XXI Congreso de Ecuaciones Diferenciales Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, 1- septiembre 9 (pp. 1 8) Exponential time differencing methods for nonlinear PDEs F. de la Hoz 1, F. Vadillo 1 Dpto. de Matemática Aplicada, UPV/EHU. E-mail: francisco.delahoz@ehu.es Dpto. de Matemática Aplicada, Estadística e I.O., UPV/EHU. E-mail: fernando.vadillo@ehu.es Kewords: ETD, Nonlinear waves, Schrödinger, Semi-linear diffusion equations, Blow-up Abstract Spectral methods offer ver high spatial resolution for a wide range of nonlinear wave equations. Because of this, it should be desirable, for the best computational efficienc, to use also high-order methods in time, but without ver strict restrictions on the step size, due to numerical instabilit. In this communication, we consider the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method. This scheme was derived b Cox and Matthews in [4] and modified b Kassam and Trefethen in [1]. We have studied its amplification factor and its stabilit region, which gives us an explanation of its good behavior for dissipative and dispersive problems. In [1], we have applied this method to the nonlinear cubic Schrödinger equation in one and two space-variables. Later, in [11], we have simulated the blow-up of semi-linear diffusion equations. 1. Introduction The spectral methods have been shown to be remarkabl successful when solving timedependent partial differential equations (PDEs). The idea is to approximate a solution u(x,t) b a finite sum v(x,t) = N k= a k(t)φ k (x), where the function class φ k (x),k =, 1,..., N, will be trigonometric for x periodic problems and, otherwise, an orthogonal polnomial of Jacobi tpe, with Chebshev polnomials being the most important special case. To determine the expansion coefficients a k (t), we will focus on the pseudo-spectral methods, where it is required that the coefficients make the residual equal to zero at as man (suitabl chosen) spatial points as possible. Three books [6], [] and [16] have been contributed to supplement the classic references [8] and [3]. When a time-dependent PDE is discretized in space with a spectral discretization, the result is a coupled sstem of ordinar differential equations (ODEs) in time; this is the 1
F. de la Hoz, F. Vadillo notion of the method of lines (MOL), and the resulting set of ODEs is stiff. The stiffness problem ma be even exacerbated sometimes, for example, when using Chebshev polnomials (see Chapter 1 of [16] and its references). The linear terms are primaril responsible for the stiffness with rapid exponential deca of some modes (as with a dissipative PDE) or a rapid oscillation of some modes (as with a dispersive PDE). Therefore, for a time-dependent PDE which combines low-order nonlinear terms with higher-order linear terms, it is desirable to use a higher-order approximation in space and time. For the sake of space, we onl show a numerical simulation for the blow-up of semilinear diffusion equations, referring to [1] for the nonlinear Schrödinger equation.. Exponential Time Differencing fourth-order Runge-Kutta Method The numerical method considered in this communication is an exponential time differencing (ETD) scheme. These methods arose originall in the field of computational electrodnamics [1]. Later on, the have recentl received attention in [1] and [14], but the most comprehensive treatment, and in particular the ETD with Runge-Kutta time stepping, is in the paper b Cox and Matthews [4]. The idea of the ETD methods is similar to the method of the integrating factor (see for example [] or [16]): we multipl both sides of a differential equation b some integrating factor, then we make a change of variable that allows us to solve the linear part exactl and, finall, we use a numerical method of our choice to solve the transformed nonlinear part. When a time-dependent PDE in the form u t = Lu + N(u,t), (1) where L and N are the linear and nonlinear operators respectivel, is discretized in space with a spectral method, the result is a coupled sstem of ordinar differential equations (ODEs): u t = Lu + N(u,t). () Multipling () b the term e Lt, known as the integrating factor, gives e Lt u t e Lt Lu = e Lt N(u,t). (3) Following [1], we integrate (3) over a single time step of length h, getting h u n+1 = e Lh u n + e Lh e Lτ N(u(t n + τ),t n + τ)dτ. (4) The various ETD methods come from how one approximates the integral in this expression. Cox and Matthews derived in [4] a set of ETD methods based on the Runge-Kutta time stepping, which the called ETDRK methods. In this communication we consider the
Exponential time differencing methods for nonlinear PDEs ETDRK4 fourth-order scheme, which has the following expression: a n = e Lh/ u n + L 1 (e Lh/ I)N(u n,t n ), b n = e Lh/ u n + L 1 (e Lh/ I)N(a n,t n + h/), c n = e Lh/ a n + L 1 (e Lh/ I)(N(b n,t n + h/) N(u n,t n )), u n+1 = e Lh u n + h L 3 {[ 4I hl + e Lh (4I 3hL + (hl) )]N(u n,t n ) + [I + hl + e Lh ( I + hl)](n(a n,t n + h/) + N(b n,t n + h/)) + [ 4I 3hL (hl) + e Lh (4I hl)]n(c n,t n + h)}. More detailed derivations of the ETD schemes can be found in [4]. Unfortunatel, in this form ETDRK4 suffers from numerical instabilit when L has eigenvalues close to zero, because disastrous cancellation errors arise. Kassam and Trefethen have studied in [1] those instabilities and have found that the can be removed b evaluating a certain integral on a contour that is separated from zero. The procedure is basicall to change the evaluation of the coefficients, which is mathematicall equivalent to the original ETDRK4 scheme of [4], although in [] it has been shown to have the effect of improving the stabilit of integration in time. Moreover, it can be easil implemented and the impact on the total computing time is small. In fact, we have alwas incorporated this idea in our MATLAB c codes. The stabilit analsis of the ETDRK4 method is as follows (see [1], [7] or [4]). For the nonlinear ODE du = cu + F(u), () dt being F(u) the nonlinear part, we suppose that there exists a fixed point u ; this means that cu +F(u ) =. Linearizing about this fixed point, if u is the perturbation of u and λ = F (u ), then u t = cu + λu (6) and the fixed point u is stable if Re(c + λ) <. The application of the ETDRK4 method to (6) leads to a recurrence relation involving u n and u n+1. Introducing the previous notation x = λh, = ch and using the Mathematica c algebra package, we obtain the following amplification factor: where c = e, c 1 = 4 + 8e u n+1 u n = r(x,) = c + c 1 x + c x + c 3 x 3 + c 4 x 4, (7) c = 8 4 + 16e 4 e 1 + 4e 8e 3 16e 3 4 + 4e 3e, + 8e 4 1 + 4e + 1e 6e + 4e 1e 3 e, + 4e 3 3
F. de la Hoz, F. Vadillo c 3 = 4 16e + 16e 4 1e 3 4 c 4 = 8 6 4e 6 + e 1e 3 + 16e + 6e 4 + 16e 6 + 6e + 8e 3 e 4 + 16e 3 6 e e e 4e 6 + 8e + 4 1e 4 + 4e e 3, + 8e 6 + 4 6e 4 + 6 18e + 6e 4 e 3 4. It is important to remark that computing c 1,c, c 3 and c 4 b the above expressions suffers from numerical instabilit for close to zero. Because of that, for small, we will use instead their asmptotic expansions: c 1 = 1 + + 1 + 1 6 3 + 13 3 4 + 7 96 + O( 6 ), c = 1 + 1 + 1 4 + 47 88 3 + 131 76 4 + 479 96768 + O( 6 ), c 3 = 1 6 + 1 6 + 61 7 + 1 36 3 + 1441 419 4 + 67 196 + O( 6 ), c 4 = 1 4 + 1 3 + 7 64 + 19 11 3 641 4 311 8616 + O( 6 ). 3. The semi-linear diffusion equations Man mathematical models have the propert to develop singularities in a finite time T: for instance, the formation of shocks in Burgers equation without viscosit. Often, this singularit represents an abrupt change in the properties of the models, so it is extremel important for a chosen numerical method to reproduce that change accuratel. In this section, we consider the semi-linear parabolic equation u t = u + u p, p > 1, x R N,t >, (8) with the initial condition u(x,) = u (x), x R N, (9) where u (x) is continuous, nonnegative and bounded. The local (in time) existence of positive solutions of (8),(9) follows from standard results, but the solution ma develop singularities in finite time. In [13], it is proved that there exists a critical exponent p c (N) = 1+ N, such that for 1 < p < p c(n), an nontrivial solution of (8),(9) blows up at a finite time T. However, if p > p c (N), there exist global solutions if the initial value is sufficientl small. The one-dimensional semi-linear parabolic equation is u t = u xx + u p, p > 1, x R,t >. (1) 4
Exponential time differencing methods for nonlinear PDEs The critical exponent is p c = 3 and for 1 < p < 3 the onl nonnegative global (in time) solution is u =. Another question is the asmptotic behavior of the solutions as the blow-up time is approached, for which we refer to [9]. Although problem (1) is not mathematicall periodic, we consider that the solution is close to zero at the ends of the interval [ L,L] for L sufficientl large, so it can be regarded as periodic in practice and we can use the Fourier transform. Hence, in the Fourier space we have û t = ξ û + û p, ξ, (11) where ξ is the Fourier wave-number and the coefficients c = ǫξ < span over a wide range of values when all the Fourier modes are considered. For high values of ξ, the solutions are attracted to the slow manifold quickl because c < and c 1. In figure 1 we displa the boundar stabilit regions in the complex plane x, for =,.9,, 1, 18, which are similar to figures 3., 3.3 and 3.4 of []. When the linear part is zero ( = ), we recognized the stabilit region of the fourth-order Runge- Kutta methods and, as, the region grows. Of course, these regions onl give an indication of the stabilit of the ETDRK4 method. 1 1 18 1.9 1 1 1 1 1 1 Figure 1: Boundar of stabilit regions for several negative In fact, for <, 1, the observed boundaries approach to ellipses whose parameters have been fitted numericall with the following expression: ( ) Im(x) (Re(x)) + =. (1).7 Then, the spectrum of the linear operator increases as ξ, while the eigenvalues of the linearization of the nonlinear part la on the imaginar axis and increase as ξ. On the other hand, according to (1), when Re(x) =, the intersection with the imaginar axis Im(x)
F. de la Hoz, F. Vadillo increases as, i.e., as ξ. Since the boundar of stabilit grows faster than x, the ETDRK4 method should have a ver good behavior to solve dissipative equation, which confirms the results of paper [1]. A similar analsis is applicable to other dissipative equations like, for instance, the Kuramoto-Sivashinsk equation or the Allen-Cahn equation of [4] or [1], where h = 1/4. In our first example, we consider p = and the initial condition u (x) = 6.exp( x ), (13) which is smmetric with respect to the origin and has a single maximum at x =, hence satisfing the hpotheses of [9] under which the asmptotic behavior is known. On the left-hand side of figure we have displaed the evolution of this initial condition from t = to t =.9; a bit later, u(,1) 6 1 1. The computer time was about 1.83 seconds. On the other hand, on the right-hand side of figure, we have displaed the numerical solution at t =.99 with continuous line and the estimate from [9] with discontinuous line. The resemblances near the origin are evident. 1 1 1 8 1 6 1 4.8.6.4. t x 4 3 1 1 3 4 Figure : Numerical solution with u (x) = 6.exp( x ) In the following case, we look for an example where the solution blows up in two points. In figure 3, we represent the numerical solutions for t 3. with the initial condition u (x) = 3exp( (x + 4) ) + exp( (x + 1) ) +.exp( 1(x 4) ). (14) On the right-hand side, we have drawn the solution at t = 3. and the initial condition. In this case we do not know the asmptotic estimates, but in theorem 3 of [9], the have proved an asmptotic behavior close to the Hermite polnomials; in fact, the shape of the plot is similar to that of H 4 (x) (see figure 17.3 of []), although we ignore the scale that should be applied. Bearing in mind the good behavior of our simulation for the one-dimensional problem and the fact that MATLAB c also implements higher dimensional discrete Fourier transforms and their inverses (in two variables, we havefft andifft), we thought that small modifications of our program would allow us to simulate the blow-up for the semi-lineal parabolic equation in two space variables: u t = u xx + u + u p, p > 1, (x,) R,t >. (1) 6
Exponential time differencing methods for nonlinear PDEs 9 8 1 7 8 6 6 4 4 4 3 1 1 1 3 1 1 1 Figure 3: Numerical solution for initial condition (14) In this case, the critical exponent is p c =, so we have taken p = 1. for our numerical experiments. For the sake of space, we onl displa one numerical example. In figure 4, we can observe the evolution of the initial condition u (x,) = 1exp( 1((x + 1) + ( + 1) ) + 1exp( 1((x 1) + ( 1) ), (16) which is smmetric respect to the origin. Figure 4: Numerical solution for initial condition (16) In http://www.ehu.es/~mepvaarf, the reader can find the movies corresponding to the previous experiments, as well as the original MATLAB c code etdrk4dchoise.m, which offers the option to introduce other initial conditions. 7
F. de la Hoz, F. Vadillo Acknowledgments This work was supported b MEC (Spain) with project MTM7-6186, and b the Basque Government with project IT-3-7. References [1] G. Belkin, J.M. Keiser, L. Vozovoi, A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs, J. Comput. Phs. (1998), 147, 36 387. [] J.P. Bod, Chebshev and Fourier Spectral Methods. Second Edition (Revised), Dover, 1. [3] C. Canuto, M.Y. Hussain, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dnamics, Springer, 1988. [4] S.M. Cox, P.C. Matthews, Exponential Time Differencing for Stiff Sstems, J. Comput. Phs. 176 (), pp. 43 4. [] Q. Du, W. Zhu, Analsis and Applications of the Exponential Time Differencing Schemes and their Contour Integration Modifications, BIT Numerical Mathematics (), 4, 37 38. [6] B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge Universit Press, 1998. [7] B. Fornberg, T.A. Driscoll, A fast spectral algorithm for nonlinear wave equations with linear dispersion, J. Comput. Phs. (1999), 1, 46 467. [8] D. Gottlieb, S.A. Orszag, Numerical Analsis of Spectral Methods: Theor and Applications, SIAM, 1977. [9] M.A. Herrero, J.J.L. Velázquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincaré (1993), 1, 131 189. [1] F. de la Hoz, F. Vadillo, An exponential time differencing method for the nonlinear Schrödinger equation, Comput. Phs. Commun. 179 (8), pp. 449 46. [11] F. de la Hoz, F. Vadillo, A numerical simulation for the blow-up of semi-linear diffusion equations, Int. J. Comput. Math. 86 (9), pp. 493. [1] A. Kassam, L.N. Trefethen, Fourth-Order Time Stepping for Stiff PDEs, SIAM J. Sci. Comput. 6 (), pp. 114 133. [13] H.A. Levine, The role of critical exponents in blow-up theorems, SIAM Review (199), 3, 6 88. [14] D.R. Mott, E.S. Oran, B. van Leer, A Quasi-Stead-State Solver for the Stiff Ordinar Differential Equations of Reaction Kinetics, J. Comput. Phs. (), no. 164, 47 48. [1] A. Taflove, Computational Electrodnamics: The Finite Difference Time-Domain Method, Artech House, Boston, 199. [16] L.N. Trefethen, Spectral Methods in MATLAB, SIAM,. 8