Int. J. Open Problems Compt. Math., Vol., No., June ISSN 99-; Copyright ICSRS Publication, www.i-csrs.org Fourier spectral methods for solving some nonlinear partial differential equations Hany N. HASSAN Hassan K. SALEH Department of Basic Science, Benha Faculty of Engineering, Benha University, Benha, 5, Egypt Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Giza, Egypt E-mails, : h_nasr77@yahoo.com, hassan.kamel@yahoo.com Abstract The spectral collocation or pseudospectral (PS) methods (Fourier transform methods) combined with temporal discretization techniques to numerically compute solutions of some partial differential equations (PDEs). In this paper, we solve the Korteweg-de Vries (KdV) equation using a Fourier spectral collocation method to discretize the space variable, leap frog and classical fourth-order Runge-Kutta scheme (RK) for time dependence. Also, Boussinesq equation is solving by a Fourier spectral collocation method to discretize the space variable, finite difference and classical fourth-order Runge- Kutta scheme (RK) for time dependence. Our implementation employs the Fast Fourier Transform (FFT) algorithm. Keywords: Fourier spectral method, Fast Fourier transform, Boussinesq equation, Korteweg-de Vries equation; leap frog, finite difference. Introduction Let us consider the equation u u u u, () t The equation () is known as the Korteweg-de Vries (KdV) equation (with non linearity uu, and dispersion u ) with real constants,. The KdV equation is one of the popular soliton equations. More than a century later the KdV equation
Hany N. HASSAN et al. 5 has been found in many other applications such as magnetohydro dynamic waves in a cold plasma, longitudinal vibrations of an enharmonic discrete-mass string, ion-acoustic waves in a cold plasma, pressure waves in liquids-gas bubble miture, rotating flow down a tube, and longitudinal dispersive waves in elastic rods. The eact solutions of () is c c u, t sec h ( ct ), () where is an arbitrary integration constant. We investigate the numerical solution of the KdV equation using the Fourier-Leap Frog methods, due to Fornberg and Whitham [], and the Fourier based fourthorder Runge-Kutta (RK) method for better stability of the solution. Consider the equation utt u ( u ) u, () The equation () is known as the Boussinesq equation, where subscripts and t denote differentiation, was introduced to describe the propagation of long waves in shallow water. The Boussinesq equation also arises in several other physical applications including one-dimensional nonlinear lattice waves, vibrations in a nonlinear string, and ion sound waves in a plasma. It is well known that The Boussinesq equation () has a bidirectional solitary wave solution u(, t) b sec h ( b( ct)). () representing a solitary wave, where c b is the propagation speed and b, arbitrary constants determining the height and the position of the maimum height of the wave, respectively. From the form of c it is apparent that the solution can propagate in either direction (left or right). In the present work, we have applied to () two numerical methods to study soliton solutions and investigate their interactions upon collision: a combination of finite differences and a Fourier pseudospectral method Analysis the method for the Korteweg-de Vries (KdV) equation Now, consider the standard KdV equation in the form ut u u u, [ L, L ], (5) where u = u(, t), subscripts and t denote differentiation, where L is a given number, usually large. We have changed the solution interval from [ L, L ] to [,π], with the change of variable. L Thus, the equation (5) becomes
Hany N. HASSAN et al. ( ) ut uu u L L, [, ], () u(, t) is transformed into Fourier space with respect to, and derivatives (or other operators) with respect to. Applying the inverse Fourier transform n u n F {( ik) F( u)}, n n,,. (7) using this with n = and n =, u { ( )} F ikf u, u F { ik F( u)}. The equation () becomes ut uf { ikf( u)} ( ) F { ik F( u)}, L L () In practice, we need to discretize the equation (). For any integer N >, we consider j j j, j,, N, N. Let u (, t) be the solution of the KdV equation (5). Then, we transform it into the discrete Fourier space as N ik N N j uˆ( k, t) F( u) u( j, t) e, k. N (9) J From this, using the inversion formula, we get ( N, ) ( ˆ) ˆ j (, ) ik j,. kn u t F u u k t e j N () where we denote the discrete Fourier transform and the inverse Fourier transform by F and F, respectively. Now, replacing F and F in () by their discrete counterparts, and discretizing () gives du( j, t) u( j, t) F { ikf( u)} ( ) F { ik F( u)}, j N. () dt L L T T [ u(, t), u(, t),, u( N, t)]. Letting U = [ u(, t ), u(, t ),, u( N, t )]. The equation () can be written in the vector form U t = F (U) () where F defines the right hand side of ().. Fourier Leap-Frog Method for KdV Equation A time integration known as a Leap-Frog method (a two step scheme) is given by n n n U U t F where the superscripts denote the time level at which the term is evaluated. In this subsection, we use a Fourier collocation method and this scheme to solve the
Hany N. HASSAN et al. 7 system given in () numerically. Here, we follow the analysis of Fornberg and Whitham []. Using the Leap-Frog scheme to advance in time, we obtain U(t +Δt) = U(t -Δt)+ΔtF(U(t)) This is called the Fourier Leap-Frog (FLF) scheme for the KdV equation (5). FLF needs two levels of initial data. Usually, the first one is given by the initial condition u(, ) and the second level u(, Δt) can be obtained by using a higherorder one-step method, for eample, a fourth order Runge- Kutta method. Thus, the time discretization for () is given by u(, t t) u(, t t) t u( j, t) F { ikf( u)} t( ) F { ik F( u)}. () L L where we denote ( ) k. L In general, the Fourier-Leap Frog scheme defined in () is accurate for low enough wave numbers, but it loses accuracy rapidly for increasing wave numbers.. Fourier Based RK Method for KdV Equation Notice that the Fourier-Leap-Frog method is a second-order scheme and has some disadvantage in the way that the solution of the model problem is subject to a temporal oscillation with period Δt a commonly suggested alternative method is the Runge-Kutta methods. The classical fourth-order Runge-Kutta methods for the system () is given by K = F(U n,t n ) K = F(U n + ΔtK,t n + Δt) K = F(U n + ΔtK,t n + Δt) () K = F(U n + ΔtK,t n + Δt) Δt U n+ = U n + K + K + K + K where the superscripts denote the time level at which the term is evaluated.. Linear stability analysis For an analysis of stability we use the standard Fourier analysis to find the condition imposed on the time step Δt. For simplicity we let and consider the KdV equation in the form u u u, (5) t
Hany N. HASSAN et al... By using FLF scheme We approimate this equation by u(, t t) u(, t t) tf { i F( u)} tf { i F( u)}. () We look a solution to () of the form t t i u(, t) e Substitution in () gives ( t t ) t i ( t t ) t i t t i t t i e e i t e i t e, i.e. it i t if ( t,, ), where f ( t,, ) t t The scheme is conditionally stable if and only if f ( t,, ) is real and less than one in magnitude. Let us again assume for simplicity that the period is [,π] and that this interval is discretized with N equidistant mesh points, that is N The wave number takes the values,,,, N. We want to find the largest value of t such that f ( t,, ) < is true for all.the most severe restriction on t is imposed for the,which are largest in magnitude, i.e. for ma, ma N. Thus, we obtain f ( t, ma, ) t( ) t( ) If we assume that >, then f ( t,, ) < when < t ( ) So, the stability condition becomes t <.55.. By using RK method To do stability analysis of the RK scheme for the KdV equation, we could use the approach used in analyzing the stability of the Fourier Leap-Frog schemes as t t i in the previous subsection. To do this, we substitute u(, t ) e into ().
Hany N. HASSAN et al. 9 After a very tedious and long derivation, we are led to the stability condition or after we numerically eperiment with the scheme, we see that the appropriate time step is the one satisfying the condition t. The right-hand side of the system of ODES in time given in () F (U) is F(U j) u( j, t) F { ikf( u)} F { i k F( u)}, j N. L L. Numerical Results and Eamples In order to show how good the numerical solutions are in comparison with the eact ones, we will use the L and L error norms defined by N eact num eact num L u u u, i ui i L eact num eact num u u ma u u. (7) i To implement the performance of the method, three test problems will be considered: the motion of a single solitary wave, the interaction of two positive solitary waves, the interaction of three positive solitary waves and other solutions... The motion of a single solitary wave Consider the KdV equation (5) with =, = and L =. ut uu u, [ L, L ]. The simplest eact solution is u(, t) sec h ( t). and initial condition u(,) sec h ( ). Eample In this eample we compute the numerical solutions u(, ) using the Fourier Leap-Frog scheme. The numerical solution at t =,.5,,.5 and in Figure. with N =. It is clear from Figure that the proposed method performs the motion of propagation of a solitary wave satisfactorily, which moved to the right with the preserved amplitude. In Figure, we plot the eact solution and Numerical solution at t = with N =, Figure show the error at each collection point at t = with N =, Table displays the values of error norms obtained at N =, N = and N = 5. i i
u(,t) u(,t) Hany N. HASSAN et al. 5.5 t = t =.5 t = t =.5 t =.5 -.5 - -5 - -5 5 5 Fig. : Fourier spectral solution of the KdV equation using FLF scheme with N = at t =,.5,,.5 and. N t L L.779.55 7.9.9.557.59 5..9.95 Table : Error norms of FLF scheme of the KdV equation at t = with N =, and 5. Numerical Eact.5.5 - -5 - -5 5 5 Fig. : Fourier spectral solution of the KdV equation using FLF scheme with N = at t =
Error Hany N. HASSAN et al. 5 5 - -5 - -5 - -5 5 5 Fig. : The error (error = eact numerical) distributions in FLF scheme for the KdV equation with N = at t =. Eample Now, we solve the same problem using the RK scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. Figure shows simulation of the solution computed using N =. In Figure 5, we plot the eact solution and Numerical solution at t = with N =, Figure show the error at each collection point at t = with N =. Table displays error norms obtained at N =, N = and N = 5. From these results we can see that by carefully choosing the time steps, RK is more accurate than the Fourier Leap-frog, but Fourier Leapfrog is easier than RK. Fig. : Numerical simulation of Fourier spectral solution of the KdV equation using RK scheme with N =.
Error u(,t) Hany N. HASSAN et al. 5 N Δt L L.5 59.5 9..9.75.7 5..99.5 Table : Error norms of Fourier spectral solution of the KdV equation using RK scheme at t = with N =, and 5. Numerical Eact.5.5 - -5 - -5 5 5 Fig. 5: Fourier spectral solution of the KdV equation using RK scheme at t = with N =. - - - - - - -5 - -5 5 5 Fig. : The error distributions in Fourier spectral solution of the KdV equation using RK scheme at t = with N =... The interaction of two positive solitary waves Consider the KdV equation (5) with =, =, and L =. The initial data is
u(,t) u(,t) u(,t) Hany N. HASSAN et al. 5 u(,) n( n )sec h ( ) () results in n solitons solution that propagate with different velocities. The initial for n = is u(,) sech ( ) (9) The eact solution can be epressed as [] cosh( t) cosh( t) u(, t) [cosh( ) cosh( )] t t Eample We solve equation (5) with initial solution (9) using the FLF scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. We plot the eact solution and the numerical solutions at t = -.,-.,,., and. at N = 5 and t. in Figure 7. 7 t = -. Numerical Eact 7 t = -. Numerical Eact 5 5 - -5 - -5 5 5 - -5 - -5 5 5 (a) (b) 7 Numerical Eact t = 5 - -5 - -5 5 5 (c)
u(,t) u(,t) Hany N. HASSAN et al. 5 7 Numerical Eact t =. 7 Numerical Eact t =. 5 5 - -5 - -5 5 5 - -5 - -5 5 5 (d) (e) Fig. 7: Fourier spectral solution for interaction of two waves of the KdV equation using FLF scheme with initial condition (9) and N =5. Eample We solve eample using the RK scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. Figure show simulation of the solution computed. We plot the eact solution and the numerical solutions at t = -.,-.,,., and. with N = 5 and t.7 in Figure 9. Fig. : Numerical simulation of Fourier spectral of interaction of two waves of the KdV equation using RK scheme with initial condition (9) and N =5.
u(,t) u(,t) u(,t) u(,t) u(,t) Hany N. HASSAN et al. 55 7 t = -. Numerical Eact 7 t = -. Numerical Eact 5 5 - - -5 - -5 5 5 (a) - - -5 - -5 5 5 (b) 5 Numerical Eact t = - - -5 - -5 5 5 (c) 7 Numerical Eact t =. 7 Numerical Eact t =. 5 5 - - -5 - -5 5 5 (d) - - -5 - -5 5 5 (e) Fig. 9: Fourier spectral solution for interaction of two waves of the KdV equation using RK scheme with initial condition (9) and N =5.
u(,t) u(,t) Hany N. HASSAN et al. 5 In net method, we can study the interaction of n solitary waves by using the initial condition given by the linear sum of n separate solitary waves of various amplitudes n i i i. () u(,).5c sec h.5c i Interaction of two positive solitary waves can be studied by using the initial condition given by the linear sum of two separate solitary waves of various amplitudes u(,) u (,) u (,) u(,) sec h ( ).5sec h (.5( )) () Eample 5 The calculation is carried out with the time step t =.9 and N = 5 over the region -, we solve using the FLF scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. We plot the numerical solutions at t =,.5,.,. and.5 with N = 5, in Figure, respectively. The initial function was placed on the left side of the region with the larger wave to the left of the smaller one as seen in the Figure, both waves move to the right with velocities dependent upon their magnitudes. The shapes of the two solitary waves is graphed during the interaction at t =. and after the interaction at time t =.5, which is seen to have separated the larger wave from the smaller one. According to the figure, the larger wave catches up the smaller wave at about t =.5, the overlapping process continues until the time t = 5, then two solitary waves emerge from the interaction and resume their former shapes and amplitudes..5 t =.5 t =.5.5.5 -.5 - - - - (a) -.5 - - - - (b)
u(,t) u(,t) u(,t) u(,t) Hany N. HASSAN et al. 57 t =. t =..5.5.5.5 -.5 - - - - (c) -.5 - - - - (d).5 t = 5.5 t =.5.5.5 -.5 - - - - -.5 - - - - (e) (f) Fig. : Fourier spectral solution for interaction of two waves of the KdV equation using FLF scheme with initial condition () and N = 5. Eample We solve eample 5 using the RK scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. The calculation is carried out with the time step t =.9 and N = 5 over the region, Figure shows simulation of the solution computed. We plot the numerical solutions at t =,.5,.,. and.5 with N = 5, in Figure, respectively.
u(,t) u(,t) u(,t) u(,t) Hany N. HASSAN et al. 5 Figure : Numerical simulation of Fourier spectral of interaction of two waves of the KdV equation using RK scheme with initial condition () and N =5..5 t =.5 t =.5.5.5 -.5 - - - - (a) -.5 - - - - (b).5 t =..5 t =..5.5 -.5 - - - - -.5 - - - - (c) (d)
u(,t) u(,t) Hany N. HASSAN et al. 59.5 t = 5.5 t =.5.5.5 -.5 - - - - -.5 - - - - (e) (f) Fig. : Fourier spectral solution for interaction of two waves of the KdV equation using RK scheme with initial condition () and N =5... The interaction of three positive solitary waves Consider the KdV equation (5) with =, =, and L = ut uu u, [,], The initial for n = in () is u(,) sec h ( ) () Eample 7 We solve (5) with initial condition ().The calculation is carried out with the time step t =. and N = 5 over the region -, we solve using the FLF scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. We plot the numerical solutions at t = -., -., -.5,,.5 and. with N =5, in Figure, respectively. Eample We solve eample 7 using the RK scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. The calculation is carried out with the time step t =.7 and N = 5 over the region -, we plot the numerical solutions at t = -., -., -.5,,.5, and. with N =5, in Figure, respectively.
u(,t) u(,t) u(,t) u(,t) u(,t) u(,t) Hany N. HASSAN et al. t = -. t = -. - -5 - -5 5 5 (a) - -5 - -5 5 5 (b) t = -.5 t = - -5 - -5 5 5 (c) - -5 - -5 5 5 (d) t =.5 t =. - -5 - -5 5 5 - -5 - -5 5 5 (e) (f) Fig. : Fourier spectral solution for interaction of three waves of the KdV equation using FLF scheme with initial condition () and N =5.
u(,t) u(,t) u(,t) u(,t) u(,t) u(,t) Hany N. HASSAN et al. t = -. t = -. - -5 - -5 5 5 (a) - -5 - -5 5 5 (b) t = -.5 t = (b) - -5 - -5 5 5 (c) - -5 - -5 5 5 (d) t =.5 t =. - -5 - -5 5 5 - -5 - -5 5 5 (e) (f ) Fig. : Fourier spectral solution for interaction of three waves of the KdV equation using RK scheme with initial condition () and N = 5.
u(,t) u(,t) Hany N. HASSAN et al. In this interaction of three positive solitary waves is studied by using the initial condition given by the linear sum of three separate solitary waves of various amplitudes from () u(,) u (,) u (,) u (,) (,) sec ( 5) sec (.5() ( )).5sec (.5( 5)) u h h h () Eample 9 We solve (5) using initial condition ().The calculation is carried out with the time step t =.57 and N = 5 over the region -7 7, we solve using the FLF scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. Figure 5 shows the numerical solution at t =, 5,,,,.As can be seen the three pulses travel with time to the right. But the taller soliton moves faster, hence the three occasionally merge and then split apart again. Eample We solve eample 9 using the RK scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. The calculation is carried out with the time step t =. and N = 5 over the region 7 7, we solve using the RK scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. Figure and 7 shows simulation and plane view of the solution computed.. t =. t = 5.............. - - - (a) - - - (b)
u(,t) u(,t) u(,t) u(,t) Hany N. HASSAN et al.. t =. t =.............. - - - (c) - - - (d). t =. t =.............. - - - - - - (e) (f) Fig. 5: Fourier spectral solution for interaction of three waves of the KdV equation using FLF scheme with initial condition () and N = 5. Fig. : Numerical simulation of Fourier spectral of interaction of three waves of the KdV equation using RK scheme with initial condition () and N =5.
Hany N. HASSAN et al. Fig. 7: Plan view of Fourier spectral of interaction of three waves of the KdV equation using RK scheme with initial condition () and N = 5... Other solutions Now suppose that the initial condition is such that is does not just produce one or more solitons. Let us choose as initial condition [] u(,) sec h ( ) () with =, =, L =,Δt =.9 and N = using the RK scheme to march the solution in time and the Fourier spectral method to take care of the spatial domain. Four is not of the form n (n + ).The peak moving to the right. In addition, there are waves moving to the left. These will disperse and lose their form with time.
u(,t) u(,t) u(,t) Hany N. HASSAN et al. 5 Fig. : Numerical simulation of Fourier spectral solution of the KdV equation using RK scheme with initial condition () and N =. 5 t = - - (a) 5 t =. 5 t = - - - - - - (b) (c) Fig. 9: Fourier spectral solution of the KdV equation using RK scheme with initial condition () and N =.
Hany N. HASSAN et al. Analysis the method for the Boussinesq equation. Fourier based RK method for Boussinesq equation Consider the Boussinesq equation () in the following form: utt u uu u u, [ a, z] (5) We solve this equation by combination of RK method with respect to t and a Fourier pseudospectral method with respect to. To prepare the equation for numerical solution we introduce the auiliary variable v ut. This reduces the second-order equation in time to the first order system ut v () vt u uu u u We need two initial conditions. The initial conditions we use to numerically solve equation (5) can thus be etracted from the above relation () for t = at u(, t) and ut (, t ) u(,) b sec h ( b( )) v u b c h b b (7) (,) t (,) sec ( ( )) tanh( ( )) We have changed the solution interval from [a, z] to [,π], with the change of variable. a L where L = z a, thus the equation () become u v t () vt u uu u u L L L L u(, t) is transformed into Fourier space with respect to, and derivatives (or other operators) with respect to. Applying the inverse Fourier transform using u F { ikf ( u )} The equation () becomes u F k F u u F { k F( u)} { ( )}
Hany N. HASSAN et al. 7 u t v vt F { k F( u)} uf { k F( u)} L L F { ikf( u)} F { k F( u)}, L L In practice, we need to discretize the equation (). For any integer N >, we consider j j, j,,, N. N Let u (, t), v(, t) be the solution equation (9). Then, we transform it into the Discrete Fourier space as N ik N N j uˆ( k, t) F( u) u( j, t) e, k N J N ik N N j vˆ( k, t) F( v) v( j, t) e, k. N J from this, using the inversion formula, we get ( N, ) ( ˆ) ˆ j (, ) ik j, kn u t F u u k t e j N ( N, ) ( ˆ) ˆ j (, ) ik j, kn v t F v v k t e j N replacing F and F in (9) by their discrete counterparts, and discretizing (9) gives du( j, t) v( j, t) dt dv( j, t) F { k F( u)} u( j, t) F { k F( u)} dt L L () F { ikf( u)} F { k F( u)} L L (9) Letting U = [ u(, t), u(, t),, u(, )] T N t, V = [ v(, t), v(, t),, v(, )] T N t, U Ut w=, w t = V. V t The system of equations () can be written in the vector form w t = F (U, V) () where F defines the right hand side of ().
Hany N. HASSAN et al.. Combination of finite differences and a Fourier pseudospectral method The numerical scheme used is based on a combination of finite differences and a Fourier pseudospectral method. After we have changed the solution interval from [a, z] to [,π] and u(, t) is transformed into Fourier space with respect to, and derivatives (or other operators) with respect to, equation (5) become utt F { k F( u)} uf { k F( u)} L L F { ikf( u)} F { k F( u)}. L L In practice, we need to discretize the equation (). For any integer N >, Let u (, t) be the solution equation (). Then, we transform it into the Discrete Fourier space using the inversion formula, we get d u( j, t) { ( )} (, ) { ( )} F k F u u j t F k F u dt L L () F { ikf( u)} F { k F( u)}. L L T [ u(, t), u(, t),, u( N, t)]. Letting U = The equation () can be written in the vector form U tt = F (U) () where F defines the right hand side of (). The time derivative in equation () is discretized using a finite difference approimation, in terms of central differences n n n u u u F (U) ( t) or u(, t t) u(, t) u(, t t) ( t) F { k F( u)} L L u( j, t) F { k F( u)} F { ikf( u)} F { k F( u)}. L L we need two level initial conditions u (,-Δt),u (,). The initial condition u (, ) given in (7), from the central difference of ut (, t ) () (5)
Hany N. HASSAN et al. 9 n n u u ut (, t) t where Δt is time step, we have the approimation n n u u tut (, t) we can get u (,-Δt) u(, t) u(, t) tu (,) t u(, t) t b c sec h ( b( )) tanh( b( )). then we substitute u (,-Δt) and u (,) in () to get u (, Δt) ( t) u(, t) u(,) t ut (,) F { k F( u)} L L () u( j,) F { k F( u)} F { ikf( u)} F { k F( u)}. L L So we substitute u (,) and u (, Δt) in () to get u (, Δt) and so on until we get u(, t) at time t. Various values of N ( to ) and time step Δt =. to... Numerical results and eamples In order to show how good the numerical solutions are in comparison with the eact ones, we will use L and L error norms. L u u u u N eact num eact num i i i eact num eact num L u u ma u u. To implement the performance of the method, test problems will be considered: the motion of a single solitary wave in right direction, the motion of a single solitary wave in left direction. Several tests have been made for the wave solution of the Boussinesq verifying that for various values of N ( to ) and time step t =. to. using Combination of finite differences and a Fourier pseudospectral method. As b increases, the stability of the wave propagating in time breaks down and for values of b close to.5 the wave blows up. For stability, we have found that the maimum value of b is.. We take the Boussinesq equation of form (5) with periodic boundary condition u( a, t) u( z, t) i i i
u(,t) Hany N. HASSAN et al. 7 Eample We solve the equation (5) with the initial conditions (7),with b =.,, c proposed method performs the motion of propagation of a solitary wave satisfactorily, which moved to the right with the preserved amplitude. b and Δ =, with Δt =..It is clear from Figure that the t L L..9..7 9.57.9.. Table : Error norms of Fourier spectral solution of the Boussinesq equation using RK scheme at t =,, and with N =. N L L Amplitude...7955 5..7.795 5..95.797 Table : Error norms of Fourier spectral solution of the Boussinesq equation using RK scheme with N =, 5 and 5...5 t - - - Fig. Numerical simulation of Fourier spectral solution of the Boussinesq equation using RK scheme with c b and N =.
Error u(,t) t Hany N. HASSAN et al. 7-5 5 Fig.. Plan view of Fourier spectral solution of the Boussinesq equation using RK scheme with c b and N =...7. Numerical Eact.5.... -. - - - Fig.. Fourier spectral solution of the Boussinesq equation using RK scheme at t = with c b and N =. 5 - -5 - - - Fig.. The error distributions in Fourier spectral solution of the Boussinesq equation using RK scheme at t = with c b and N =.
u(,t) u(,t) Hany N. HASSAN et al. 7..7..5 t = t = t = t = t =.... -. - - - Fig.. Fourier spectral solution of the Boussinesq equation using a finite difference scheme at t =,,, and with c b and N =...7 Numerical Eact..5.... -. - - - Fig.5. Fourier spectral solution of the Boussinesq equation using a finite difference scheme at t = with c b and N =.
Error Hany N. HASSAN et al. 7 5 - - - - - -5 - - - Fig..The error distributions of Fourier spectral solution of the Boussinesq equation using a finite difference scheme at t = with c b and N =. t L L.5.5.7.77 9.5.9.7.95 Table 5. Error norms of Fourier spectral solution of the Boussinesq equation using a finite difference scheme at t =,,, and with N =. N L L Amplitude.7.95.7955 5.55..795 5.957.5.797..55.799 Table. Error norms for the single soliton of Fourier spectral solution of the Boussinesq equation using a finite difference scheme with N =,5, 5 and. Now, we solve the same problem with b =., =, c b and Δ = with Δt =.. It is clear from Figure 7 that the proposed method performs the motion of propagation of a solitary wave satisfactorily, which moved to the left with the preserved amplitude.
t Hany N. HASSAN et al. 7 u(,t)..5 t - - - Fig. 7. Numerical simulation of Fourier spectral solution of the Boussinesq equation using RK scheme with c b and N =. -5 5 Fig.. Plan view of Fourier spectral solution of the Boussinesq equation using RK scheme with c b and N =.
Error u(,t) Hany N. HASSAN et al. 75..7 Numerical Eact..5.... -. - - - Fig.9. Fourier spectral solution of the Boussinesq equation using RK scheme at t = with c b and N =. 5 - -5 - - - Fig.. The error distributions of Fourier spectral solution of the Boussinesq equation using RK scheme at t = with c b and N =.
u(,t) u(,t) Hany N. HASSAN et al. 7..7..5 t = t = t = t = t =.... -. - - - Fig.. Fourier spectral solution of the Boussinesq equation using a finite difference scheme at c b, t =,,, and with N =...7 Numerical Eact..5.... -. - - - Fig.. Fourier spectral solution of the Boussinesq equation using a finite difference scheme at c b, t = with N =.
Error Hany N. HASSAN et al. 77 5 - - - - - -5 - - - Fig.. The error distributions in Fourier spectral solution of the Boussinesq equation using a finite difference scheme with c b at t = with N =. Conclusions In Our study, we applied Fourier spectral collocation methods to solve partial differential equations of the form ut F ( u) particularly the Korteweg-de Vries equation (KdV), and the form utt F ( u) particularly the Boussinesq equation. A finite differences or a fourth-order Runge-Kutta method is used to march our solution in time, with the help of the fast Fourier transform algorithm our methods cost only (number of operations) O N log N, provided that N is a power of. We applied the leap-frog scheme combined with the Fourier spectral collocation, called the Fourier Leap-Frog method, to find numerical solution of the KdV equation, with α =, and β =. Also we applied the fourth-order Runge-Kutta (RK) method combined with the Fourier spectral collocation to the same problem. We presented stability conditions for these schemes, and we found out that the Fourier Leap-Frog method is less stable and far less accurate compared to the classical RK scheme. Even though it is quite costly to use, RK is easy to implement and needs only one level of initial data, whereas two levels of the initial data are needed in the Fourier Leap-Frog methods. We then applied the finite difference scheme combined with the Fourier spectral collocation to find numerical solution of the Bossinesq equation. Also we applied the fourth-order Runge-Kutta (RK) method combined with the Fourier spectral collocation to the same problem. In order to show how good the numerical solutions are in comparison with the eact ones, we will use L and L error
Hany N. HASSAN et al. 7 norms. It is apparently seen that Fourier spectral collocation method is powerful and efficient technique in finding numerical solutions for wide classes of nonlinear partial differential equations. 5 Open Problem The work presented in this paper transform some types of partial differential equations like the Korteweg-de Vries equation (KdV), Boussinesq equation to simple ordinary differential equations using Fourier spectral method, can be solved by simple techniques ( Leap frog, finite difference, Runge Kutta, ), the question here what the results of using other spectral methods to solve these equations instead of Fourier spectral method. The other question, in this paper we applied the used method to solve solitons, what happened when we solve equations by non solitons spectral methods. References [] Fornberg B. A Practical Guide to Pseudospectral Methods. Cambridge University Press, New York. 99. [] Fornberg B and Whitham G B. A numerical and Theoretical Study of Certain Nonlinear Wave Phenomena. Philos. Trans. Soc. London, 9, May 97, pp. 7-. [] Hany N. Hassan and Hassan K. Saleh, The Solution of the Regularized Long Wave Equation Using the Fourier Leap-Frog Method, Z. Naturforsch. 5a, 7 () [] Boyd J P. Chebyshev and Fourier Spectral Methods. Second Edition Dover Publications Inc., New York,. [5] Whitham G B. Linear and Nonlinear Waves. John Wiley &Sons, 97. [] Weideman, J.A.C., and Reddy, S.C., A MATLAB Differentiation Matri Suite, ACM Transactions on Mathematical Software,, No., December, pp. 5 59. [7] Efstratios, E., Tzirtzilakis, Charalampos D., Skokos, and Tassos C. Bountis, Numerical Solution of the Bossinesq equation using Spectral Methods and Stability of Solitary Wave Propagation, first International Conference From Scientific Computing to Computational Engineering, Athens, - September,. [] Eric Gourgoulhon, Introduction to spectral methods, th EU Network Meeting, Palma de Mallorca, Sept.. http://www.luth.obspm.fr [9] Tzirtzilakis, E., Marinakis, V., Apokis, C., and Bountis, T., Soliton-like solutions of higher order wave equations of the Korteweg de Vries type, Journal of Mathematical Physics, No., December, pp. 5 5.
Hany N. HASSAN et al. 79 [] Bengt Fornberg and Tobin A. Driscoll, A Fast Spectral Algorithm for Nonlinear Wave Equations with Linear Dispersion, Journal of Computational Physics55, 999, pp.5 7. [] Randall, J. LeVeque, Finite Difference Methods for Differential Equations, A Math 55, Winter Quarter, University of Washington, Version of January,. [] Drazin, P.G. and Johnson, R.S., Solitons: an introduction, Cambridge University Press, Cambridge, 99. [] Idris Dag, M. Naci Ozer, Approimation of the RLW equation by the least square cubic B-spline finite element method Applied Mathematical Modelling 5 () - [] H. Jafari, H. Hosseinzadeh and S. Mohamadzadeh Numerical Solution of System of Linear Integral Equations by using Legendre Wavelets Int. J. Open Problems Compt. Math., Vol., No. 5, December [5] Hany N. Hassan and M. A. El-Tawil, A new technique of using homotopy analysis method for solving high-order nonlinear differential equations, Mathematical Methods in the Applied Sciences,, 7 7 ().