An Efficient Multiscale Runge-Kutta Galerkin Method for Generalized Burgers-Huxley Equation

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1 Applied Mathematical Sciences, Vol. 11, 2017, no. 30, HIKARI Ltd, An Efficient Multiscale Runge-Kutta Galerkin Method for Generalized Burgers-Huxley Equation Jian Chen Department of Mathematics, Foshan University Foshan , P. R. China Copyright c 2017 Jian Chen. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, an efficient multiscale Runge-Kutta Galerkin method (MRKGM) efficient numerical solution for the generalized Burgers-Huxley equation is presented. Firstly, the spacial variable is discretized by multiscale Galerkin method with the multiscale orthonormal bases in H0 1 (0, 1). This method yields Burgers-Huxley equation to a system of ordinary differential equations. Secondly, the strong stability preserving third-order Runge-Kutta method is employed to solve the system of ordinary differential equations. The numerical results obtained in this way are compared with the exact solution to demonstrate the high accuracy of the method even in the case of a small number of bases. Mathematics Subject Classification: 65J15; 65M60 Keywords: multiscale Galerkin method, multiscale orthonormal bases, Runge-Kutta method, generalized Burgers-Huxley equations 1 Introduction Mathematical modeling of many physical systems leads to nonlinear partial differential equations in various fields of science and engineering. Generalized Burgers-Huxley equation u t + αu δ u x u xx = βu(1 u δ )(u δ γ), a x b, t 0, (1.1)

2 1468 Jian Chen where α, β, γ and δ are constants with β 0, δ > 0, γ (0, 1), being a nonlinear partial differential equation is of great importance for describing different mechanisms. This equation investigated by Satsuma shows a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transport [1]. In recent years, many researchers have used various numerical methods in the literatures to solve the Burgers-Huxley equation (1.1). For instance, Adomian decomposition method was studied in [2], spectral collocation method, pseudospectral method and spectral domain decomposition method were studied in a series of paper [3, 4, 5, 6, 7], wavelet method was introduced [8]. More recently, a three-step Talyor-Galerkin method was presented for the singularly perturbed generalized Burgers-Huxley equation [9]. And the local discontinuous Galerkin method has been used to solve the Burgers-Huxley and Burgers-Fisher equations [10]. Multiscale or multilevel numerical methods received much attention recently. They have considerable advantages and become the standard approaches in solving integral equations([11, 12, 13, 14, 15]). The methods lead to a numerically sparse matrix presentation of the integral operator. A proper truncation of such a sparse matrix will result in a fast numerical algorithm for solving the equation. What s more, the methods allow us to design multilevel augmentation method (MAM) for fast solving the system resulting from discretization of the integral equation because of the multiscale structure of the coefficient matrix [13, 14, 15]. This method was then applied to solve differential equations [16, 17, 18]. The main advantage of multiscale methods for solving differential equations is the stability of the multiscale bases. It can reduce the condition number of the resulting discrete systems largely and obtain the uniform boundedness of the condition number under some conditions[16]. In this paper, the generalized Burgers-Huxley equation with Dirichlet boundary condition is solved by combination of the multiscale Galerkin method and the strong stability preserving Runge-Kutta method. After discreting the spacial variables by Galerkin method with the multiscale orthornomal bases constructed in [16], we apply the strong stability preserving three-oder Runge- Kutta method [19] to solve the resulting ordinary differential equations. The computed results show that the method is effective for obtaining numerical approximations of Burgers-Huxley equation. Furthermore, the use of considerably small number of bases and strong stability make the method advantageous. The remainder of this paper is organized as follows: In section 2 we introduce the multiscale orthornomal bases in Sobolev spaces. We describe the model problem and the proposed algorithm in detail in section 3 and section 4, respectively. In section 5 numerical results of several test experiments are reported, which show that the algorithm is stable and accurate to solve

3 Multiscale Runge-Kutta Galerkin method 1469 Burgers-Huxley equations. At last a short conclusion is drawn in section 6. 2 Multiscale orthonormal bases for H 1 0(0, 1) In this section, we introduce the multiscale orthonormal bases constructed in [16] for Sobolev spaces H0(E), 1 E := [0, 1]. Readers can refer to [16] for the details. Let X := H0(E) 1 denote the Sobolev space of elements u vanishing at both 0 and 1, the inner product and norm of H0(E) 1 are equipped by u, v 1 := u (x)v (x)dx, u 1 := u, u 1, u, v H0(E). 1 E For k 2, we denote by X n the finite dimensional subspace of X whose elements are the piecewise polynomials of order less than k with knots j/µ n, j 1 Z µ n 1, where µ > 1 and the notation Z n := {0, 1, 2,, n 1}. It is easily seen that the subspaces X n are nested, i.e., X n 1 X n, n N 0 := {0, 1, 2, }. By the definition of X n, the dimension of the space X n is dim X n = (k 1)µ n 1. Because of the nestedness property, X n can be expressed as an orthogonal direct sum of X n 1 and W n. It follows that for n N 0, X n = X 0 W 1 W n, where the notation S 1 S 2 stands for the orthogonal direct sum of spaces S 1 and S 2. It can be computed that the dimension of the subspace W n is obtained by ω(n) := dim W n = dim X n dim X n 1 = (k 1)µ n 1. When subspace W 1 has been given, W n can be recursively constructed via the linear operators T e : L 2 (E) L 2 (E), e Z µ, defined by: { µ (T e u)(x) := 1/2 u(φ 1 e (x)), x φ e (E), (2.1) 0, x / φ e (E), where φ e (x) := x+e are affine mappings which define the partitions of interval µ E. Specially, we introduce the construction of linear multiscale orthonormal bases which will be used to discretize the spatial variables of Burgers-Huxley equation. Choose k = 2, µ = 2. It is easily seen that X 0 =, dim W i = 2 i 1, i > 1. The desired basis of W 1 is given by { x, x [0, 1 ω 1,0 = ), 2 1 x, x [ 1, 1]. 2 Let {ω i,j (x), j Z ω(i) } be the bases of W i, then the operators (2.1) for bases of W i+1 have concrete forms: { 2 1 ω i,j (2x), x [0, 1 2 ω i+1,j = ), 0, x [ 1, 1], j = 0, 1, 2,, 2 i 1 1, 2

4 1470 Jian Chen and { 0, x [0, 1 ω i+1,2 i 1 2 +j = ), 1 2 ω i,j (2x 1), x [ 1, 1], j = 0, 1, 2,, 2 i Therefor, functions {ω i,j (x) : j Z ω(i) } form an orthonormal basis for W i, and X n = {ω i,j (x) : (i, j) J n }, J n := {(i, j) : i Z n+1, j Z ω(i) }. The graph of bases for X 3 are shown in Fig ω i,j (x) x Fig. 1: H 1 0 linear multiscale orthogonal bases for X 3. For an arbitrary interval [a, b], the multiscale orthonormal bases of Sobolev space H 1 0(a, b) can also be constructed just by a simple affine transformation. 3 The model problem As mentioned above, the generalized Burgers-Huxley equation represents behaviors of many physical systems encountered in models of reaction mechanisms, convection effects and diffusion transport. The governing initial boundary value generalized Burgers-Huxley equation under consideration is: u t + αu δ u x u xx = βu(1 u δ )(u δ γ), a x b, 0 < t T, u(x, 0) = u 0 (x) = [ γ + γ tanh(θ 2 2 1x))] 1/δ, a x b, u(a, t) = u a (t) = [ γ + γ tanh(θ 2 2 1(a θ 2 t))] 1/δ, 0 < t T, u(b, t) = u b (t) = [ γ + γ tanh(θ 2 2 1(b θ 2 t))] 1/δ, 0 < t T. (3.1)

5 Multiscale Runge-Kutta Galerkin method 1471 The exact solution of the problem (3.1) is given by ([20]) where u(x, t) = [ γ 2 + γ 2 tanh(θ 1(x θ 2 t))] 1/δ, θ 1 := αδ + δ α 2 + 4β(1 + δ) 4(1 + δ) θ 2 := αγ 1 + δ, (1 + δ γ)( α + α2 + 4β(1 + δ). 2(1 + δ) The aim of this paper is to numerically solve this equation by multiscale Galerkin method. The computed results are compared with the exact solutions, showing that the presented method is capable of achieving high accuracy and stable solution for problem (3.1). 4 The multiscale Galerkin method for Burgers- Huxley equation In this section, we combine the multiscale Galerkin method and the strong stability preserving third-order Runge-Kutta method to solve the generalized Burgers-Huxley equation. First we convert (3.1) to a zero boundary problem. To this end, we let ũ(x, t) = u(x, t) h(x, t) = u(x, t) b x b a u a(t) x a b a u b(t). Then ũ(a, t) = ũ(b, t) = 0, and the problem (3.1) can be rewritten as an initial-boundary value problem of functions ũ(x, t) ũ t + h t + α(ũ + h) δ (ũ x + h x ) ũ xx = f(ũ), a < x < b, 0 < t T, ũ(x, 0) = u 0 (x) h(x, 0), a x b, ũ(a, t) = ũ(b, t) = 0, 0 < t T, (4.1) where f(ũ) = β[1 (ũ + h)][1 (ũ + h) δ ][(ũ + h) δ γ]. Let X n be a finite dimensional subspace of H 1 0(E). The multiscale orthonormal basis functions {ω i,j (x) : (i, j) J n } introduced in section 2 are chosen as bases for X n. Then the approximate solution ũ n in space X n can be written as: ũ n (x, t) = (i,j) J n ũ i,j (t)ω i,j (x). (4.2)

6 1472 Jian Chen Then, the multiscale Galerkin method of problem (4.1) can be posed as: Find ũ n = ũ n (, t) X n such that (ũ n,t, ω) + (h t, ω) + (g n, ω) = (f n, ω), ω X n, ũ n (x, 0) = ũ i,j (0)ω i,j (x), (i,j) J n (4.3) where (u, v) := b a u(x)v(x)dx, g n = g(ũ n ) := α(ũ n + h) δ (ũ n,x + h x ) ũ n,xx and f n = f(ũ n ) := β[1 (ũ n + h)][1 (ũ n + h) δ ][(ũ n + h) δ γ]. The coefficients ũ i,j (0) can be computed by ũ i,j (0) = ũ(x, 0), ω i,j (x) 1, (i, j) J n. By substituting the above representation of ũ n (4.2) into the multiscale Galerkin scheme (4.3), we obtain an initial value problem of ordinary differential equation system d = L(Ũn, t), dtũn Ũ n (0) = [ũ i,j (0) : (i, j) J n ]. (4.4) Choose τ as the time step and apply the SSP-RK3 scheme to solve ordinary differential equation system (4.4), i.e., for m = 1, 2,, T/τ, compute K 1 = Ũ(m 1) n K 2 = 3 4Ũ(m 1) n Ũ (m) n = 1 3Ũ(m 1) + τl(ũ(m 1) n, (m 1)τ), + 1K τL(K 4 1, mτ), (4.5) n + 2K τL(K 3 2, (m 1)τ). 2 Thus for m = 1, 2,, T/τ, we obtain the coefficients Ũ m n = [ũ i,j (mτ) : (i, j) J n ]. Therefor, the approximate solution of the equation(1.1) can be expressed as: u n (x, mτ) = ũ n (x, mτ) + h(x, mτ) = ũ i,j (mτ)w i,j (x) + b x b a u a(mτ) + x a b a u b(mτ). (i,j) J n

7 Multiscale Runge-Kutta Galerkin method Numerical experiments In this section we obtain numerical solutions of generalized Burgers-Huxley equation in the form (1.1) by MRKGM. To verify the efficiency of the proposed method for the current problem in comparison with the exact solution, errors for various values of α, β, δ and γ are reported in the following tables. For all of the experiments, n and N := N(n) = 2 n 1 stand for the level and the corresponding dimension of X n. And the maximum error norm L at time level t = mτ, m = 0, 1, 2, is defined by L = u n (x, t) u(x, t) = max 1 j N u n(x j, t) u(x j, t), where u n (x, t) and u(x, t) denote the approximate solution and theoretical solution respectively. For the computational work, we select the following examples from [5, 8]. In all of the examples, n is taken as 3 and τ is taken as Example 1. Consider Burgers-Huxley equation (1.1) in the domain 0 x 1 with α = β = 1 and γ = The maximum error obtained by our method and domain decomposition algorithm in [5] for various δ and t are given in Table 1. Notice that the number of basis functions in [5] and in our method are (N + 1) M = 5 2 = 10 and = 7, respectively. But the time step in [5] is much less than this in present method. Furthermore, as indicated in this table, the present method is more accurate. To show the solitary wave evolution with time, we expand the computation domain to [ 10, 20] and plot the numerical solution and exact solution in Fig.2 for the values α = β = 1, γ = 2, δ = 1 at times t = 0, 5, 10. Table 1: Maximum error with α = β = 1, γ = for Example 1. t δ Error in [5] MRKGM t = t = δ = e e-8 δ = e e-5 δ = e e δ = e e-8 δ = e e-5 δ = e e-5

8 1474 Jian Chen t=0 t=5 t=10 u(x,t) x Fig.2: Numerical solution( o ) and exact solution( - ) at time t = 0, 5, 10 of Example 1 for α = β = δ = 1, γ = 2. Example 2. The proposed method is applied to equation (1.1) with α = 0.1, β = 0.001, γ = in [0, 1] at t = 0.2, 1.0. In Table 2, we show maximum error for various values of δ and compare with the corresponding results in [5]. Table 2: Maximum error with α = 0.1, β = 0.001, γ = for Example 2. t δ Error in [5] MRKGM t = t = δ = e e-13 δ = e e-10 δ = e e δ = e e-13 δ = e e-10 δ = e e-9 Example 3. Table 3 shows the absolute errors for various values of δ and x with α = 0.1, β = 0.1 and γ = at t = 0.9. The results are compared with [8] and it is found that these are better than the results presented in [8].

9 Multiscale Runge-Kutta Galerkin method 1475 Table 3: Absolute errors for various values of δ and x at t = 0.9 for Example 3. x δ = 1 δ = 2 δ = 8 Error in [8] MRKGM Error in [8] MRKGM Error in [8] MRKGM t = t = t = t = t = t = 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 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-7 Example 4. In Table 4 we present the absolute errors for various values of β and x with α = 1, δ = 1 and γ = at t = 0.9. The results are compared with [8]. For t = 0.9 and β = 50, graphical presentation of the absolute error between u n and u(x, t) is depicted in Fig. 3. Table 4: Absolute errors for various values of β and x at t = 0.9 for Example 4. x β = 1 β = 10 β = 50 Error in [8] MRKGM Error in [8] MRKGM Error in [8] MRKGM t = t = t = t = t = t = 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 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-7

10 1476 Jian Chen 3 x u n (x,t) u(x,t) x Fig.3: The error curve of u n (x, t) u(x, t) with α = δ = 1, β = 50, γ = at t = 0.9. Example 5. Table 5 gives the absolute errors for various values of γ and x with α = 5, β = 10 and δ = 2 at t = 0.9. Also, the results are compared with [8] and it is found that these are better than the results presented in [8]. Table 5: Absolute errors for various values of γ and x at t = 0.9 for Example 5. x γ = 10 2 γ = 10 3 γ = 10 4 Error in [8] MRKGM Error in [8] MRKGM Error in [8] MRKGM t = t = t = t = t = t = 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 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-8

11 Multiscale Runge-Kutta Galerkin method Conclusion In this paper, we combine the multiscale Galerkin method with multiscale orthonormal bases and third-order Runge-Kutta method to solve generalized Burgers-Huxley equation. Equation (1.1) is computed for various paraments by the present method, and the numerical examples demonstrate that the presented method gives highly accurate results even in the case of a small number of grid points. Compared with analytical solution and other numerical scheme, the proposed method can give more accurate results. Acknowledgements. This research was supported in part by the Natural Science Foundation of China under grant and , the Natural Science Foundation of Guangdong Province of China under grant 2016A and the Training Programme Foundation for Excellent Young Scholar of Guangdong Province under grant YQ References [1] J. Satsuma, M. Ablowitz, B. Fuchssteiner and M. Kruskal (Eds.), Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, [2] H.N.A. Ismail, K. Raslan and A.A.A. Rabboh, Adomian decomposition method for Burger s-huxley and Burger s-fisher equations, Appl. Math. Comput., 159 (2004), [3] M. Javidi, Spectral collocation method for the solution of the generalized Burger-Fisher equation, Appl. Math. Comput., 174 (2006), [4] M. Javidi, A numerical solution of the generalized Burger s-huxley equation by pseudospectral method and Darvishi s preconditioning, Appl. Math. Comput., 175 (2006), [5] M. Javidi and A. Golbabai, A new domain decomposition algorithm for generalized Burger s-huxley equation based on Chebyshev polynomials and preconditioning, Chaos Solitons Fract., 39 (2009), [6] M. Javidi, A numerical solution of the generalized Burger s-huxley equation by spectral collocation method, Appl. Math. Comput., 178 (2006),

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13 Multiscale Runge-Kutta Galerkin method 1479 [19] S. Gottlieb, C. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), [20] X. Wang, Z. Zhu and Y. Lu, Solitary wave solutions of the generalised Burgers-Huxley equation, J. Phys. A Math. General., 23 (1990), no. 3, Received: February 14, 2017; Published: May 23, 2017