit ν → 0 allows the study of these shock wave solutions as limits of the solutions of the viscous Burgers equation.In particular, although t e inviscid Burgers equation has an infinite dimensional group of symmetries (being a first order equation), the limit of the symmetry group of the viscous Burgers up of the whole group of symmetries of the inviscid Burgers equation which is a useful tool in the study of the equation and in particular chemes for finding numerical solutions on invariant lattices [1,10].In some of these works, explicit comparison has been made, showing the higher accuracy and stability of these methods [4].

In this paper we do not intend to present this kind of numerical results but ra ariant discrete schemes and present explicit invariant quantities which ca n the previous section the discrete scheme which pres t symmetries of the potential Burgers equation
u y − u xx − u 2 x = 0, (20)
in the simple case when the Clairaut-Schwarz-Young theorem is satisfied.The point symmetries of the potential Burgers equation are [18]
V1 = ∂ x , V2 = ∂ y , V3 = ∂ u , V4 = x∂ x + 2y∂ y ,(21)V5 = 2y∂ x − x∂ u , V6 = 4yx∂ x + 4y 2 ∂ y − (x 2 + 2y)∂ u , Vα = α(x, y)e −u ∂ u ,
where the function α(x, y) satisfies the heat equation α y = α xx .The infinitesimal generator Vα is the one responsible for the linearizability of the potential Burgers equation.A function F (x, y, u, u x , u y , u xx ) is invariant under the infinite imal generators V1 , V2 , V3 V4 , easy to see that ( 22) is invariant also under V6 and Vα , i.e.

pr (2)  V6
I (1) = 2 u xx (I (1) − 1), pr(2)
Vα I (1) = − e −u u xx αu 2

x + 2α x u x + α y (I (1) − 1).

The potential Burgers equation ( 20) is then given by
I (1) = 1(23)
Taking into account the discrete prolongation (15) and the definition of the in initesimal coefficients ( 16) we can construct the discrete prolongation of the vector fields (21).It is easy to show that the conditions (17) are satisfied for all symmetries (21) except for V6 when
ξ n,m+1 − ξ n,m − ξ n+1,m+1 + ξ n+1,m = −4[h x n,m h y n,m + σ x n,m σ y n,m ] = 0 and τ n,m+1 − τ n,m − τ n+1,m+1 + τ n+1,m = −8h x n,m σ x n,m = 0.
hus in the following we will consider just the discrete prolongation of the first five Lie point algebra generators necessary for the case of the discrete Burgers equation.

Taking into ) we have:
pr d
The commutation table of this algebra appears in Table 3 (it is, obviously the same as in the continuous case).

It is immediate to see that a discrete potential Burgers equation preserving the Lie algebra of the continuous potential Burgers given by the generators V1 , V2 , V3 V4 , V5 is given by
I (1) = D 2 x u n,m D y u n,m − (D x u n,m ) 2 = 1, i.e. D y u n,m − D 1 σ y n − h y m (u n+1,m − u n,m ) − σ y n (u n,m+1 − u n,m ) h x n h y m − σ x m σ y n (h x n h y m − σ x m σ y n ) −1 − (h y m (u n+1,m − u n,m ) − σ y n (u n,m+1 − u n,m )) 2 (h x n h y m − σ x m σ y n ) 2 = 0
To complete the diff y n,m , K 6 = 1 (h y n,m ) 3/2 h y n,m+1 σ x n,m − h y n,m σ x n,m+1 , K 7 = 1 (h y n,m ) 3/2 h x n,m (h y n+1,m − h y n,m ) − σ y n,m (h x n,m+1 − h x n,m ) , K 8 = 1 (h y n,m ) 3/2 h x n,m σ y n+1,m − h x n+1,m σ y n,m , K 9 = 1 (h y n,m ) 1/2 h x n,m + 2σ y n,m D x u n,m , K 10 = h y n,m D y u n,m − (D x u n,m ) 2 , K 11 = h y n,m D xx u n,m
Let us notice that in the simple case when the Clairaut-Schwarz-Young theorem is satisfied the conditions (12) imply that K (1) ≡ 1 and K (7) ≡ 0.Moreover, if we shift the in ) 3/2 h y m+1 σ x m − h y m σ x m+1 , K 8 = 1 (h y m ) 3/2 h x n σ y n+1 − h x n+1 σ y n , K 9 = 1 (h y m ) 1/2 (h x n + 2σ y n D x u n,m ) , K 10 = h y m D y u n,m − (D x u n,m ) 2 , K 11 = h y m D xx u n,m
Then we can construct the equations for the lattice in terms of the these invariants (28).

,m+1 −y n,m
dα m , y n+1,m −y n,m = 0 while (36) can be integrated once and gives x n,m h x n = β(dα m ) 3/2 D x u n,m .As to an increase of u n,m there should correspond a decrease of h x n , as an indication of the adaptiveness of the lattice, we must have βd > 0.


Conclusions

In this work we have shown that also in the case of partial difference equation we can introduce a set of variables which are in one to one correspondence with the grid points and which substitute them by the lattice differences and the derivatives on the lattice of the dependent function.This correspondence allows us to write down the invariance equations by using the knowledge of only the continuous invariants.The construction of the lattice scheme, however, requires the construction of a set of invariants written mainly in terms of lattice variables.We also remark that the schem we have obtained is a first order in h and σ approximation of the partial differential equation since D x and D y are the first order approximations of the first partial derivatives.In this paper these results have been applied to the case of the Burgers equation.

It would be good to see what happ