Crank Nicolson Implicit Method For The Nonlinear -Books Pdf

Crank Nicolson Implicit Method For The Nonlinear
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Since the analytical solutions for the NLS equation are limited numerical methods have become important in. order to understand the physical behavior of the equation Many researchers are devoted to study the numerical. solution of NLS equation Numerous numerical methods have been investigated such as the discrete Adomian. decomposition 7 the finite difference 8 the finite element 9 and the multi symplectic Runge Kutta 10. In the present work we proposed the Crank Nicolson implicit method for solving the NLS equation with. variable coefficient The truncation errors for the present method are second order in time and space directions The. stability analysis shows that the present method is unconditionally stable and satisfies discrete conservation laws. The numerical results obtained by the present method are compared with the exact solutions 1 It shows that the. Crank Nicolson method is compatible with the analytical result. Our paper is organized as follows in Section 2 we will introduce the Crank Nicolson implicit method to solve. the NLS equation with variable coefficient We investigated the stability analysis for equation 1 in Section 3 In. Section 4 we present the numerical results and compared them with the analytical results Section 5 ends this paper. with a conclusion,THE CRANK NICOLSON IMPLICIT METHOD. The analytical solution of the equation 1 mention U as a function of and W U W where both and W. are continuous variables For the finite difference method we seek approximation U mn to the original function. U W at a set of points m W n on a rectangular grid in the 2 dimensional plane and W where m 0 m. Wn W 0 n W and W are the grid spacing in and W respectively For simplicity 0 and W 0 are set to be. 0 0 and W 0 0 in our following discussion, The main idea of Crank Nicolson implicit method is to produce the same order of truncation error in W and. variables For this reason the forward difference for W derivative in equation 1 is replaced with the backward. difference approximation this gives,wU W U W U W W W w 2U y. wW W 2 wW 2, for some y in W W W The derivative in equation 1 is the usual central difference approximation. w 2U W U W 2U W U W w 4U W x, for some x in Then the equation 1 can be written as.
U n U mn 1 U n 2U n U n,P2 U mn U mn P3 h1 W U mn,for m M M 1 0 1 M n 0 1 2 N. Equation 4 is the implicit backward difference equation at the nth step in W with the truncation error of order. O W 2 For the backward difference equation at the n 1 th step in W the equation 4 becomes. U n 1 U mn U n 1 2U n 1 U n 1,P2 U mn 1 U mn 1 P3 h1 W U mn 1. for m M M 1 0 1 M n 0 1 2 N, The order of the error in W and variables can be made equal by taking the average of equation 5 and forward. difference equation for equation 1 11 this gives, This article is copyrighted as indicated in the article Reuse of AIP content is subject 77. to the terms at http scitation aip org termsconditions Downloaded to IP 103 31 34 2. On Mon 23 Feb 2015 07 52 04, U n 1 U mn P1 U mn 1 2U mn U mn 1 U mn 11 2U mn 1 U mn 11 P2 n 2 n P.
U m U m U mn 1 U mn 1 3 h1 W U mn U mn 1,for m M M 1 0 1 M n 0 1 2 N. where the truncation error is of order O W 2 2 Rearrange the above expression yield the difference equation. for the Crank Nicolson implicit method,rU mn 11 i P1 r 3 W h1 W U mn 1 1 rU mn 11. P1 P3 n P2,i P1r 2 W h1 W U m 2 W U m U m U m U m,U n n2 n n 1 2 n 1. The solution for the above difference equation 7 is sought in the region M d m d M u W n d N W. m M M n 0 1 2 N Since equation 1 is a boundary value problem so the values of U at time. step n 0 are known The right hand side of equation 7 consists of both of the known values U at time step n. as well as the unknown values U at time step n 1 In order to solve the above equation 7 we apply the. Functions Arguments which can be referred to using MATLAB package. STABILITY ANALYSIS, In this section we investigate the stability analysis of the Crank Nicolson implicit method that we have. discussed in Section 2 First we linearized the NLS equation with variable coefficient 1 We later obtained the. difference equation by applying the Crank Nicolson implicit method to the linearized NLS equation To analyze the. stability of the numerical scheme we use Fourier series method which leads to the analysis known as von Neumann. stability test, The most common linearization methods are Taylor s series expansion optimal linearization method and global.
linearization method In this paper we adopt the method of Taylor s series expansion Consider the function f U. of a single variable U Suppose that U is a point such that f U 0 The point U is called equilibrium point of. the system U f U By expanding the NLS equation with variable coefficient 1 in Taylor series expansion of. f U this gives the linearized NLS equation with variable coefficient. P1 i P3U 0 8, Employing the Crank Nicolson implicit method to the linearized NLS equation with variable coefficient 8 we. have the difference equation,P1 n 1 P P P P P, rU m 1 i P1r 3 W U m n 1 1 rU m n 11 1 rU m n 1 i P1r 3 W U m n 1 rU m n 1 9. 2 2 2 2 2 2, with r For the von Neumann method a harmonic decomposition is made of the error E at grid points at. a given time level leading to the error function, This article is copyrighted as indicated in the article Reuse of AIP content is subject 78. to the terms at http scitation aip org termsconditions Downloaded to IP 103 31 34 2. On Mon 23 Feb 2015 07 52 04, where the frequencies E j and j are arbitrary To investigate the error propagation as time increases it is.
necessary to find a solution of the finite difference equation which reduces to eiE x when time is zero Apply. equation 10 into equation 9 and the Emn satisfies the same finite difference equation so we get. P1 P P P1 P P, rEmn 11 i P1r 3 W Emn 1 1 rEmn 11 rEmn 1 i P1r 3 W Emn 1 rEmn 1 11. 2 2 2 2 2 2,D nk iE mh,Substitute Emn e e equation 11 becomes. reD n 1 k eiE m 1 h i P1r 3 W eD n 1 k eiE mh 1 reD n 1 k eiE m 1 h. 1 reD nk eiE m 1 h i P1r 3 W eD nk eiE mh 1 reD nk eiE m 1 h. Cancellation of eD nk eiE mh leads to,i W 2P1 r sin 2. i 3 W 2P1 r sin 2, The quantity eD k is called the amplification factor For stability eD k d 1 for all values of E h Clearly the. modulus is at most one for all positive values of r Thus the Crank Nicolson implicit method is unconditionally. stable according to linear analysis However in the actual condition the simulation may become unstable because. nonlinear terms may play a dominant role in the dynamics. RESULTS DISCUSSION, In this section we apply the scheme of equation 7 to solve equation 1 Furthermore we also compared our.
proposed Crank Nicolson implicit method with the exact solution of NLS equation with variable coefficient In. comparing the numerical results with the exact results we calculate the maximum absolute error Lf at certain m. which is defined as,Lf max U exact U numerical 14, The exact solution for the NLS equation with variable coefficient 1 is given by 1. U W a tanh 2 a9 exp i K W P3 h1 s ds, where 2P1 KW and P1 K 2 P2 a2 The coefficients of P1 P2 and P3 can be obtained in 1 The h1 W. is the stenosis function which is defined as sech 0 30W In order to obtain a numerical solution we need the initial. condition by assuming 1 W 0 in equation 15 2 spatial step W 0 01 3 travelling wave profile step. 0 01 and 4 artificial boundary conditions U W 2 U W 5 0 The numerical results are presented over. the travelling wave profile interval 2 5 and the space interval 6 1 by choosing the parameter as a 1 K 2. TABLE 1 Maximum absolute errors of the NLS equation with variable coefficient 1 at W 0 01 0 01 for. Travelling wave profile 0 0 1 0 1 50 2 00 2 50 3 00. Lf 0 2630 0 2790 0 2818 0 2781 0 2790 0 2754, This article is copyrighted as indicated in the article Reuse of AIP content is subject 79. to the terms at http scitation aip org termsconditions Downloaded to IP 103 31 34 2. On Mon 23 Feb 2015 07 52 04,Radial displacement U W. 6 5 4 3 2 1 0 1, FIGURE 1 Crank Nicolson solution of the NLS equation with variable coefficient 1 with space W at certain travelling.
wave profile,Radial displacement U W,6 5 4 3 2 1 0 1. FIGURE 2 Exact solution of the NLS equation with variable coefficient 1 with space W at certain travelling wave profile. The maximum absolute errors Lf between the exact and numerical solutions of the NLS equation with variable. coefficient are shown in Table 1 Figure 1 depicts the Crank Nicolson implicit solution for the NLS equation with. variable coefficient It is seen that the numerical solution in Figure 1 is exactly the same as the exact solution in. Figure 2 in terms of position and amplitude, This article is copyrighted as indicated in the article Reuse of AIP content is subject 80. to the terms at http scitation aip org termsconditions Downloaded to IP 103 31 34 2. On Mon 23 Feb 2015 07 52 04, TABLE 2 Maximum absolute errors of the NLS equation with variable coefficient 1 at W 0 01 0 01 for. different W,Space W 6 4 2 0 1,Lf 0 4403 0 2839 0 2462 0 0000 0 0703. Radial displacement U W,2 1 0 1 2 3 4 5, FIGURE 3 Crank Nicolson solution of the NLS equation with variable coefficient 1 with travelling wave profile at.
certain space W,Radial displacement U W,2 1 0 1 2 3 4 5. Travelling wave profile, FIGURE 4 Exact solution of the NLS equation with variable coefficient 1 with travelling wave profile at certain space. This article is copyrighted as indicated in the article Reuse of AIP content is subject 81. to the terms at http scitation aip org termsconditions Downloaded to IP 103 31 34 2. On Mon 23 Feb 2015 07 52 04, Table 2 illustrates the maximum absolute errors Lf between the numerical and exact solutions of the NLS. equation with variable coefficient with different W The maximum absolute errors in Tables 1 and 2 are quite. large for most of the cases This is due to the number of spatial grid points are less and the step size of travelling. wave profile is large In Table 2 for W 0 the maximum error is small because it is the initial condition To. maintain the stability and to achieve the high accuracy of the approximation solution the number of spatial grid. points must be large and the step size of travelling wave profile must be small However this numerical scheme. required large computational cost if number of grid points for spatial or travelling wave profile increases. Comparing Figures 3 and 4 it is shown that the numerical results show good approximation with the analytical. results However this numerical scheme required extremely large computational cost. CONCLUSION, The Crank Nicolson implicit method with second order accurate in time and space direction is proposed for. solving the NLS equation with variable coefficient This method is shown to be unconditionally stable for the. linearized NLS equation with variable coefficient Numerical tests presented for the NLS equation with variable. coefficient show that our method is in agreement with the analytical method. ACKNOWLEDGEMENTS, The authors gratefully acknowledge financial support from the Registrar Office UTHM.
REFERENCES, 1 Y Y Choy C T Ong and K G Tay Mathematics Journal of Universiti Teknologi Malaysia 28 1 13 2012. 2 H Demiray J of Maths and Math Sci 60 3205 3218 2004. 3 G K Tay Y Y Choy C T Ong and H Demiray Int J Eng Sci and Tech 2 708 723 2010. 4 H Demiray Int J Eng Sci 36 1061 1082 1998,5 H Demiray Int J Eng Sci 41 1387 1403 2003. 6 H Demiray Applied Maths And Computation 145 179 184 2003. 7 A Bratsos M Ehrhardt and I T Famelis Applied Maths And Computation 197 190 205 2008. 8 M Dehghan and A Taleei Computer Physics Communications 181 43 51 2010. 9 J Argyris and M Haase Comput Methods Appl Mech Engrg 61 71 122 1987. 10 S Reich J Comput Phys 157 473 499 2000, 11 W N Tan Theory of Bichromatic Wave Groups Amplitude Amplification Using Implicit Variational Method Ph D. Thesis University of Technology Malaysia 2006, This article is copyrighted as indicated in the article Reuse of AIP content is subject 82. to the terms at http scitation aip org termsconditions Downloaded to IP 103 31 34 2.


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