A COMPARATIVE ANALYSIS OF THE COMPUTATIONAL STABILITY FOR GALERKIN FINITE ELEMENT AND DIFFERENCE SCHEMES OF NONLINEAR ADVECTION EQUATION

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume, Number 3, Pages c 006 Institute for Scientific Computing and Information A COMPARATIVE ANALYSIS OF THE COMPUTATIONAL STABILITY FOR GALERKIN FINITE ELEMENT AND DIFFERENCE SCHEMES OF NONLINEAR ADVECTION EQUATION XIAOZHONG YANG, GUANGHUI WANG, AND YANGGUO LIU, XIAOJUN HOU Abstract. The problem of nonlinear stability of computational schemes is very important in numerical weather forecasts and climate simulations. In this paper, using the heuristic stability analysis method, and concerning a nonlinear advection equation, we compare and contrast the stability criterion of Galerkin Finite Element Method scheme with that of general difference schemes, and especially, point out that the stability criterion of Galerkin FEM scheme approximation is consistent with that of original partial differential equation. Further, the work affirms the superiority of the Galerkin FEM scheme to the general difference schemes on the point-view of computational stability. Finally, several numerical examples on nonlinear stability are given, from which the conditions for nonlinear instability are highlighted. Key Words. nonlinear computational stability, Galerkin FEM scheme, difference scheme, stability criterion, heuristic stability analysis 1. Introduction Most problems concerning unsteady, low-speed fluid dynamics such as simulations of climate changes and numerical weather forecasting involve searching for solutions of nonlinear partial differential equations. No matter what kind of numerical methods are used, including difference method, FEM and spectral method, nonlinear computational instability may occur. It can not be overcome by reducing the time step. Usually, the approximate solution displays a character of mutation or rapid exponential increase. So, it is necessary to do some nonlinear computational stability analysis to some typical advection equations. Much work[1-9] has been done, all those problems were attacked step by step, but still it is far from complete resolution. In this paper, on the base of our preceding work, we discuss and contrast the nonlinear computational stability criterion of two kinds of numerical schemes of nonlinear advection equation, Galerkin FEM and difference Received by the editors October 15, 005 and, in revised form, February 1, 006. The Proect Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry No. 383(005) and North China Electric Power University (005). 336

2 COMPUTATIONAL STABILITY OF GALERKIN FEM AND FDM 337 approximation, with the aids of the heuristic Method analysis, special solution and numerical examples, therefore, extend our preceding results. It shows that the nonlinear computational stability is closely connected with the structure of numerical scheme and the initial condition. In addition, the difference schemes should keep as much as possible the physical characteristics of original differential equations, especially the conservation of total energy, which is an important measure to depress the nonlinear computational instability. Finally, several numerical examples are given, from which the conditions for nonlinear instability are highlighted.. Equations and semi-discrete Galerkin FEM scheme.1. Stability of semi-discrete Galerkin FEM scheme. Simply, we mainly discuss the uniform grid case. In order to approach one-dimensional nonlinear advection equation, (1) () where i i t (N i, N ) + i (N i, N ) = t + u = 0 xb x a N xb i (N i, N ) = x a i u i u i (N i N i, N ) = 0 N i (x)n (x)dx N i (x) N i (x) N (x)dx (x a, x b ) is a segment on x-axis. Accordingly, () is the Galerkin FEM scheme of formula (1). Especially, we set x i constant, and take ax + b as the basic functions, on the region (x i, x i+1 ), () takes the form, x i (3) 3 t + x i t 3 u iu i 1 6 u i+1u i u iu i u i+1u i+1 = 0 At the same time, on the region (x i 1, x i ), () takes the form, x i 1 (4) + x i 3 t 3 t 1 6 u i 1u i u iu i u i 1u i u iu i = 0 Plus (3) and (4) together, we have the Galerkin FEM scheme on (x i 1, x i+1 ), ( 1 i 1 (5) + 4 i 6 t t + ) i t 3 (u i+1 + u i + u i 1 ) u i+1 u i 1 = 0 By taking the lumping technique, we have, i (6) t [(u i+1 u i 1) + u i (u i+1 u i 1 )] = 0 It can be proved directly that (6) always ensures the conservation of total energy 1 u under the periodical boundary, and ensures the limit of u

3 338 X. YANG, G. WANG, AND Y. LIU AND X. HOU. Therefore, (6) will never lead to nonlinear computational instability. It yields that Theorem 1. The semi-galerkin FEM scheme formula (5) of the equation t + u = 0 is absolutely computational stable... The stability of general semi-discrete schemes. For the convenience of constructing the difference scheme, we generalize (1) to (7) + (1 )u t + = 0 (0 1) By differentiating (7) spatially with centric finite difference scheme, we have (8) du (t) t + (1 )u (t) u +1(t) u 1 (t) + u +1 (t) u 1 (t) = 0 It is easy to see that in the case = 3, (8) degenerate into (6), and is nonlinear computational stable. While for the case 3, it has to be discussed in other ways. Similar to [3], we set (9) u(t) = C(t)cos π + S(t)sinπ + U(t)cosπ + V (t) By substituting (9) into (7), we have a general differential equation group, (10) dc dt ds = 1 x S(U + V ) + x S(U V ) x C(U V ) + x C(U + V ) = 1 dt = 1 du dt dv dt = 0 x SC 3. The nonlinear stability analysis of full-discrete Leap-frog scheme 3.1. The stability analysis of full-discrete Galerkin FEM scheme. Firstly, we give the explicit leap-frog scheme of (6), (11) u n+1 +1 un un+1 u n 1 + un+1 1 un x (un +1+u n +u n 1)(u n +1 u n 1) = 0 t t t We analyze the stability of (11) with heuristic method. By expanding the spatial terms with Taylor series, u n +1 = u n + n x + 1 u n x u n 6 3 x u n 4 4 x4 + O( x 5 ) u n 1 = u n n x + 1 u n x 1 3 u n 6 3 x u n 4 4 x4 + O( x 5 ) We have, (1) 1 x (un +1 [ + un + un 1 )(un +1 un 1 ) ] n [ = 1 x 3u + u x u 1 x 4 + O( x 5 ) 4 x u 3 x 3 + O( x 5 ) [ ] 3 n = 6u + u 3 u x + u 3 x 3 + O( x 4 ) ] n

4 COMPUTATIONAL STABILITY OF GALERKIN FEM AND FDM 339 So, we have the modified partial differential equation of (11) { (13) + u = t u( [ )3 3 u t x] u + [ 1 6 ( t u 3 + x u) 1 6 x u ] 3 u + O( t x + x 4 ) 3 It is well known that a numerical scheme is stable when it is dissipative. In other words, the stability of (11) requires that the dissipative coefficient of the right term in (13) must be greater than zero, ( 3 (14) u t + 1 ) 6 x > 0 It requires (x,0) > 0 for t > 0, especially, requires > 0 for t = 0. therefore, we have Theorem. For the stability of the full-discrete Galerkin FEM scheme (11) of advection equation (1), the necessary condition is (x,0) > 0, the initial function u(x, 0) must be a monotonously increasing. 3.. The stability analysis of Leap-Frog full-discrete difference scheme. By taking the central difference scheme to the temporal term in (8), we have the Leap-frog full discrete difference scheme, u n+1 u n 1 (15) + (1 )u n u n +1 un 1 + (u n +1 ) (u n 1 ) = 0 t By expanding its terms as Taylor series, u n+1 u n 1 t (1 )u n = t u 6 t 3 t + O( t 4 ) ( = (1 )u n ) u 6 3 x + O( t 4 ) u n +1 un 1 (u n +1 + un 1 )(un +1 un 1 ) = [ u 6 3 x + O( x 4 ) ] [ ] u n + u x + O( x 4 ) We have the modified differential equation of (15), { (16) t + u = u x + 3 u t u + u ( 3 ) t u3 3 u t u 3 u 3 6 x + O( t 4, x 4 ) 3 The scheme (15) is stable only if the second-order dissipative coefficient in (16) is positive, (3u t x ) > 0. There are two cases in which the condition is matched, 1) In the case that < 0 and u t x < 3 for any t > 0, the scheme is stable. In the case that (x,0) > 0 and 3 < u t x < 1 where 0 < 1 for any t > 0, the scheme is also stable. Therefore, we have

5 340 X. YANG, G. WANG, AND Y. LIU AND X. HOU Theorem 3. For the Leap-frog full-discrete difference scheme (15), the necessary condition of computational stability is (x,0) < 0 and u t x < 3, or (x,0) > 0 and 3 < u t x < 1 (0 < 1). 4. Several Numerical examples In order to clarify the relationship among the nonlinear computational instability, scheme and initial values further, we have the following numerical examples. By differentiating the time-derivative terms in (10), we have (17) C n+1 C n 1 = t x [( 1)U n V n ]S n S n+1 S n 1 = t x [( 1)U n + V n ]C n U n+1 U n 1 = t x ( 1)Sn C n V n+1 V n 1 = 0 Having x = 0.1, t = 0.004, and initial values C 1 = C 0, S 1 = S 0, U 1 = U 0, V 1 = V 0 (the values of C 0, S 0, U 0, V 0 displayed in table 1) The results of the numerical examples show that, the nonlinear computational stability is very closely related to the parameter and initial values. Especially, when = 1 and = 3 the schemes are stable for any initial values we taken. When = 3 and = 1, the cases are ust corresponding to the degenerated Galerkin FEM scheme and a transient energy conservation scheme respectively. These results show that the energy conservation scheme and FEM scheme are much more easier stable. 5. Summary and Discussion Through the above stability analysis and numerical examples, we get the results which extend the analysis in [5-6] concerning computational stability. They are simply summarized as following. (1) In this paper, all time-differential terms are approximated as explicit Leap-frog schemes, results show that the nonlinear computational instability more likely occur in these schemes. () From the qualitative analysis of differential equations, we know that, when the initial function u 0 (x) is monotonous non-decreasing, u 0 (x) 0, the global classical solution of nonlinear advection equation t + u = 0 exists in upper half-space (u(x, 0) = u 0 (x), t 0, ). But while u 0 (x) < 0, there is no global classical solution, rather, the blow-up must will occur in finite time. So the criterion ( (x,0) > 0) obtained from heuristic stability analysis of Galerkin FEM approximation is ust consistent with that of stability theories of differential equation. From both the qualitative theories analysis and numerical examples, we can see that generally the stability of Galerkin FEM approximation is better than that of difference scheme. Further, the computational precision of Galerkin FEM approximation is better than that of difference scheme. The point is ust we always emphasize that the difference equation should keep as much as possible the physical nature of original differential equations, It is also the precondition of computational stability.

6 COMPUTATIONAL STABILITY OF GALERKIN FEM AND FDM 341 Table 1. The computational stability cases of (17) Group Initial values stability C 0 S 0 U 0 V 0 I unstable 1/ stable /3 stable II unstable 1/ stable /3 stable III unstable 1/ stable /3 stable IV unstable 1/ stable /3 stable V stable 1/ stable /3 stable 1 stable V I unstable 1/ stable /3 stable 1 stable (3) It is shown in Table 1 that the scheme is more likely stable when = 1 and 3 than when = 0 and 1, because at the cases = 1 and 3, the energy conservation is almost kept. So, constructing the schemes which keep energy conservation is useful way to overcome the nonlinear computational instability. In other words, destroying the energy conservation may lead to nonlinear computational instability more easily. (4) The instability criteria show that whether the nonlinear computational instability occur is closely related to initial function u(x, 0). (5) It should be noted that the stability criteria obtained by heuristic method are only necessary conditions for computational stability. Although they are weak condition, we can use them to analyze the stability of some schemes which are too complex to apply delicate analysis and save huge unnecessary computation. (6) The nonlinear advection equation we discussed in this paper is named as shock equation in high-speed fluid dynamic, generally, its global classical solution does not exists, and blow-up will occur in finite time. That is the

7 34 X. YANG, G. WANG, AND Y. LIU AND X. HOU necessary condition of nonlinear computational instability. Some work has shown that there is not nonlinear computational instability if the global classical solution of the equation exist. In another word, if the solution of original differential equation is not smoothly (showing discontinuity and bifurcation), and matches the instability conditions (especially the improper initial conditions), then the nonlinear computational instability may occur. References [1] Zeng Qingcun, Several problems of the computational stability.chinese Journal of Atmospheric Sciences (in Chinese), :3,181. [] Shuman,F.G., Analysis and experiment in nonlinear computational stability. NMC office Note. 94. [3] Zeng Qingcun, Ji Zhongzhen, On the computational stability of evolution equations. Chinese Journal of Computational Mathematics (in Chinese). 3:1, pp [4] Ji Zhongzhen, Wang Bin, Further discussion on the construction and application of difference scheme of evolution equations Chinese Journal of Atmospheric Sciences (in Chinese). 5:(1991), pp [5] Lin Wantao, Ji Zhongzhen, Li Shuanglin, Yang Xiaozhong, 000. Research on the comparative for conservation schemes and non-conservation schemes. Progress in Natural Science (in Chinese). 10, pp [6] Ji Zhongzhen, Lin Wantao, Yang Xiaozhong, 001. Problems of nonlinear computational instability in evolution equations. Advances in Atmospheric Sciences. 3, pp [7] Xin Xiaokang, Liu Ruxun, Jiang Bocheng, Computational Fluid Dynamics (in Chinese) Changsha: University of Science and Technology for National Defense Press. pp [8] Wu Jianghang, Han Qingshu, The Theory, Method and Application of Computational Fluid Dynamics (in Chinese). Beiing: Science Press, pp [9] Hirt, C. W., Henistis stability theory for finite difference equation. J. Comp. Phys.., pp School of Mathematics and Physics, North China Electric Power University, Beiing 1006, China Center for Numerical Prediction Research, Chinese Academy of Meteorological Sciences, Beiing , China School of Mathematics and Physics, North China Electric Power University, Beiing 1006, China