1 Chien-Hong Cho ( 卓建宏 ) (National Chung Cheng University)( 中正大學 ) 2010/11/11 in NCKU On the finite difference approximation for hyperbolic blow-up problems

2 Contents  A brief introduction for blow-up problems.  Review on the parabolic blow-up problem.  CLM equation Spectral method Finite difference method  Numerical results of the semi-linear wave equation. Sufficient condition for blow-up. Numerical scheme for semi-linear wave equation and blow-up result  Future work.

3 Blow-up problem for  Broadly speaking, singularity occurs in the solution, its derivative, or its high-order derivatives in finite time.  Here, we use blow-up in a narrow sense. We say that the solution of an initial-value problem blows up in finite time T if the solution becomes infinity as where T is called the blow-up time.

4 Example The solution of the ODE is. The solution tends to infinity as t tends to 1.

5 More examples Semi-linear heat equation (Fujita, Weissler, Friedman, Mcleod, …) The porous medium equation with a nonlinear source (Samarskii, Galaktionov, …) Semi-linear wave equation (John, Kato, Glassey, Levine, …) CLM equation (Constantin, Lax, Majda, …) Hyperbolic type (H denotes the Hilbert transform)

Problem Occurred in Numerical Calculation for a Blow-up Problem Ex: Consider the ODE and the difference scheme The numerical solution exists for all That is, the numerical solution exists globally, or equivalently, the numerical solution does not blow up in finite time T. or equivalently,

7 Main purpose We want to construct an appropriate finite difference scheme for blow-up problems. By the word ‘ appropriate ’ we mean a scheme that satisfies the following: convergence. blow-up in finite time. convergence for the numerical blow-up time.

8 1-dim semi-linear heat equation where. Then the solution blows up in finite time T : (Fujita(1966), Weissler(1984), Friedman and Mcleod(1985), etc..)

9 Finite difference scheme where ： the approximation of at ： the spatial mesh point. ： the time grid point. ： fixed.

10 (numerical blow-up time) Blow-up cannot be reproduced if we use uniform

11 The semi-linear case (a) H is monotone increasing and (b) is monotone increasing. (c) Then Namely, the finite difference solution blows up in finite time & the numerical blow-up time converges to T. Let T be the blow-up time. Assume that H satisfies

12 Remark  This was a toy problem. Although the convergence of the numerical blow-up time is proved, the convergence rate still remains open.  More fluid related problem? The finite difference approximation becomes much more difficult for blow-up problems of hyperbolic type than the parabolic ones. We consider two simple equations -- CLM equation and the semi-linear wave equation, which show the difficulties. Our recent results will also be reported.

13 CLM equation The Constantin-Lax-Majda equation Here H denotes the Hilbert transform: where the integral denotes Cauchy’s principal part. is periodic in x and

14 Remark  A 1-dim model for the vorticity equation. (see Constantin, Lax, Majda (1985))  The solution to CLM equation is given explicitly by Thus, the solution blow-up in finite time if and only if is nonempty, where

15 Let Then Thus, the solution is given by

16 Spectral method for CLM equation Let N be a positive integer. Consider the following spectral approximation : where, for is given by denotes the approximation of the solution u.

17 For simplicity, we consider the case of odd function: whence Then we have which tells that is a polynomial in t of order ( n-1). Theorem does not blow up for all N.

18 Finite difference method Let Then one has (Constantin, Lax, Majda, 1985) We define a finite difference approximation in such a way that approximates for. Here, is by definition the imaginary part of. Then we have The solution does not blow up.

19 Other difference scheme  Discretize the Hilbert transform in a different way.  Difficulties: (a) The equalities derived from the properties of the Hilbert transform are not true in the discrete version. (b) It is difficult to show that the maximum values of the finite difference solution propagate to the zero of I.D. due to the complexity of the discretized Hilbert transform.

20 1-dim semi-linear wave equation  The solution blows up in finite time T. That is,  Many sufficient conditions for blow-up were given. For example, Glassey, John, Kato, Levine, etc..

21 Levine ’ s Result (1974) Let the nonlinear term be and the initial data satisfy Then the solution blows up in finite time T.

22 Another sufficient condition for blow-up (Cho) Let Assume that Then blows up in finite time T. Independent of the Levine ’ s condition. More convenient for numerical analysis.

23 By Jensen’s inequality, This implies the blow-up of. Sketch of the proof Multiply to both sides, one has where K is a constant decided by the initial data.

24 Numerical Scheme where ： the approximation of at ： the spatial mesh point. ： the time grid point. ： fixed.

25 Remark If we put, then the scheme is in fact

26 Moreover, assume that converges to u (t, x) while u is smooth. Then the numerical blow-up time also converges. Namely, Theorem (Cho) Define Assume that H satisfies (H1) is monotone increasing. (H2) (H3) Then the solution blows up in finite time. That is, : constant decided by I.D.

27 Convergence while u is smooth  Suppose the solution u blows up at t = T.  u is smooth in  u j n converges to u in That is, as long as

28 Numerical examples

29 Numerical examples

30 Difficulty in proving convergence  Two level time meshes appear in the scheme and thus their relation plays an important role in the stability.  To show the convergence (while u is smooth), we need some a priori estimates or stability in some norms, which can be derived from the well- known “ energy conservation property ” of the wave equations for the uniform time mesh ( ), while the energy need not be conserved for non-uniform time mesh.

31 Remark  In fact, to prove the convergence (while u is smooth), we only need the stability for the finite difference solution of the linear wave equation. But for the non-uniform time mesh, only a little is known.  Samarskii & Matus ’ s scheme(2001); Matsuo ’ s ( 松 尾宇泰 ) scheme(2007): strong restriction on the spatial part of their scheme.

32 Linear wave equation We consider the initial-boundary-value Problem Then we have That is,

33 Finite difference scheme We consider and the well-known discrete energy which corresponds to the energy Then we have

34 Samarskii ’ s and Matsuo ’ s scheme  Samarskii et. al considered a difference equation in a finite dimensional space and then applied to the linear wave equation.  Matsuo used the so-called discrete variational method. Namely, he defined the discrete energy first, and then derived the finite difference scheme whose solution conserves the given discrete energy.  Neither of which can be applied to our scheme.

Semi-discrete scheme Moreover, we have that, for any There exist such that and that Theorem (Cho)

Remark  It should be noted that we can prove the convergence of the numerical solution for the semi-discrete scheme by using the energy conservation property of the linear wave equation, which does not hold in the full-discrete case.

2-nd order ODE We consider the 2-nd ODE blow-up problem where and the finite difference analogue

Theorem (Cho) Under certain assumption on H, we have where T denotes the blow-up time of denotes the numerical blow-up time. are constants independent of. C. H. Cho, On the convergence of numerical blow-up time for a 2 nd order nonlinear ordinary differential equation, Appl. Math. Lett., 24, 2011,

39 Future work  Stability for finite difference schemes of the linear wave equation with non-uniform time meshes.  A rigorous proof for the convergence of the finite difference solution to the semi-linear wave equation.  Convergence order for the numerical blow-up time.

40

41 Thank you for your attention.

References C.-H. Cho, S. Hamada, and H. Okamoto, On the finite difference approximation for a parabolic blow-up problem, Japan J. Indust. Appl. Math., 24 (2007), pp C.-H. Cho, A finite difference scheme for blow-up problems of nonlinear wave equations, Numerical Mathematics:TMA, 3 (2010), pp C.-H. Cho, On the convergence of numerical blow-up time for a second order nonlinear ordinary differential equation, Appl. Math. Lett., vol.24 no.1 (2011), pp