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Academic research paper on topic "Multisymplectic structure of numerical methods derived using nonstandard finite difference schemes"



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Journal of Physics: Conference Series 490 (2014) 012205


Multisymplectic structure of numerical methods derived using nonstandard finite difference schemes

Odysseas Kosmas1 and Dimitrios Papadopoulos2

1 Chair of Applied Dynamics, University of Erlangen-Nuremberg, Germany 2Delta Pi Systems Ltd., 57022 Thessaloniki, Greece



Abstract. In the present work we investigate a class of numerical techniques, that take advantage of discrete variational principles, for the numerical solution of multi-symplectic PDEs arising at various physical problems. The resulting integrators, which use the nonstandard finite difference framework, are also multisymplectic. For the derivation of those integrators, the necessary discrete Lagrangian is expressed at the appropriate discrete jet bundle using triangle and square discretization. The preliminary results obtained by the resulting numerical schemes show that for the case of the linear wave equation the discrete multisymplectic structure is preserved.

1. Introduction and motivation

In the present paper we investigate the geometric structure for multisymplectic integrators using nonstandard finite difference schemes for variational partial differential equations (PDEs). The approach we consider is first to develop a Veselov type discretization for PDEs in variational form [1, 2] (see also [3]) and second to combine this approach with nonstandard finite difference schemes of [6, 7]. These resulting multisymplectic-momentum integrators have, under appropriate circumstances, very good energy performance in the sense of the conservation of a nearby Hamiltonian up to exponentially small error [5]. Furthermore, the nonstandard finite difference schemes was developed for compensating the weaknesses that may be caused by standard finite difference methods, for example, the numerical instabilities.

Following the steps of the derivation of the Euler-Lagrange equations in the continuous formulation of Lagrangian dynamics, the discrete Euler-Lagrange equations can be derived [3]. Denoting the tangent bundle of the configuration manifold Q by TQ, the continuous Lagrangian L : TQ ^ R can be defined. In the discrete setting, considering approximate configurations qk œ q(tk) and qk+1 ~ q(tk+1) at the time nodes tk,tk+1 with h = tk+1 — tk being the time step, a discrete Lagrangian Ld : Q x Q ^ R is defined to approximate the action integral along the curve segment between qk and qk+1, i.e.

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Journal of Physics: Conference Series 490 (2014) 012205


(i +1,j + 1)

(i,j + 1)

Figure 1. Triangle discretization. a) Three ordered triple and b) the triangles which touch (i,j).

Defining the discrete trajectory Yd = (Qo,... ,Qn), N e N one can obtain the action sum

Sd(Yd) = Ld(qk,qk+1).

Discrete Hamilton's principle states that a motion Yd of the discrete mechanical system extremizes the action sum, i.e.¿Sd = 0. By differentiation and rearrangement of the terms, holding the end points q0 and qN fixed, the discrete Euler-Lagrange equations are obtained

D2Ld(Qk-\,Qk)+ DiLd(Qk, Qk+i) = 0, k = 1,..., N - 1 (3)

where the notation D^Ld indicates the slot derivative with respect to the i-th argument of Ld, see [3].

2. Triangle discretization

Just as the ODEs of classical mechanics can be described by a Lagrangian variational structure, PDEs theory can as well. For that, in order to express the discrete Lagrangian function, we need to define an appropriate generalization of the tangent bundle TQ and cotangent bundle T*Q to fields over the higher-dimensional manifold X. Following the work of [3] we consider fields over X as sections of some fiber bundle B ^ X, with fiber Y, and then we consider the first jet bundle J1B and its dual (JiB)* as the appropriate analogs of the tangent and cotangent bundles.

We now generalize the Veselov discretization given [1, 2] to multisymplectic field theory, by discretizing the spacetime X. For simplicity we restrict to the discrete analogue of dimX = 2. Thus, we take X = Z x Z = (i, j) and the fiber bundle Y to be X x F for some smooth manifold F, see [3].

2.1. Triangle discretization

Assume that we have a uniform quadrangular mesh in the base space, with mesh lengths Ax and At. The nodes in this mesh are denoted by (i, j) e Z x Z, corresponding to the points (xi,tj) := (iAx, j At) e R2. We denote the value of the field u at the node (i,j) by uj. We label the triangle at (i, j) with three ordered triple ((i,j), (i,j + 1), (i + 1, j + 1)) as Aij-, and we define Xa to be the set of all such triangles, see Figure 1a. Then the discrete jet bundle is defined as follows [3]

JiY := {(uj,uj+1,uj+i) e R3 : ((i, j), (i,j + 1), (i + 1, j + 1)) e Xa} (4)

which is equal to Xa x R3. The field u can be now defined by averaging the fields over all vertices of the triangle (see Figure 1a)

j i j+i i j+i uj + uj + uj+i

u ^-3-— (5)

while the derivatives can be expressed using nonstandard finite differences

du^ uj+i - uj duu^ uj+ii - uj+i (6)

dt 0(At) , dx ^(Ax) ( )

0(At) = 1 sin (A?) , ^(Ax) = 2 sin (AX) • (7)

For a more comprehensive study on the above expressions see [6, 7]. Using the above expressions, we can obtain the discrete Lagrangian at any triangle, that depends on the edges of the triangle, i.e. Ld(uj, uj+1,uj+:l1)- When using triangle discretization, the discrete Euler-Lagrange field equations are

DiLd(uj ,uj+1,uj+1) + D2Ld(u^1,uj ,uj+i) + D3 Ld^,^,^) = 0 (8)

see Figure 1b.

3. Numerical examples

To illustrate the behavior of the proposed method, we will consider the linear wave equation see e.g. [4], i.e. the second order partial differential equation of the form

d2u d2u . .

du + ^ = 0 (9)

For the special case that c = -1, the corresponding Lagrangian is

L(u, ut, ux) = 2u2 - 22uX (10)

where du/dt = ut and du/dx = ux. The triangle discretization of Section 2.1 leads to the discrete Lagrangian

Ld(uj ,uj+i,uj+1i) = 1AtAx ( 2 (ji^ )2 - 1 ( j^f )2) dD

where At and Ax are the mesh lengths for time and space respectively. Applying the above discrete Lagrangian to the discrete Euler-Lagrange field equations (8) we get

4+1 - 2uj + uj-1 uj+i - 2uj + uj-

(0(At))2 (^(Ax))2

which is the expression of the variational integrator for the linear wave equation (9) using nonstandard finite difference schemes.

Figure 2. The waveforms of linear wave equation (9).

In Figure 2 the solution of the above equation is presented. On that we have considered initial conditions 0 < x < 1, u(x, 0) = 0.5[1 — cos(2nx)], ut(x, 0) = 0.1 and periodic boundary conditions u(0,t) = u(1,t), ux(0,t) = ux(1,t). The grid discretization has been considered to be At = 0.01 and Ax = 0.01. Figure 2 shows that the time evolution of the solution is continues and the periodicity is also preserved.

Furthermore in Figure 3 the evolution of the discrete energy, both temporal and spatial,

uj+1 -0(Ai)

uj+1 -

' j+1 j+1 s ^(Ax)

' j+1 j+1 '

is presented. Figure 3 shows clearly that during the numerical simulation no energy loss or blow-up appears.

4. Summary and conclussions

The derivation of multisymplectic numerical methods derived from nonstandard finite difference schemes is investigated. Even though the results are preliminary, the numerical examples which have been already tested (e.g. the linear wave equation) show that these numerical schemes have good energy behavior and preserve the discrete multisymplectic structure. Thus, applying these schemes in the field of incompressible fluid dynamics, see e.g. [11] and [12], would be interesting. Furthermore, in complex geometries which appear in real world problems, it is necessary to extend this methodology to non-uniform grids.

Journal of Physics: Conference Series 490 (2014) 012205


Figure 3. a) The temporal evolution of the discrete energy (13) and b) the spatial evolution of the discrete energy of (14).


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