ANOTHER CLOSE ENCOUNTER: WHEN PAINLEV I MEETS THE WAVE EQUATION

ANOTHER CLOSE ENCOUNTER:WHEN PAINLEV I MEETS THEWAVE EQUATIONR. GLOWINSKI & A. QUAINIINTRODUCTION Few months ago, while cleaning my office desk, I found a 2010 issue of the Notices of the AMS mentioning a newly started Painlev Project, this project being a cyber meeting point for those scientists interested by all aspects of the six transcendent Painlev equations. Intrigued by such a project, I went to WIKIPEDIA where I learnt that the 6 Painlev equations read as (with j = 1, 2, , 6)

Rj being a complex-valued rational function of its three arguments.

In this lecture, we will focus on the 1st Painlev equation (Painlev I), namely:

If y(0) = 0 and (dy/dt)(0) = 0, the graph of the solution looks like (with explosion (blow-up) at t 2.5):

Despite the fact that the 6 Painlev equations occur in many applications, from Mechanics and Physics in particular, it seems that that there is still much to do concerning their numerical solution. A recent contribution in that direction is:B. FORNBERG & J.A.C. WEIDEMAN, A numerical methodology for the Painlev equations, J. Comp. Phys., 230(15), 2011, 5957-5973.

Our goal here is more modest: it consists in investigating the numerical solution of the following nonlinear wave equation, where Is a bounded domain of R2:

and to study the dependence of the solutions with respect to c and the boundary conditions.

Paul Painlev (1863-1933) was twice France Prime Minister (it is reasonable to assume that Painlev went to Politics because Mathematics were too easy for him*, the same way that J. Von Neumann went to Physics, since, according to P. Lax, Mathematics were also too easy for him). It is worth mentioning that the Emile Borel was Secretary of the Navy in both Painlev cabinets ministeriels, a most important information since J.L. Lions, and therefore his many PhD students (several of them are attending this conference), are E. Borel descendants (as are their own PhD students). Paul Painleve

2. RELATED PROBLEMS In the Chapter I of his celebrated 1969 book Quelques Mthodes de Rsolution des Problmes aux Limites Non Linaires, J.L. Lionspresents existence and non-existence results from D. Sattinger, J.B. Keller & H. Fujita concerning the solutions of the following nonlinearwave equation

Concerning the solution of nonlinear parabolic equations with blow-up let us mention

A.A. SAMARSKII, V.A. GALAKTIONOV, S.P. KURDYUMOV & A.P. MIKHAILOV, Blow-Up in Quasi-Linear Parabolic Equations, 1995

3. An operator-splitting approach to the numerical solution of the Painlev I Wave Equation problem The problem under consideration being multi-physics (reaction-propagation type) and multi-time scales, an obvious candidate for its time discretization is the Strangs Symmetrized Operator-Splitting Scheme (SSOS Scheme), that scheme being a reasonable compromise between simplicity, robustness and accuracy (more sophisticated, but more complicated, O.S. schemes are available). In order to apply the SSOS scheme to the solution of our Painlev I Wave Equation problem, the 1st step is to write the above problem as a 1st order in time PDEsystem. To do so, we introduce p = u/t, obtaining thus if we take u = 0 on (0, Tmax) as boundary condition: With t > 0 a time-discretization step, tn+ = (n+ )t, and , (0, 1) with + = 1, we obtain by application of the SSOS scheme:

(1) u0 = u0 , p0 = u1.For n 0, {un, pn} {un+1, pn+1} via (2.1) {un+1/2, pn+1/2} = {u(tn+1/2), p(tn+1/2)},{u, p} being the solution of

(2.2)

(3.1)

{u, p} being the solution of

(3.2)

(4.1)

{u, p} being the solution of

(4.2)

By (partial) elimination of p, we obtain the following O.S. scheme:

(5) u0 = u0 , p0 = u1.For n 0, {un, pn} {un+1, pn+1} via

(6.1)

u being the solution of

(6.2)

(7.1)

with u the solution of

(7.2)

(8.1)

u being the solution of

(8.2)

4. On the solution of the linear wave-suproblemsAt each time-step, we have to solve a linear wave problem of the following type:

(LWE)

We assume that 0 H10() and 1 L2(). A variational formulation of (LWE), well-suited to finite element implementation, is given by (LWE-V):

where < . , . > denotes the duality pairing between H 1() and H10(). Next, assuming that is a bounded polygonal sub-domain of R2, we introduce a triangulation T h of and the following finite dimensional finite element approximation of the space H10():

P1 being the space of the two variable polynomials of degree 1. We approximate (LWE-V) by (LWE-V)h defined as follows:

with 0h and 1h both belonging to Vh and approximating 0 and 1, respectively.

Let us denote by Nh the set of the interior vertices Pj of T h (we have Nh = dim Vh) and by h(t) the Nh dimensional vector

We have then

Above, the mass matrix Mh and the stiffness matrix Ah are both symmetric and positive definite. Now, let Q be a positive integer and define by

For the time-discretization, we advocate the following non-dissipativesecond order accurate centered scheme (the subscripts h have been omitted):

The stability condition of the above scheme is given by

where N is the largest eigenvalue of M 1A ( = O(h2 ) here).

5. On the solution of the nonlinear suproblems At each time step of the Strang symmetrized scheme and for every vertex of Th we have to solve two initial value problems of the following type:

(NLOD2)

Let M be a positive integer; we denote (tf t0)/M by and t0 + m by tm. We approximate then (NLOD2) by

The above scheme can be obtained by limiting to the second order the following Taylor expansion

(TE)m

We can use the 3rd order term in (TE)m to adapt by observing that

We advocate then the following adaptation strategy:

(i) If

keep integrating with (a typical value of tol being 10 4 ).

(ii) If the above inequality is not verified, divide by 2, as many times as necessary to have the above error estimator lessthan 0.2 tol.

6. NUMERICAL EXPERIMENTSAll with = (0, 1)2, x1 = x2 = 1/100 and 1/150, c t x, Q = 3, and M = 3 (initially), = = .Initial conditions: u0 = 0, u1 = 0. 6.1. Dirichlet Boundary conditionsRoughly speaking, the blow up time is of the order of c2 (we stopped computing as soon as the max of the approximate solution reached 104). The results for both space discretization steps are essentially identical.

c = 0

u, c = 0.8

p, c = 0.8

f = 0.9 Hz

6.2. Dirichlet-Sommerfeld Boundary conditionsConsider

1 = {{x1, x2}| x1 = 1, 0 < x2 < 1}, 0 = \1

and take as boundary conditions

For the same value of c the blow-up time is shorter than for pure homogeneous Dirichlet boundary conditions.

u, c = 0.8

p, c = 0.8

I dont want to be polemical but I think that monolithic (un-split) schemes will have troubles at handling this nonlinear wave problem. By the way the approach discussed here is highly parallelizable.

Thank you for your attention