Du Fort Frankel finite difference scheme for Burgers equation

Transcription

1 Arab J Math (13) :91 11 DOI 1.17/s RESEARCH ARTICLE K. Pandey Lajja Verma Amit K. Verma Du Fort Frankel finite difference scheme for Burgers equation Received: 8 February 1 / Accepted: 1 September 1 / Published online: 11 October 1 The Author(s) 1. This article is published with open access at Springerlink.com Abstract In this paper we apply the Du Fort Frankel finite difference scheme on Burgers equation and solve three test problems. We calculate the numerical solutions using Mathematica 7. for different values of viscosity. We have considered smallest value of viscosity as 1 4 and observe that the numerical solutions are in good agreement with the exact solution. Mathematics Subject Classification 65N6 (Du Fort-Frankel) Mathematica 7. 1 Introduction The Burgers equation which is going to be examined is with the initial condition w t and the boundary conditions + w w x ν d w =, (x, t) (, 1) (, T ] (1.1) x w(x, ) = f (x), x 1, (1.) w(, t) = g 1 (t), w(1, t) = g (t), t T, (1.3) K. Pandey Department of Mathematics and Astronomy, University of Lucknow, Lucknow 67, India pandey_kanti@yahoo.co.in L. Verma (B) A. K. Verma Department of Mathematics, BITS Pilani, Pilani 33331, Rajasthan, India lajjamaths@yahoo.co.in A. K. Verma amitkverma@yahoo.co.in 13

2 9 Arab J Math (13) :91 11 where ν d > is the coefficient of viscous diffusion, and f (x), g 1 (t) and g (t) are the sufficiently smooth given functions in space and time domain. Burgers equation (1.1) can model several physical phenomenon such as traffic flow, shock waves, turbulence problems, cosmology, seismology and continuous stochastic processes. It can also be used to test the variousnumericalalgorithm.dueto itswiderangeofapplicability,researchers [3,5,7] were attracted to it and studied properties of its solution using various numerical techniques. In [3] the Group-Explicit method is introduced which is semi explicit, unconditionally stable and is of order O ( t + ( x) + x t ) with consistency condition t x as t, x. In [5] using the Hopf Cole transformation the Burgers equation is reduced into linear heat equation and a standard explicit finite difference approximation is derived. Then assuming that this explicit finite difference scheme has product solution, they derived exact explicit finite difference solution which converges to the Fourier solution as mesh size tends to zero. In [7] Douglasfinitedifferencescheme is consideredwhichis unconditionallystable. In [6] a numerical method is proposed to approximate the solution of the nonlinear Burgers equation which is based on collocation of modified cubic B-splines over finite elements. They computed the numerical solutions to the Burgers equation without transforming the equation and without using the linearization. In a recent review article [] different techniques for the solution of nonlinear Burgers equation are presented. In this paper we consider Du Fort Frankel [8, p. 1] finite difference scheme which is unconditionally stable and has local truncation error O(( x) +( t) +( t/ x) ) which tends to zero as ( x, t) (, ) provided ( t/ x). Du Fort Frankel method is explicit, so matrix inversions are not required for computations and it is therefore simpler to program and cheaper to solve (on a per time-step basis). We compare the absolute errors for our results to the absolute errors of Douglas finite difference scheme [7] and present this comparison using graphs. It can be observed that if we can have little control over the mesh sizes then the results are promising even for very small coefficient of viscosity (ν d = 1 4 ). Exact solution Hopf [4]andCole[1] suggested that (1.1) can be reduced to the linear heat equation φ t = ν d φ xx (.1) by the non-linear transformation ψ = ν d (log φ) and w = ψ x. The Fourier series solution to the linearized heat equation (.1)is with Fourier coefficients as and φ(x, t) = A + A n = A = n=1 ( A n exp ν dn π ) t cos nπ x (.) exp 1 ν d exp 1 ν d x x w (ξ)dξ dx w (ξ)dξ cos(nπ x)dx where w (ξ) = w(ξ, ). Using the Hopf Cole transformation we have the exact solution ( ) n=1 A n exp ν dn π t n sin nπ x w(x, t) = πν d A + ( ). (.3) n=1 A n exp cos nπ x ν dn π t 13

3 Arab J Math (13) : Description of the method The solution domain is discretized into uniform mesh. We divide the interval [, 1] into N equal sub-intervals and dividetheinterval [, T ] into M equal sub-intervals. Let h = 1/N be the mess width in space and x i = ih for i = 1,,...,N. Letk = T/M be the mess width in time and t j = jk for k =, 1,...,M. Du Fort Frankel discretization [8, p. 1] to linearized heat equation (.1)isgivenby (1 + r)φ i, j+1 = (1 r)φ i, j 1 + r(φ i 1, j + φ i+1, j ) (3.1) where r = ν dk h is the discrete approximation to φ(x i, t j ) at the point (i, j). The approximate solution of Burgers equation (1.1) in terms of approximate ( solution of ) heat equation (.1) using Hopf Cole transformation is given by w i, j (x, t) = ν dφ x φ = ν d φi+1, j φ i 1, j i, j hφ i, j. It is stable for all values of r and has the truncation error of O (k + h + ( ) k ) h which will tend to zero as (h, k) (, ) provided h k. Initial data are given on one-line only; the first row ( j = 1) of values must be calculated by another method. Here we use Schmidt process φ i,1 = rν d (φ i+1, + φ i 1, ) + (1 rν d )φ i, to obtain the values at first row ( j = 1). 4 Numerical results and discussion In this section we demonstrate the accuracy of the present method by solving three test problems and compare it with the exact solution at different nodal points.the computed results are displayed in Tables 1,, 3, 4, 5, 6, 7, 8, 9, 1, 11, 1, 13, 14, 15, 16 and Figs. 1,, 3, 4, 5, 6 at different nodal points for different values of viscosity. 4.1 Problem 1 Consider Equation (1.1) with boundary conditions and initial condition as w(, t) = w(1, t) =, t >, (4.1) w(x, ) = sin π x. (4.) The exact solution of the Burgers equation (1.1)is(.3) with given Fourier coefficients A = ( ) 1 exp (1 cos π x) dx, A n = πν d ( ) 1 exp (1 cos π x) cos nπ x dx. πν d In Table we have compared our computed numerical solutions with the exact solution for N = 1, k =.1 and ν d =. In Table 3 we compare the numerical solutions with the exact solutions for different values of ν d (.5,.15,.315) and N (, 4, 8). InTables5, 8 and 11 we have displayed the numerical and analytical solutions for very small ν d values 1,1 3 and 1 4.InTable14 we compare our results with results of Kutluay et al. [5] forν d =., h =.15 and k =.1. This comparison shows that Du Fort Frankel is giving good results. 4. Problem As a second example consider (1.1) with the boundary condition (4.1) and initial condition w(x, ) = 4x(1 x), < x < 1. The exact solution (.3) can be obtained in the similar fashion as in Problem 4.1 with the Fourier coefficients as follows: A = ( x ) exp (3 x) dx, A n = 3ν d ( x ) exp (3 x) cos(nπ x)dx. 3ν d 13

4 94 Arab J Math (13) :91 11 Table 1 Comparison of numerical solutions with the exact solutions at different space points of Problems 4.1, 4., 4.3 at T =.1, for ν d =, h =.1andk =.1 x Problem 4.1 Problem 4. Problem 4.3 Computed solution Exact solution Computed solution Exact solution Computed solution Exact solution Table Comparison of numerical solutions with the exact solutions at different space points at T =.1 andk =.1 for different values of ν d for Problem 4.1 x h =.5, ν d =.5 h =.5, ν d =.15 h =.15, ν d =.315 Computed solution Exact solution Computed solution Exact solution Computed solution Exact solution Table 3 Comparison of numerical solutions with the exact solutions at different space points at T =.1 andk =.1 for different values of ν d for Problem 4. x h =.5, ν d =.5 h =.5, ν d =.15 h =.15, d =.315 Computed solution Exact solution Computed solution Exact solution Computed solution Exact solution Table 4 Comparison of numerical solutions with the exact solutions at different space points at T =.1 andk =.1 for different values of ν d for Problem 4.3 x h =.5, ν d =.5 h =.5, ν d =.15 h =.15, ν d =.315 Computed solution Exact solution Computed solution Exact solution Computed solution Exact solution Can not be computed using Mathematica

5 Arab J Math (13) : Table 5 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4.1 at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N = Table 6 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4. at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N = Table 7 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4.3 at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N = Table 8 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4.1 at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N =

6 96 Arab J Math (13) :91 11 Table 9 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4. at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N = Table 1 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4.3 at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N = Table 11 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4.1 at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N = Table 1 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4. at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N = E

7 Arab J Math (13) : Table 13 Comparison of the numerical solutions with the exact solutions at different space points of Problem 4.3 at T = 1 for ν d =.1 and k =.1 N = 1 N = N = 4 N = Table 14 Comparison of the numerical solutions with the exact solutions at different times of Problem 4.1 for ν d =., h =.15 and k =.1 x T Numerical solutions Kutluay et al. [5] Du Fort Frankel Exact solution Table 15 Comparison of the numerical solutions with the exact solutions at different times of Problem 4. for ν d =, h =.5 and k =.1 x T Numerical solutions Kutluay et al. [5] Du Fort Frankel Exact solution

8 98 Arab J Math (13) :91 11 Table 16 Comparison of the numerical solutions with the exact solutions at different times of Problem 4. for ν d =., h =.15 and k =.1 x T Numerical solutions Kutluay et al. [5] Du Fort Frankel Exact solution Fig. 1 Numerical solution at different times for N = 4, ν d =.15 and k =.1 for a Problem 4.1 and b Problem 4. Fig. Numerical solutions of Problem 4.1 at different times for N = 8, ν d =.315 and k =.1 for a Problem 4.1 and b Problem 4. 13

9 Arab J Math (13) : Fig. 3 Numerical solutions of Problem 4.3 at different times for a N = 4, ν d =.15 and k =.1, b N = 8, ν d =.315 and k =.1 Fig. 4 Comparison of the absolute error for N = 1, T =.1, ν d = andk =.1 for a Problem 4.1 and b Problem 4. Fig. 5 Comparison of the absolute error for N = 4, T =.1, ν d =.15 and k =.1 for a Problem 4.1 and b Problem 4. In Table we have compared our computed numerical solutions to the exact solutions for N = 1, k =.1 and ν d =. In Table 4 we compare the numerical solutions with the exact solutions for different values of ν d (.5,.15,.315) and N (, 4, 8).InTables6, 9 and 1 we have displayed the numerical 13

10 1 Arab J Math (13) :91 11 Fig. 6 Comparison of the absolute error for N = 8, T =.1, ν d =.315 and k =.1 for a Problem 4.1 and b Problem 4. and analytical solutions for very small ν d values 1,1 3 and 1 4.InTables15 and 16 we show that our results are as good as the results of Kutluay et al. [5] forν d =, h =.5, k =.1 and ν d =., h =.15, k = Problem 3 π sin π x Consider Equation (1.1) with boundary condition (4.1) and initial condition as w(x, ) = +cos π x. The exact solution (.3) of the equation (1.1) can be obtained in the similar fashion with the Fourier coefficients as follows: A = ( + cos π x 3 ) νd dx, An = ( + cos π x 3 ) νd cos nπ xdx. In Table 1 we have compared our computed numerical solutions to the exact solutions for N = 1, k =.1 and ν d =. In Table 4 we compare the numerical solutions with the exact solutions for different values of ν d (.5,.15,.315) and N (, 4, 8). InTables7, 1 and 13 we have displayed the numerical and analytical solutions for very small values of ν d, e.g., 1,1 3 and Figures In this section we describe the physical properties of the solutions using D and 3D plots (Figs. 1,, 3). In Figs. 4, 5 and 6 we verify the accuracy of the present method by comparing the absolute error of our result to the absolute error of Douglas finite difference scheme [7] for different N (1, 4, 8), ν d (,.15,.315) and T =.1 for Problems 4.1 and 4.. The truncation errors of Du Fort Frankel method is O(h + k + (k/h) ) and that of Douglas scheme is O(h 4 + k ). Since we have kept values of time step k smaller than h, i.e., k < h so truncation error is more for Douglas method. This can easily be observed in Fig. 4a, b when h =.1 and k =.1. Similar trends can be observed in Figs. 5 and 6. In these two figures for chosen values of h and k we obtain same accuracy (order of truncation error is O(1 6 ) for both) so we observe that Du Fort Frankel method gives values as accurate as that of Douglas. 6 Conclusions Since exact solutions fail to converge if ν d or T is small. Therefore, while computing numerical solutions for small values of ν d we have kept value of T high so that we can compute exact solutions and thus numerical solutions are verified. Computed results show that if we can keep the ratio k/h sufficiently small, good results can be obtained. Since Du Fort Frankel method does not require matrix inversion, it is easy to program and takes less time to compute. 13

11 Arab J Math (13) : Acknowledgments We are thankful to the reviewers for their expert comments and valuable suggestions. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. References 1. Cole, J.D.: On a quasilinear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 5 36 (1951). Dhawan, S.; Kapoor, S.; Kumar, S.; Rawat, S.: Contemporary review of techniques for the solution of nonlinear Burgers equation. J. Comput. Sci. doi:1.116/j.jocs Evans, D.J.; Abdullah, A.R.: The group explicit method for the solution of Burgers equation. Computing 3, (1984) 4. Hopf, E.: The partial differential equation u t + uu x = νu xx. Commun. Pure Appl. Math. 3, 1 3 (195) 5. Kutluay, S.; Bahadir, A.R.; Ozdes, A.: Numerical solution of one-dimensional Burgers equation: explicit and exact explicit methods. J. Comput. Appl. Math. 13, (1999) 6. Mittal, R.C.; Jain, R.K.: Numerical solutions of nonlinear Burgers equation with modified cubic B-splines collocation method. Appl. Math. Comput. 18(15), (1) 7. Pandey, K.; Verma, L.; Verma, A.K.: On a finite difference scheme for Burgers equation. Appl. Math. Comput. 15, 6 14 (9) 8. Smith, G.D.: Numerical Solution of Partial Differential Equations. Oxford University Press, New York (1978) 13