A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 22, A Numerical Solution of the Lax s 7 th -order KdV Equation by Pseudospectral Method and Darvishi s Preconditioning M. T. Darvishi a,, S. Kheybari a and F. Khani b a Department of Mathematics, Razi University Kermanshah 67149, Iran b Department of Mathematics, Ilam University P. O. Box , Ilam, Iran Abstract In this paper, we solve the Lax s seventh-order Korteweg-de Vires (KdV) equation by pseudospectral method. To enhance the accuracy in matrix vector multiplication method we use Darvishi s preconditioning. We compare the numerical results with exact solution of the equation. They are in a good agreement with each other because absolute errors are very small. Mathematics Subject Classification: 65N55 Keywords: Lax s seventh-order KdV equation; Pseudospectral method; Darvishi s preconditioning 1 Introduction The Lax s seventh-order KdV equation is taken as [14] u t +(35u 4 +70(u 2 u xx +uu 2 x)+7(2uu xxxx +3u 2 xx +4u x u xxx )+u xxxxxx ) x =0. (1) Equation (1) is a nonlinear problem. The study of nonlinear problems is of crucial importance in mathematics and physics. Many authors had paid attention to study the solution of nonlinear equations using different schemes. Among these methods, we can point to Backlund transformation [1], inverse scattering scheme [10], the tanh-function [13], Hirota s bilinear method [12] and the homogeneous method [18]. Fan [9] described an extended tanh-function method and symbolic computation to solve the new coupled modified KdV

2 1098 M. T. Darvishi, S. Kheybari and F. Khani equations. Soliman [15] by means of variational iteration method solved the Lax s seventh-order KdV equation. The aim of this study is to solve the Lax s seventh-order KdV equation with known initial and boundary conditions by pseudospectral method using Darvishi s preconditioning. 2 Pseudospectral method We can approximate the arbitrary function f(x) by polynomials in x. However, it is well known that the Lagrange interpolation polynomial based on equally spaced points does not give a satisfactory approximation to general smooth f. In fact, it converges for analytic functions with poles far enough from the interval of interpolation. This poor behavior of polynomial interpolation can be avoided for smoothly differentiable functions by removing the restriction to equally spaced collocation points. Good results are obtained by relating the collocation points to the structure of classical orthogonal polynomials, like Chebyshev and Legendre polynomials. In the most common pseudospectral Chebyshev method, the interpolation points in the interval [ 1, 1] are the Chebyshev-Gauss-Lobatto collocation points x j = cos( jπ ) for j =0,,N, N which are the extreme of the Nth order Chebyshev polynomials T N (x) = cos(ncos 1 x). In order to construct the interpolant of f(x) at the point x the following polynomials are defined g j (x) = ( 1)j+1 (1 x 2 )T N (x), j =0,,N, (2) c j N 2 (x x j ) where c 0 = c N = 2 and c j = 1 for j =1, 2,...,N 1. The interpolation polynomial, P N (x), to f(x) is given by P N (x) = f(x j )g j (x). (3) Then to obtain a pseudospectral approximation we have to express the derivatives of P N (x) in terms of f(x) at collocation points x j. This can be done by differentiating (3), that is d r P N (x) dx r = f(x j )g (r) j (x), (4) so that d r P N (x k ) dx r = f(x j )d (r) kj, (5)

3 Numerical solution of the Lax s 7 th -order KdV equation 1099 where d (r) kj = g(r) j (x k ), (6) are the elements of differentiation matrix D r. The elements of D r can be obtained analytically. For example if r = 1 and j, k =0,,N elements of D are [11] d kj = c k ( 1) j+k, k j cj (x k x j ) d kk = 1 x k 2 (1 x 2), k 0,N k (7) d 00 = 2N 2 +1, 6 d NN = 2N One of the basic steps in pseudospectral methods involves finding an approximation for the differential operator in terms of the grid point values of u N. The derivative of u(x) at collocation points x j can be computed using the matrix-vector multiplication method. 2.1 Matrix-Vector multiplication method If u = {u(x i )} is the vector consisting of values of u(x) at the N +1 collocation points and u = {u (x i )} consists of values of the derivative at the collocation points, then the collocation derivative matrix D is the matrix mapping u u. Two matrix multiplications yield u, where u is the vector containing the second derivatives evaluated at the collocation points. More efficiently, the matrix D 2 = D 2 maps u u, and so on. Collocation matrix methods are asymptotically less efficient than the Chebyshev transform method (see [16]), because it needs O(N 2 ) operations. It can be shown that the Chebyshev derivative, when computing the derivative using the matrix vector multiplication method, is a rather ill-conditioned operator, and inaccuracies in the function can be magnified by as much as O(N 4 ) where N is the number of collocation points. Hence attempts have been made to improve the method. These studies have concentrated on the problem of the roundoff error in Chebyshev collocation methods and various algorithms have been suggested to reduce it. The best result for the matrix vector multiplication algorithm managed to reduce the roundoff error from O(N 4 ε)too(n 3 ε), where ε is the machine precision [7, 8]. Some researchers have studied the problem of reducing roundoff errors in Chebyshev collocation derivative methods. Baltensperger and Trummer [3] demonstrated that naive algorithms for computing these matrices suffer from severe loss of accuracy due to roundoff errors. Breuer and Everson [5] introduced a preconditioning to reduce roundoff error by making the value of the function on the boundaries vanish. Tang and Trummer [17] used trigonometric identities and a flipping trick to reduce roundoff errors. A different approach was suggested in [2, 4].

4 1100 M. T. Darvishi, S. Kheybari and F. Khani Don and Solomonoff attempted to reduce the roundoff error using trigonometric identities for rewriting components of the derivative matrix as follows [7]: d kj = 1 c k 2 cj sin[ π 2N d kk = 1 x k 2 d 00 = 2N 2 +1, 6 d NN = 2N sin 2 ( kπ N ), ( 1) j+k π (k+j)] sin[ 2N (k j)], k j, k 0,N, Formula (8), which avoids the differencing of nearly equal numbers was introduced to reduce this source of error from O(N 4 ε)too(n 3 ε). Preconditioning. From (8) for k j we have d kj = c k 1 2c j sin((k + j) π π ) sin((k j) ) (9) 2N 2N since the value of sin((k j) π ) is near zero when k is near j, hence this causes 2N d kj to become large for k near j. This means that the entries of derivative matrix D with large absolute values are on a band near the main diagonal. That is, large values of d kj correspond to values of k near j. Therefore, the matrix vector multiplication method causes large roundoff errors. In [6], to reduce roundoff error in the kth node, Darvishi and Ghoreishi defined h k (x) as follows: h k (x) =u(x) u(x k ), (10) where x k = cos( kπ N ). We can interpolate h k(x) as hence from (4)-(6) the derivative of h k at x k is or as h k = u we have (8) h k (x) = g j (x)h k (x j ) (11) u (x k )= h k(x k )= d kj h k (x j ) (12) d kj (u(x j ) u(x k )). (13) j k Therefore, using this preconditioning, we can vanish the influence of large values of d kj, that is d kk for all k in the matrix vector multiplication method.

5 Numerical solution of the Lax s 7 th -order KdV equation Governing equation The Lax s seventh-order equation is defined as u t + (35u (u 2 u xx + uu 2 x ) + 7(2uu xxxx +3u 2 xx +4u xu xxx )+u xxxxxx ) x =0, a x b,t 0 (14) where a and b are real numbers. The initial and boundary conditions of equation (14) are taken as u(x, 0) = 2k 2 sech 2 (kx) =H(x) (15) u(a, t) =2k 2 sech 2 (k(a 64k 6 t)) = g 1 (t) (16) and u(b, t) =2k 2 sech 2 (k(b 64k 6 t)) = g 2 (t). (17) The exact solution of equation (14) is taken as u(x, t) =2k 2 sech 2 (k(x 64k 6 t)), (18) where k is a known constant [15]. To solve equation (14) by pseudospectral method we discretize the equation in space u = t (70D3 u(d 2 u + u 2 ) + 140uDu(2D 2 u + u 2 )+ 42DuD 4 u + 70(Du) 3 +14uD 5 u + D 7 u) (19) where D is the differentiation matrix and if D r is the matrix mapping u u (r), then we have D r = D r, [11]. From equation (19) in the kth collocation point we have u (x t k,t)= (70 N d (3) kj u(x j,t){ N d (2) kj u(x j,t)+u 2 (x k,t)} +140u(x k,t) N d kj u(x j,t){2 N d (2) kj u(x j,t)+u 2 (x k,t)} +42 N d kj u(x j,t) N d (4) kj u(x j,t)+70{ N d kj u(x j,t)} 3 +14u(x k,t) N d (5) kj u(x j,t)+ N d (7) kj u(x j,t)) (20) where D r = (d (r) kj ) for r =2, 3,...,7 and D = D 1. For simplicity we set u(x k,t)=v k (t), k =0,...,N. Hence v k (t) is a function of t, where v 0 (t) and v N (t) are boundary functions, namely g 1 (t) and g 2 (t). Equation (20) can be rewritten as v k (t) = (70 N d (3) kj v j(t){ N d (2) kj v j(t)+vk 2(t)} + 140v k(t) N d kj v j (t){2 N d (2) kj v j(t)+vk 2(t)} +42 N d kj v j (t) N d (4) kj v j(t)+70{ N d kj v j (t)} 3 +14v k (t) N d (5) kj v j(t) + N d (7) kj v j(t)), k =1,...,N 1 (21)

6 1102 M. T. Darvishi, S. Kheybari and F. Khani Preconditioned system. The kth row of the preconditioned system of equation (20) by Darvishi s preconditioning is as follows d v dt k(t) = (70 N 1 j=1 d (3) kj (v j(t) v k (t)){ N 1 j=1 d (2) kj (v j(t) v k (t)) + vk(t)} v k (t) N 1 j=1 d kj (v j (t) v k (t)){2 N 1 j=1 d (2) kj (v j(t) v k (t)) + vk(t)} N 1 j=1 d kj(v j (t) v k (t)) N 1 j=1 d(4) kj (v j(t) v k (t)) + 70{ N 1 j=1 d kj(v j (t) v k (t))} 3 +14v k (t) N 1 j=1 d(5) kj (v j(t) v k (t)) + N 1 j=1 d(7) kj (v j(t) v k (t))) (70{d (3) k0 (g 1(t) v k (t)) + d (3) kn (g 2(t) v k (t))} + {d (2) k0 (g 1(t) v k (t)) + d (2) kn (g 2 (t) v k (t))} + 140v k (t){d k0 (g 1 (t) v k (t)) + d kn (g 2 (t) v k (t))}{2d (2) k0 (g 1 (t) v k (t)) + 2d (2) kn (g 2(t) v k (t)) + vk 2(t)} +42{d k0(g 1 (t) v k (t))+ d kn (g 2 (t) v k (t))}{d (4) k0 (g 1(t) v k (t)) + d (4) kn (g 2(t) v k (t))} +70{d k0 (g 1 (t) v k (t)) + d kn (g 2 (t) v k (t))} 3 +14v k (t){d (5) k0 (g 1(t) v k (t))+ d (5) kn (g 2(t) v k (t))} + {d (7) k0 (g 1(t) v k (t)) + d (7) kn (g 2(t) v k (t))}) k =1,...,N 1. (22) Note that system (22) is a system of ordinary differential equations (ODEs). Hence we can use any ODE solver to solve it. In this paper we use the standard fourth-order Runge-Kutta method. 4 Numerical results In this section we solve equation (14) for different values of a and b. Note that as Chebyshev-Gauss-Lobatto collocation points are in interval [ 1, 1], therefore we have to map interval [a, b] to[ 1, 1] by a linear mapping. To demonstrate the efficiency of our method we report the absolute errors in some arbitrary points in tables 1 and 2. To obtain the numerical results we used MATLAB 7.0 software. Problem 1. We solve equation (14) for a = b = 100 and N = 8 collocation points. Numerical results are shown in Table 1. Problem 2. We set b = a = 200 and N = 8 collocation points. The results are shown in Table 2.

7 Numerical solution of the Lax s 7 th -order KdV equation 1103 Table 1. Comparison of the exact and numerical solutions by pseudospectral method using Darvishi s preconditioning for the Lax s seventh-order KdV equation with t =2,N = 8 and Δt = t x Exact solution Numerical solution Abs. error E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 09

8 1104 M. T. Darvishi, S. Kheybari and F. Khani Table 2. Comparison of the exact and numerical solutions by pseudospectral method using Darvishi s preconditioning for the Lax s seventh-order KdV equation with t = 15, N = 8 and Δt =0.01. t x Exact solution Numerical solution Abs. error , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 08 5 Conclusion In this study, the Lax s seventh-order Korteweg-de Vries equation is solved by pseudospectral method using a preconditioning scheme. As numerical results show, the errors are very small. References [1] M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge: Cambridge University Press, [2] R. Baltensperger and J.P. Berrut, The errors in calculating the pseudospetral differentiation matrices for Chebvshev-Gauss-Lobatto points, Comput. Math. Appl. 37 (1999), [3] R. Baltensperger and M.R. Trummer, Spectral differencing with a twist, SIAM J. of Sci. Comput., 24 (5) (2003),

9 Numerical solution of the Lax s 7 th -order KdV equation 1105 [4] A. Bayliss, A. Class and B. Matkowsky, Roundoff error in computing derivatives using the Chebyshev differentiation matrix, J. Comput. Phys., 116 (1995), [5] K.S. Breuer and R.S. Everson, On the errors incurred calculating derivatives using Chebyshev polynomials, J. Comput. Physics, 99 (1992), [6] M.T. Darvishi and F. Ghoreishi, Error reduction for higher derivatives of Chebyshev collocation method using preconditioning and domain decomposition, Korean J. Comput. and Appl. Math., 6 (2) (1999), [7] W.S. Don and A. Solomonoff, Accuracy and speed in computing the Chebyshev collocation derivative, SIAM J. of Sci. Comput., 16 (4) (1995), [8] W.S. Don and A. Solomonoff, Accuracy enhancement for higher derivatives using Chebyshev collocation and a mapping technique, SIAM J. of Sci. Comput., 18 (4) (1997), [9] E. Fan, Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation, Phys. Lett. A, 282 (2001), [10] C.S. Gardner, J.M. Green, M.D. Kruskal and R.M. Miura, Method for solving the Korteweg-de Vires equation, Phys. Rev. Lett., 19 (1967), [11] D. Gottlieb, M.Y. Hussaini and S.A. Orszag, Theory and application of spectral methods, in spectral methods for partial differential equations, ed. by R.G. Voigt, D. Gottlieb and M.Y. Hussaini, (SIAM-CBMS, Philadelphia) (1984), [12] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions, Phys. Rev. Lett., 27 (1971), [13] W. Malfiet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), [14] S. M. El-Sayed, D. Kaya, An application of the ADM to seven-order Sawada-Kotara equations, Applied Mathematics and Computation 157 (2004), [15] A.A. Soliman, A numerical simulation and explicit solutions of KdV- Bursers and Lax s seventh-order KdV equations, Chaos, Solitons and Fractals, 29 (2) (2006), [16] A. Solomonoff and E. Turkel, Global properties of pseudospectral methods, J. Comput. Phys., 81 (1989), [17] T. Tang and M.R. Trummer, Boundary layer resolving pseudospetral methods for singular perturbation problems, SIAM J. Sci. Comput. Phys., 17 (1996),

10 1106 M. T. Darvishi, S. Kheybari and F. Khani [18] M. L. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 213 (1996), Received: March 20, 2007