Relaxation Newton Iteration for A Class of Algebraic Nonlinear Systems

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.8 (009) No., pp Relaxation Newton Iteration for A Class of Algebraic Nonlinear Systems Shulin Wu, Baochang Shi, Chengming Huang School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan , P. R. China (Received 8 November 008, accepted 7 January 009) Abstract. Relaxation Newton algorithm, introduced in [Shulin Wu, Chengming Huang, Yong Liu, Newton waveform relaxation method for solving algebraic nonlinear equations, Applied Mathematics and Computation, 0 (008), pp. 3 60], is a method derived by combining the classical Newton s method and the waveform relaxation iteration. It has been shown that, with a special choice of the so called splitting function, this algorithm takes advantages of global convergence, less storage and absolute stability and can be implemented simultaneously. In this paper, we investigate a class of nonlinear equations which are well suited to be solved by this algorithm. These nonlinear equations are derived from the implicit discretization of nonlinear ordinary differential equations and nonlinear reaction diffusion equations. Several examples are tested to illustrate our theoretical analysis and the results show the advantages of this algorithm in the sense of iterative number and CPU time very well. Keywords: relaxation Newton algorithm; waveform relaxation methods; Newton s method; nonlinear algebraic equations; reaction diffusion equations AMS (MOS) subject classifications: 6Y0, 6Y0, 6Y0, 68Q60. Introduction Consider the following equations f(x) = 0, (.) where f : D R n R n. It is well known that to solve (.) efficiently is a very important problem in many fields, such as management science, industrial and financial research, data mining and numerical simulation of nonlinear systems, etc. There are numerous methods to solve (.) and the fundamental one is the classical Newton s method and its modifications. The classical Newton s method is an iterative method which is written as f (x k ) x k = f(x k ), x k+ = x k + x k, k = 0,,..., (.) where x 0 is the initial approximation of the solution x. It is known long that method (.) converges quadratically and locally. We know that two severe drawbacks counteract the direct application of method (.) in practice. One is that this algorithm converges only locally and this implies that one should choose the initial approximation x 0 which is sufficiently close to the unknown solution x ; the other is that the Jacobian matrix f (x) This work was supported by NSF of China(No , ) and by Program for NCET, the State Education Ministry of China. Corresponding author(shulin Wu). address: wushulin ylp@63.com (Shulin Wu), sbchust@6.com (Baochang Shi), chengming huang@hotmail.com (Chengming Huang). Copyright c World Academic Press, World Academic Union IJNS /7

2 44 International Journal of Nonlinear Science,Vol.8(009),No.,pp must be nonsingular in D and one needs to calculate [f (x)] for every iteration. The latter will bring in unacceptable burden in both storage and computation time. To overcome these drawbacks, many modifications of the classical Newton s method have been investigated and many excellent results have been obtained. For example, we can treat (.) as linear problem Ax = b to obtain Δx k ; there are many methods to treat this linear problem, such as Jacobi, Gauss Seidel, Conjugate Gradient[6, 7], GMRES[3], AOR iteration [30], etc. There are so many prominent results in this field that we can not recount them detailedly. For a description of state of the art, we refer the reader to the classical books [3, 9, ] and papers [7, 0,, 9, 3, 3], etc. In [4] the authors introduce another variation of the classical Newton s method, the relaxation Newton algorithm, to solve equations (.). This method is called Newton waveform relaxation in [4], but here we call it relaxation Newton, since through each iteration we obtain a set of discrete values but not a set of continuous functions which is an important character of the waveform relaxation methods [0 3, 7, 8]. The key idea of the relaxation Newton algorithm is to choose a splitting function F : D D R n which is minimally assumed to satisfy the consistency condition F (x, x) = f(x) (.3) for any x R n. Then with an initial guess x 0 of the unknown solution x at hand, we start with the previous approximation x k to compute the next approximation x k+ by solving the problem F (x k, x k+ ) = 0, k = 0,,.... (.4) with some conventional method, such as the classical Newton s method, quasi Newton methods, Conjugate Gradient method, etc. In [4] and this paper, we adopt the classical Newton s method to solve (.4), which explains the name relaxation Newton. The combination of the notion of the waveform relaxation iteration with other methods for (.4) will lead to new algorithms and this is one of the future directions. The deduced algorithm written compactly is shown in Figure.. In figure. and hereafter for k =0,,,... with a given initial approximation x 0 of x k+ ; for m =0,,..., M F (x k, x m ) x m = F(x k, x m ), x m+ = x m + x m, end x k+ = x M, end Figure.: The relaxation Newton method F (x, y) = F (x, z) z. (.) z=y If we set x 0 = x k and M = in figure., by consistency condition (.3), the iterative scheme is equivalent to F (x k, x k ) x k = f(x k ), x k+ = x k + x k, k = 0,,.... (.6) With a special choice of F, the Jacobi matrix F (x, x) will be a diagonal or block diagonal matrix and invertible in R n, and thus iterative method (.6) can be processed stably and simultaneously with less storage compared with the classical Newton s method. IJNS for contribution: editor@nonlinearscience.org.uk

3 S.L. Wu, B.C. Shi, C.M. Huang: Relaxation Newton Iteration for A Class of Algebraic 4 In [4], an affine covariant Lipschitz condition imposed on the splitting function F (x, y) is given to guarantee the global convergence of method (.6) for general nonlinear equations (.). However, the authors say nothing about what class of nonlinear equations is suited to be solved by the relaxation Newton method; this is the topic of present paper. We will see that these nonlinear equations play an important role in the field of implicit discretization of the nonlinear ordinary differential equations(odes) and the reaction diffusion equations with nonlinear reaction term. We show in this paper that with a special choice of splitting function F (x, y), these nonlinear equations can be solved efficiently with much less storage, CPU time and iterations. The remainder of this paper is organized as follows. In section, we recall the affine covariant convergence lemma proved in [4] and then we introduce the splitting function F (x, y) and the nonlinear equations discussed in this paper. In section 3, we apply this algorithm to the reaction diffusion equations with nonlinear reaction term. In section 4, we test a set of examples to illustrate our theoretical analysis and the efficiency of this algorithm. The nonlinear equations and convergence analysis For nonlinear equations (.), the global convergence of the relaxation Newton algorithm was proven in [4] provided the splitting function F (x, y) satisfies an affine covariant Lipschitz condition. Lemma. Let F : D D R n be a continuous mapping with D R n open and convex and satisfy the consistent condition (.3). Function F (x, y) is Fr echet differentiable with aspect to the second variable y and the Jacobi matrix F (x, y) is invertible for any x, y D. Assume that and F (x, x)(f (y, y) F (x, y)) α x y with α <, (.) F (x, x)(f (x, x) F (x, y)) = 0 (.) for any x, y D. Then with an arbitrary starting approximation x 0, the sequence x k } obtained by (.6) is well defined, remains in the open ball B(x, x 0 x ) D and converges to x with F (x, x ) = 0(i.e., f(x ) = 0). Moreover, the error x k x satisfies x k x α k x 0 x. (.3) Note that condition (.) is satisfied if the matrix F (x, y) is independent of the second variable y. In present paper, we assume that the function f in (.) satisfies the following conditions. Condition Suppose that the function f in (.) has a form f(x) = x + g(x) + h(x), (.4) where g(x) = (g (x ), g (x ),..., g n (x n )) T ; moreover, each component of functions g(x), h(x) satisfies g i,max < +, h i (x) h i (y) c i x y, and c i < + g i,min, (.) for any x, y D and x i R, here and in what follows g i,min dg i (x i ) = inf, g dg i (x i ) i,max = sup, i =,,..., n. (.6) x i R dx i x i R dx i Remark Consider the following differential system y (t) + G(t, y(t)) + H(t, y(t)) = 0, t > 0, y(0) = y 0, t = 0, (.7) where G(t, y(t)) = (G (t, y (t)),..., G n (t, y n (t))) T. IJNS homepage:

4 46 International Journal of Nonlinear Science,Vol.8(009),No.,pp By applying the backward Euler method to (.7) we arrive at y n+ = y n τg(t n+, y n+ ) τh(t n+, y n+ ), n = 0,,..., N, (.8) where τ is the discretization step-size. Clearly, equations (.8) can be written into the general form x + g(x) + h(x) = 0. (.9) In fact, many implicit numerical methods applied to differential system (.7) will lead to nonlinear equations (.9). Under Condition, we consider the following splitting function F (x, y) = a y + ( a )x + [b (x )(y x ) + ]g (x ) + h (x) a y + ( a )x + [b (x )(y x ) + ]g (x ) + h (x). a n y n + ( a n )x n + [b n (x n )(y n x n ) + ]g n (x n ) + h n (x), (.0) where a i, b i (x i ) R. It is clear that the splitting function F (x, y) satisfies the consistency condition F (x, x) = f(x) with any a i and b i (x i ), i =,,..., n. Moreover, F (x, y) = diag(a +b (x )g (x ),..., a n + b n (x n )g n (x n )) and this implies that condition (.) in Lemma. holds and the Jacobi matrix F will be nonsingular with a i and b i (x i ) chosen properly. Theorem. Assume the function f in (.) satisfies conditions (.4) and (.). Then relaxation Newton method (.6) with splitting function (.0) converges globally and the Jacobi matrix F will be nonsingular, provided a i, b i (x i ) satisfy a i + b i (x i )g i (x i ) + g i,max, i =,,..., n. (.) Proof. It is clear that the Jacobi matrix F will be nonsingular since a i + b i (x i )g i (x i ) + g i,max + g i,min > c i > 0. Moreover, since the Jacobi matrix F (x, y) is independent of the second variable y, we just need to prove that condition (.) in Lemma. holds for any x, y D. By routine calculation, we have F (x, x)(f (x, y) F (y, y)) a i b i (x i )g i (x i ) + g i = max (ξ i)(x i y i ) + h i (x) h i (y) i n a i + b i (x i )g i (x i ) a i b i (x i )g i (x i ) + g i max (ξ i + c i x y i n = max i n a i + b i (x i )g i (x i ) + c i g i (ξ i) a i + b i (x i )g i (x i ) } y x, where in the first equality we used Lagrange s mean theorem g i (ξ i) = g i (x i ) g i (y i ) with some ξ i between x i and y i ; in the second inequality we used condition (.) and in the last equality we used hypothesis (.). Since c i < + g i,min and a i + b i (x i )g i (x i ) + g i,max > 0, we have c i g i (ξ i) a i + b i (x i )g i (x i ) c i g i,min a i + b i (x i )g i (x i ) < 0 (.) for any ξ i, x i R; this implies max + c i g i (ξ } i) + c i g i,min i n a i + b i (x i )g i (x i ) + g i,max = c i + g i,max g i,min + g i,max <. From this, we complete our proof. IJNS for contribution: editor@nonlinearscience.org.uk

5 S.L. Wu, B.C. Shi, C.M. Huang: Relaxation Newton Iteration for A Class of Algebraic 47 From (.3) and Theorem. we have x k x ( max i n + c i g i,min a i + b i (x i )g i (x i ) }) k x 0 x. (.3) Note that in practical implementation of relaxation Newton algorithm (.6), the Jacobi matrix F (x k, x k ) is a diagonal matrix with elements a i + b j (x i )g i (x i ), i =,,..., n. And thus, it is unnecessary to determine the parameters a i, b i (x i ) separately but just need to determine the quantities a i +b j (x i )g i (x i ). Further more, according to (.3) we know that the optimal diagonal elements of the matrix F should be + g i,max, i =,,..., n. However, in many cases we can not calculate each quality g i,max accurately, even some reliable upper bound. For these nonlinear equations we need deeper research for the convergence conditions of the relaxation Newton algorithm and this is our further work. At present moment, we roughly set a i +b j (x i )g i (x i ) = φ i + φ i g (x i ) with some φ i, φ i, i =,,..., n. Fortunately, for many such nonlinear equations this simply strategy still lead to convergence of the algorithm as shown in section 4. 3 Application to nonlinear reaction diffusion equations In this section, we apply the relaxation Newton algorithm to the implicit discretization of the reaction diffusion equations with nonlinear reaction term as u t νu xx + R(u, x, t) = 0, (x, t) (0, L) (0, T ), u(x, 0) = ψ 0 (x), x [0, L], u(0, t) = ψ (t), u(l, t) = ψ (t), t [0, T ], where ν > 0. We apply the method of lines to discretize the diffusion term u xx using the central difference discretization on M points x j = j/m, j =,..., M, Δx = L/M. Then we obtain a system of M differential equations u j (t) ν [u Δx j+ (t) u j (t) + u j (t)] + R(u j (t), x j, t) = 0, t (0, T ), (3.) u j (0) = ψ 0 (x j ), j =,,..., M, where u j (t) = u(x j, t), u 0 (t) = ψ (t), u M (t) = ψ (t). Define U(t) = (u (t), u (t),..., u M (t)) T, U(t) ( ν = Δx ψ ν ) T (t), 0,..., 0, Δx ψ (t), ( ) ν G(Δx, U) = Δx u ν T + R(u, x, t),..., Δx u M + R(u M, x M, t), H = ν Δx , U 0 = (ψ 0 (x ),..., ψ 0 (x M )) T Then we have U (t) + G(Δx, U(t)) + HU(t) U(t) = 0, U(0) = U 0, By applying some implicit method, for example the backward Euler method, to discrete system (3.3) in time we obtain U n+ + ΔtG(Δx, U n+ ) + ΔtHU n+ Δt U(t n+ ) U n = 0. (3.4) (3.) (3.3) IJNS homepage:

6 48 International Journal of Nonlinear Science,Vol.8(009),No.,pp Let g(u n+ ) = ΔtG(Δx, U n+ ), h(u n+ ) = ΔtHU n+ Δt U(t n+ ) U n. (3.) Then equations (3.4) can be rewritten as U n+ + g(u n+ ) + h(u n+ ) = 0. (3.6) Clearly, nonlinear equations (3.6) takes the form of (.4). Therefore, we may apply the relaxation Newton method with the splitting function given in (.0) to solve (3.6) from t n to t n+. The following result guarantees the convergence of the relaxation Newton method for equations (3.6). Theorem 3. Let γ(x i, t n+ ) = inf u R R(u,x i,t n+ ) u. Then for nonlinear equations (3.4) the relaxation Newton method with the splitting function defined in (.0) is convergent from t n to t n+ provided Δtγ(x i, t n+ ) + > 0 holds for i =,,..., M. Proof. With definitions (3.), it is easy to get g i,min dg i (u) = inf = νδt R(u, x i, t n+ ) u R du Δx + Δt inf = νδt u R u Δx + Δtγ(x i, t n+ ) and h(x) h(y) νδt Δx x y. Thus, we may state by applying Theorem. that the relaxation Newton method applied to nonlinear equations (3.4) is convergent if Δtγ(x i, t n+ ) + > 0 holds for i =,,..., M. Similar to the analysis given in the end of section 3, we know that the optimal ( Jacobi matrix F of the ) ν R(u,x relaxation Newton algorithm should be diagonal matrix with elements + Δt + sup i,t n+ ) Δx u, i =,,..., M. In practical implementation of the relaxation Newton algorithm, if we can not get R(u,x sup i,t n+ ) R(u,x u accurately or a reliable upper bound, we may roughly replace sup i,t n+ ) u by φ i + u R u R R(u φ i,n+,x i,t n+ ) i u with some φ i, φ i, where u i,n+ is the i th component of the vector U n+. In our numerical experiments presented in the next section, we choose φ i = φ i =, i =,,..., M. 4 Numerical results In this section, we test some problems to illustrate the efficiency of the relaxation Newton algorithm in the sense of iterative number and CPU time. 4. Relaxation Newton algorithm for nonlinear ODEs u R Consider the following system consisting of M differential equations y (t) + G(t, y(t)) + H(t, y(t)) = 0, t 0, y(0) = 0, t = 0, (4.) with and H j (t, y(t)) = cos t + G j (t, y(t)) = e M+ j arctan M i=,i =j y i (t) (e M+ M j y j (t)) IJNS for contribution: editor@nonlinearscience.org.uk

7 S.L. Wu, B.C. Shi, C.M. Huang: Relaxation Newton Iteration for A Class of Algebraic 49 j =,,..., M. Applying the implicit Euler method to (4.) we get y n+ = y n τg(t n+, y n+ ) τh(t n+, y n+ ), n = 0,,..., N. (4.) Therefore, from t n to t n+ we need to solve the following nonlinear equations x + g(x) + h(x) = 0, (4.3) with g j (x j ) = τe M+ j arctan (e M+ ) M j x j and Routine calculation yields h j (x) = τ cos t + dg j dx j = M i=,i =j ( + x i y n,j, j =,..., M. τe M j M e M+ M j x j ). From this we have g j,min = 0, g j,max = τem j M. With Lagrange s mean theorem, it is easy to get h i (x) h i (y) τ(m ) x y, (4.4) i =,..., M. Thus, by Theorem. we know that τ < M is sufficient to guarantee the global convergence of the relaxation Newton algorithm. Experiment A Let M = 0. We know that, to solve nonlinear equations (4.3), fixed point iteration method is a good choice provided the functions g, h satisfy the following contraction Lipschitz condition g i (x i ) g i (y i ) + h i (x) h i (y) η i x y, η i <, i =,..., 0. (4.) This contraction condition theoretically indicates that τ < e is necessary to guarantee the convergence of the fixed point method on each time point t n. Therefore, if we solve system (4.) in interval I = [0, ], we just need to calculate 0 steps by solving the resulting nonlinear equations with the relaxation Newton method, while with the fixed point iteration we need to calculate 0 4 steps. By our analysis stated in the end of section, we know that the Jacobi matrix F (x k, x k ) in (.6) should be diagonal matrix with elements + τe M j M, j =,,..., 0. In this experiment, we solve system (4.) numerically in interval [0, 0] by the backward Euler method and the resulting nonlinear equations (4.) are solved by the relaxation Newton method and the fixed-point method, respectively. For these two iterative methods, the initial approximation of y n+ is y n, and the tolerance of residual is 0 4, i.e., for k from 0 to k max if the stop criterion RES n = y k n+ + τg(t n+, y k n+) + τh(t n+, y k n+) y n 0 4 holds for some k, we let y n+ = y k n+, where k max is the maximal iterative number. We set k max = 0 in all experiments. Set τ = 0, The required iterative number and the finial residual at every time point t n for these two methods are plotted in Figs. 4. and 4.. We see from the left panel of figure 4. that to achieve the tolerance 0 4 the iterative number of the relaxation Newton method is at every step. However, the fixed-point method is not convergent at every time point t n. We see from the left panel of figure 4. that, for τ =. 0 4, the residual of the relaxation Newton algorithm is less than 0 after only one iteration. However, from the right panel we see that the fixed-point method is still not convergent. Experiment B Next we compare the computational efficiency of the relaxation Newton method with the classical Newton s method coupled with inner linear problem solvers, the stabilized bi-conjugate gradient IJNS homepage:

8 0 International Journal of Nonlinear Science,Vol.8(009),No.,pp Residual Iterative Number: k Residual 0 Iterative Number: k Relaxation Newton Method Fixed Point Method Figure 4.: Step size τ = Residual Iterative Number: k Residual Iterative Number: k Relaxation Newton Method Fixed Point Method Figure 4.: Step size τ =. 0 4 method (BICGSTAB) [6, 7] and the generalized minimal residual method (GMRES) [3]; for convenience, we denote the methods derived by combining the classical Newton s method and the inner solvers BICGSTAB and GMRES by Newton BICGSTAB and Newton GMRES, respectively. In this and the next three experiments, for the classical Newton s method we refer the iterative number at every time point as the number of summation of inner linear iterations; moreover, for Newton s method coupled with some linear solver and the relaxation Newton method, the tolerance T OL outer for outer nonlinear iteration is 0 4 and the tolerance T OL inner for inner linear iteration is 0 3. In this experiment, we consider system (4.) with M = 00. Similar to the analysis stated in experiment A we know that in this case τ 0.00 is sufficient to guarantee the convergence of the relaxation Newton method. We choose τ = 0 and solve system (4.) numerically in interval [0, 0]. The resulting nonlinear equations (4.) are solved by the relaxation Newton method and the methods Newton BICGSTAB and Newton GMRES. The residual and iterative number at every time point t n for these three methods are plotted in figure 4.3, left panel for the relaxation Newton method and right panel for the methods Newton BICGSTAB and Newton GMRES. We see on the left panel of figure 4.3 that the residual of the relaxation Newton method at every time point is less than 0 6 after only one iteration, while the Newton BICGSTAB and Newton GMRES methods only achieve 0 8 after and 4 linear iterations, respectively. Moreover, the computational time costed at every time point of these three methods is significantly different which is shown in figure 4.4. Actually, the average computational time costed at every time point of the Newton BICGSTAB and Newton GMRES methods is about 68 times of that of the relaxation Newton method. IJNS for contribution: editor@nonlinearscience.org.uk

9 S.L. Wu, B.C. Shi, C.M. Huang: Relaxation Newton Iteration for A Class of Algebraic Residual Iterative Number:k. 0. Residual Newton BICGSTAB Newton GMRES Iterative Number:k Newton BICGSTAB Newton GMRES Relaxation Newton Method Newton BICGSTAB/GMRES Method Figure 4.3: Step size τ = 0 Relaxation Newton Newton BICGSTAB Newton GMRES 0 0 CPU time [s] Figure 4.4: CPU time costed at every time point of the Newton-BICGSTAB /GMRES methods and the relaxation Newton method 4. Relaxation Newton algorithm for nonlinear reaction diffusion equations Next, we test some reaction diffusion equations with nonlinear reaction term to compare the efficiency of the relaxation Newton algorithm with the classical Newton s method; the comparison is focused on two aspects: the iterative number and the CPU time at every time point. For the classical Newton s method, we consider at present moment the iterative methods GMRES, SYMMLQ [8], MINRES [, 8], BICG, BICGSTAB, CGS, LSQR as the inner linear problem solver. For nonlinear system (3.4), we note that for each Newton iteration we actually need to solve a tridiagonal linear system, and thus direct solver the cyclic reduction algorithm proposed by Hockney [8] is also a good choice. We denote the methods derived by combining the classical Newton s method and the inner linear problem solvers GMRES, SYMMLQ, MINRES, BICG, BICGSTAB, CGS, LSQR and Hockney s direct method by Newton GMRES, Newton SYMMLQ, Newton MINRES, Newton BICG, Newton BICGSTAB, Newton CGS, Newton LSQR and Newton DIRECT, respectively. Moreover, to show explicitly the advantages of the relaxation Newton method in the sense of CPU time and iterative number, we define the ratios about average iterative number and CPU time of the Newton type methods to the relaxation Newton method as γ GMRES = T GMRES, γ SY MMLQ = T SY MMLQ γ BICGST AB = T BICGST AB, γ CGS = T CGS, γ MINRES = T MINRES, γ BICG = T BICG,, γ LSQR = T LSQR, γ DIRECT = T DIRECT, IJNS homepage:

10 International Journal of Nonlinear Science,Vol.8(009),No.,pp and κ GMRES = K GMRES, κ SY MMLQ = K SY MMLQ κ BICGST AB = K BICGST AB, κ CGS = K CGS, κ MINRES = K MINRES, κ BICG = K BICG,, κ LSQR = K LSQR, κ DIRECT = K DIRECT, respectively, where, stand for the average CPU time and iterative number costed at every time point of the relaxation Newton method (similar interpretation for the other symbols). Experiment C Consider the following reaction diffusion problem [6, ] where c(x) = u t = ε u xx + u( u)(u c(x)), (x, t) (0, 80) (0, 0), u(x, 0) = sin ( πx L), x [0, 80] u(0, t) = 0., u(l, t) = 0.4, t [0, 0], 0.7, x < 90, 0., x 90. Let ε = 4. We choose mesh parameters Δx = 0.0 and Δt = 0.0 to solve (4.6) numerically. In Figure 4.6 we plot the CPU time and iterative number of the Newton type method and the relaxation Newton method at every time in the left and right panels, respectively. In this and the following experiments, the relation of the line types to the methods is specified in Figure 4.. (4.6) Relaxation Newton Newton GMRES Newton SYMMLQ Newton MINRES Newton BICG Newton BICGSTAB Newton CGS Newton LSQR Newton DIRECT 0 0 Figure 4.: Relation of the line types to the methods CPU Time [s] Iterative Number Figure 4.6: CPU time (left panel) and iterative number (right panel) of the methods at every time point IJNS for contribution: editor@nonlinearscience.org.uk

11 S.L. Wu, B.C. Shi, C.M. Huang: Relaxation Newton Iteration for A Class of Algebraic 3 We see in figure 4.6 that the advantages of the relaxation method in the sense of CPU time and iterative number are obvious. Particularly, we list in tables 4. and 4. the ratios about average CPU time and iterative number of the Newton type methods to the relaxation Newton method, respectively. Table 4.: Ratios of average CPU time γ GMRES γ SY MMLQ γ MINRES γ BICG γ BICGST AB γ CGS γ LSQR γ DIRECT Table 4.: Ratios of average iterative number κ GMRES κ SY MMLQ κ MINRES κ BICG κ BICGST AB κ CGS κ LSQR κ DIRECT From the results listed in table 4., one may see that the relaxation Newton method outperforms the Newton type method even though compared with some algorithms, such as Newton SYMMLQ, Newton MINRES, Newton BICGSTAB and Newton DIRECT, more iterations are required (see the results in table 4.). Example D Our last experiment is to test the well known fisher reaction diffusion equation [,, 4, 9, 4] as u t = Du xx + ku(c u), (x, t) (0, 30) (0, 0), u(x, 0) = u 0 (x), x [0, 30], (4.7) u(0, t) = φ (t), u(l, t) = φ (t), t [0, 0], where D, k, c are positive parameters. We complete (4.7) by the following initial boundary conditions ( πx ) ( ) π u 0 (x) = 0.8 sin, φ (t) =, φ (t) = 0.8 sin. (4.8) 0 We consider the parameters D = 0.0, k = 0.78, c = 0.9, if u, 4., if u >. (4.9) We choose mesh parameters Δx = 0.08 and Δt = 0.0 to solve (4.7) (4.9) numerically. In figure 4.7, we plot the profiles of the solution u(x, t) computed by the finite difference method coupled with the relaxation Newton iteration, where one can see that (4.7) (4.9) is really a challenge problem Space: x 0 0 Time: t Figure 4.7: Profiles of the solution of problme(4.7) (4.9) computed by the relaxation Newton method We compare the computational efficiency of the relaxation Newton method with the Newton type methods by showing the CPU time and the iterative number at every time point on the left and right panels of IJNS homepage:

12 4 International Journal of Nonlinear Science,Vol.8(009),No.,pp figure 4.8, respectively. In tables 4.3 and 4.4 we list the ratios about average CPU time and iterative number of the Newton type methods to the relaxation Newton method, respectively. It is shown clearly in figure 4.8 and these two tables that the relaxation Newton algorithm has significantly advantages in the sense of CPU time and iteration number. CPU Time [s] Iterative Number Figure 4.8: CPU time (left panel) and iterative number (right panel) of the methods at every time point Table 4.3: Ratios of average CPU time γ GMRES γ SY MMLQ γ MINRES γ BICG γ BICGST AB γ CGS γ LSQR γ DIRECT Table 4.4: Ratios of average iterative number κ GMRES κ SY MMLQ κ MINRES κ BICG κ BICGST AB κ CGS κ LSQR κ DIRECT Acknowledgements The authors are grateful to the anonymous referee for the careful reading of a preliminary version of the manuscript and their valuable suggestions and comments, which really improve the quality of this paper. References [] M. Ablowitz, A. Zepetella: Explicit solution of Fisher s equation for a special wave speed. Bull. Math. Biol. 4: (979) [] N. F. Britton: Reaction diffusion equations and their applications to biology. Academic Press, New York.(986) [3] P. Deulfhard: Newton Methods for Nonlinear Problems: Affine Invariant and Adaptive Algorithms. Springer, Berlin. (004) [4] P. C. Fife: Mathematical aspects of reacting and diffusing systems, Lectures Notes in Biomathematics, vol. 8. Springer, Berlin. (979) [] A. Greenbaum, M. Rozložník, Z. Strakoš: Numerical behavior of the modified Gram Schmidt GMRES implementation. BIT. 37: (997) IJNS for contribution: editor@nonlinearscience.org.uk

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14 6 International Journal of Nonlinear Science,Vol.8(009),No.,pp [8] S. Vandewalle: Parallel multigrid waveform relaxation for parabolic problems. B. G. Teubner, Stuttgart. (993) [9] Y. Wang, L. Wang, W. Zhang: Application of the Adomian Decomposition Method to Fully Nonlinear Sine Gordon Equation. International Joural of Nonlinear Science. :9 38 (006) [30] L. Wang: Comparison Results for AOR Iterative Method with a New Preconditioner. International Journal of Nonlinear Science. :6 8 (006) [3] T. J. Ypma: Historical development of the Newton-Raphson method. SIAM Rev. 37:3 (99) [3] H. Zhu, S. Wen: A Class of Generalized Quasi Newton Algorithms with Superlinear Convergence. International Journal of Nonlinear Science. :40 46 (006) IJNS for contribution: editor@nonlinearscience.org.uk