Du Fort–Frankel finite difference scheme for Burgers equation

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 7.0 for different values of viscosity. We have considered smallest value of viscosity as 10−4 and observe that the numerical solutions are in good agreement with the exact solution.


Introduction
The Burgers equation which is going to be examined is where ν d > 0 is the coefficient of viscous diffusion, and f (x), g 1 (t) and g 2 (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 various numerical algorithm. Due to its wide range of applicability, 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) 2 + t x with consistency condition t x → 0 as t, x → 0. 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] Douglas finite difference scheme is considered which is unconditionally stable. 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 [2] different techniques for the solution of nonlinear Burgers equation are presented.
In this paper we consider Du Fort-Frankel [8, p. 102] finite difference scheme which is unconditionally stable and has local truncation error O(( x) 2 +( t) 2 +( t/ x) 2 ) which tends to zero as ( x, t) → (0, 0) provided ( t/ x) → 0. 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 = 10 −4 ).

Exact solution
Hopf [4] and Cole [1] suggested that (1.1) can be reduced to the linear heat equation by the non-linear transformation ψ = −ν d (log φ) and w = ψ x . The Fourier series solution to the linearized heat equation (2.1) is with Fourier coefficients as where w 0 (ξ ) = w(ξ, 0). Using the Hopf-Cole transformation we have the exact solution

Description of the method
The solution domain is discretized into uniform mesh. We divide the interval [0, 1] into N equal sub-intervals and divide the interval [0, T ] into M equal sub-intervals. Let h = 1/N be the mess width in space and x i = ih for i = 1, 2, . . . , N . Let k = T /M be the mess width in time and t j = jk for k = 0, 1, . . . , M.
Du Fort-Frankel discretization [8, p. 102] to linearized heat equation (2.1) is given by The approximate solution of Burgers equation (1.1) in terms of approximate solution of heat equation (2.1) using Hopf-Cole transformation is given by . It is stable for all values of r and has the truncation error of O k 2 + h 2 + k h 2 which will tend to zero as (h, k) → (0, 0) provided k h → 0. 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 to obtain the values at first row ( j = 1).

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, 2 (4. 2) The exact solution of the Burgers equation (1.1) is (2.3) with given Fourier coefficients In Table 2 we have compared our computed numerical solutions with the exact solution for N = 10, k = 0.001 and ν d = 2. In Table 3 Table 14 we compare our results with results of Kutluay et al. [5] for ν d = 0.02, h = 0.0125 and k = 0.0001. This comparison shows that Du Fort-Frankel is giving good results.

Problem 2
As a second example consider (1.1) with the boundary condition (4.1) and initial condition w( 3) can be obtained in the similar fashion as in Problem 4.1 with the Fourier coefficients as follows:     In Table 2 we have compared our computed numerical solutions to the exact solutions for N = 10, k = 0.001 and ν d = 2. In Table 4 we compare the numerical solutions with the exact solutions for different values of ν d (0.5, 0.125, 0.03125) and N (20, 40, 80). In Tables 6, 9 and 12 we have displayed the numerical In Table 1 we have compared our computed numerical solutions to the exact solutions for N = 10, k = 0.001 and ν d = 2. In Table 4 we compare the numerical solutions with the exact solutions for different values of ν d (0.5, 0.125, 0.03125) and N (20, 40, 80). In Tables 7, 10 and 13 we have displayed the numerical and analytical solutions for very small values of ν d , e.g., 10 −2 , 10 −3 and 10 −4 .

Figures
In this section we describe the physical properties of the solutions using 2D and 3D plots (Figs. 1, 2, 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 (10, 40, 80), ν d (2, 0.125, 0.03125) and T = 0.1 for Problems 4.1 and 4.2. The truncation errors of Du Fort-Frankel method is O(h 2 + k 2 + (k/ h) 2 ) and that of Douglas scheme is O(h 4 + k 2 ). 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 = 0.1 and k = 0.001. 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(10 −6 ) for both) so we observe that Du Fort-Frankel method gives values as accurate as that of Douglas.

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.