1 Introduction and Notations he simplest model to describe a hot dilute electrons moving in a fixed ion background is the following one dimensional Vl

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1 Numerical Study on Landau Damping ie Zhou 1 School of Mathematical Sciences Peking University, Beijing , China Yan Guo 2 Lefschetz Center for Dynamical Systems Division of Applied Mathematics Brown University, Providence, RI 02912, USA and Chi-Wang Shu 3 Division of Applied Mathematics Brown University, Providence, RI 02912, USA Abstract We present a numerical study of the so-called Landau damping phenomenon in the Vlasov theory for spatially periodic plasmas in a nonlinear setting. It shows that the electric field does decay exponentially to zero as time goes to infinity with general analytical initial data which are close to a Maxwellian. he time decay depends on the length of the period as well as the closeness between the initial data and the Maxwellian. Similar pattern is observed if the Maxwellian is replaced by other algebraically decaying homogeneous equilibria with a single maximum, or even by some homogeneous equilibria with small double-humps. he numerical method used is a high order accurate hybrid spectral and finite difference scheme which is carefully calibrated with the well-known decay theory for the corresponding linear case, to guarantee a reliable resolution free of numerical artifacts for a long time integration. 1 zht@sxx0.math.pku.edu.cn. Research supported by NSFGrant IN and ARO Grant DAAG while in residence at the Division of Applied Mathematics, Brown University. 2 guoy@cfm.brown.edu. Research supported in part by NSFGrant DMS and an Sloan Fellowship. 3 shu@cfm.brown.edu. Research supported by ARO Grant DAAG and DAAD , NSF grants DMS and ECS , and AFOSR Grant F

2 1 Introduction and Notations he simplest model to describe a hot dilute electrons moving in a fixed ion background is the following one dimensional Vlasov-Poisson system. Let F (t; x; v) denote the density of electrons in a collisionless plasma, E denote its electric field, then the Vlasov-Poisson system is F t + vf R x + E(t; x)f v =0; (1.1) 1 E x = 1 F (t; x; v)dv 1: (1.2) Here we have normalized all physical constants to be one. A simple steady-state solution (equilibrium) is the Maxwellian distribution F (t; x; v) = 1 p exp( v2 ) m(v); (1.3) 2ß 2 and E(t; x) 0. We may reformulate the Vlasov-Poisson system (1.1) and (1.2) as equations for the perturbations f and e of the equilibrium (1.3) so that F = m(v)+f; E =0+e: We deduce that they satisfy f t + vf x + e(t; x)f v = e(t; x)m 0 (v); (1.4) e x = Z 1 1 f(t; x; v)dv; (1.5) with its corresponding linearized equation (by dropping the term e(t; x)f v "): f t + vf x = e(t; x)m 0 (v); (1.6) e x = Z 1 1 f(t; x; v)dv: (1.7) In 1946, L. D. Landau discovered that waves in a plasma should be damped even in the absence of collisions. More precisely, he has shown that the macroscopic (collective) electric field e(t; x) to the linearized Vlasov-Poisson system (1.6) and (1.7) decays exponentially to zero as time tends to infinity. he effect of the Landau damping, as it was subsequently called, plays a fundamental role in the study of the plasma physics ever since, and it is highly significant from both physical as well as mathematical points of view: Although the Vlasov-Poisson system is time reversible on the particle level, their collective effect is time irreversible. Unfortunately, strictly speaking, the Landau damping is still a linear phenomenon up to now. Despite many significant theoretical, numerical, and experimental work along this direction [15], no rigorous justification of the Landau-damping has been given in a nonlinear, dynamical sense. As a matter of fact, recently, there have been quite some renewed interests [3] as well as controversy about the Landau damping. In [7] and [8], it was proven that even in the linear case, there is no decay at all if the physical space is the whole line. Even in a fixed spatial period, it is of fundamental importance to determine if the nonlinear effect could take over eventually and destroy the decay property of the electric field. It is the purpose of this article to use a highly 2

3 accurate and carefully calibrated numerical scheme to simulate the Vlasov-Poisson system (1.4) and (1.5) over a long time, and to study the time-decay of its electric field je(t; )j 1 = je(t; )j 1 in a fully nonlinear and dynamical setting. In order to extensively study the Landau damping phenomenon, we mainly consider the following type of initial 2aß-periodic perturbation 1 f(0;x;v) =ffl cos(x=a) exp(sin(x=a)) p exp( v2 ): (1.8) 2ß 2 We first reduce this problem to a 2ß-periodic problem. By a direct computation, we notice that if [F (t; x; v);e(t; x; v)] is one of the solutions of (1.1) and (1.2) with initial condition (1.8), so is the rescaled pair F (t; x; v) =af (t; ax; av); E(t; x) = 1 E(t; ax): (1.9) a Notice that the pair of [F (t; x; v); E(t; x)] satisfies F (0;x;v) = [1 + ffl cos x exp(sin x)]» a p exp( (av)2 ) ; (1.10) 2ß 2 or f(0;x;v)=ffl cos x exp(sin x)h a p 2ß exp( (av)2 2 ) i,and je(t; )j 1 = 1 a je(t; )j 1: (1.11) his implies that we can recover all the decay information of the solution to the 2aßperiodic problem (1.8) by studying the standard 2ß periodic problem (1.10) around a rescaled Maxwellian (depending on a) m(av) = p a exp( (av)2 ): (1.12) 2ß 2 hroughout this article, we shall only compute 2ß-periodic problems with this scaled Maxwellian. Our numerical evidence shows that Landau damping does exist for the nonlinear Vlasov-Poisson system (1.1), (1.2) with analytical initial data such as (1.8) which is close to a Maxwellian m(av). he decay rate depends on the parameter a (or equivalently, the length of the period): he larger a is, the slower is the decay rate. For such cases, our numerical simulations indicate very similar results between the nonlinear problem and the linear problem, until machine zero is reached. On the other hand, no exponential decay is observed if the initial data is far from m(av). We also observe that the Landau damping phenomenon is robust: the same conclusions hold if one replaces the Maxwellian by other algebraically decaying equilibria with a single maximum in v, or even by some equilibria close to m(av) with small double-humps! his implies that those double-humped equilibria may be dynamically stable. It is also well-known that many large double-humped equilibria which satisfy the Penrose instability criterion are indeed unstable [6], and there are arbitrarily small BGK waves close to them [5]. his implies no Landau damping is possible in this case. 3

4 In the literature there have been developments of numerical methods to solve the Vlasov-Poisson system (1.1), (1.2) and Landau damping has been used as a test case [2, 4, 10]. hese methods split (1.1) into two one dimensional systems, i.e., first solve for half a time step, and then solve F t + vf x =0 (1.13) F t + E(t; x)f v =0 (1.14) for the second half time step. Characteristic based method has been used in each of the split steps. Unfortunately, such splitting can be at most second order accurate. About Landau damping, exponentially decay in the linear cases is observed in [2] and some no-decaying phenomenon in the nonlinear cases is observed in [2, 4, 10]. he emphasis of this paper is not to develop a new numerical method, rather it is to study Landau damping in the full nonlinear, dynamic setting, by using a high order accurate hybrid spectral and finite difference method, to be described in detail in section 2, which is carefully calibrated with the well-known decay theory for the corresponding linear case, to guarantee a reliable resolution free of numerical artifacts for a long time integration. We carefully apply the principle that any computed feature which disappears after a grid refinement is very likely to be a numerical artifact rather than a phenomenon relevant to the solution of the original PDE. 2 A Description of the Numerical Method o discretize the Vlasov equation (1.4), (1.5), we use a Fourier collocation spectral method in the x direction, a ninth order upwind-biased finite difference method in the v direction to obtain a method-of-lines ODE in t, and then discretize this ODE by the classical fourth order explicit Runge-Kutta method. Several remarks are in order: 1. his method is based on a successful WENO (weighted essentially non-oscillatory, [9]) algorithm to solve the kinetic equations in semiconductor device simulations [1]. As the solutions for the Vlasov equation is quite smooth, the weights in the WENO schemes can be frozen to be the linear weights, resulting in a upwindbiased finite difference approximation in the v direction. he ninth order method we use involves the ten grid points x i 5 to x i+4 to compute the derivative f v at the grid x i, if the coefficient e(t; x i ) is positive. Otherwise, the upwind-biasing would be to the right and the ten points used would be from x i 4 to x i+5. he linear weights can be found in [14]. 2. Since the numerical solution is periodic in x and the solution is quite smooth, a Fourier spectral method is the most natural choice to discretize the x derivative. Fast Fourier ransform (FF) can be handily used to make the computation efficient. 3. he un-split method of lines approach coupled with a fourth order Runge-Kutta method in time, and a small time step required by the CFL stability condition, guarantees a global high order accuracy in space and time. We have performed careful calibrations of the numerical method with the well-known decay theory for the corresponding linear case, to guarantee a reliable resolution free of numerical 4

5 artifacts for a long time integration. Grid refinement study has been performed to make sure that any observed decay or non-decay of the electric field is not a numerical artifact. We will use a rectangular mesh to represent the x-v phase plane with the computational domain f(x; v)j0» x» 2ß; jvj» v max g. he cut-off v max is carefully chosen and closely monitored to make sure that the numerical solution is well below roundoff zero there for all t. In fact, it is found out that an erroneous choice of v max may lead to spurious numerical artifacts such as an increase in the magnitude of e, which seems to converge with a mesh refinement study but would go away when v max is enlarged. We use a uniform mesh in both x and v directions and denote the grid points as (x i ;v j ). he numerical solutions are denoted by e n i and f n ij,fori =1; 2; :::; N and j = M; M +1; :::; M. Periodic boundary conditions are enforced in the x direction, namely f n = f n and i+n;j ij en = i+n en i. Neumann boundary conditions (zero normal derivatives) are imposed in the v direction, which do not affect the accuracy of the method as we choose v max to guarantee that the numerical solution is well below round-off zero there for all t. We now outline the details of the computational procedure: 1. Given the data f n ij, for each fixed i, compute the concentration cn i (the integral on the right hand side of (1.5)) by the rectangular rule in v, which is infinitely high order accurate since f is fast decaying in v. 2. Use FF in x to find the Fourier coefficients of the concentration c n i. 3. Scale the Fourier coefficients of the concentration c n i (divided it by i), and then use inverse FF to find the point values of the electric field e n i. 4. In order to compute the x-derivative, for fixed n and j, use FF in x to find the Fourier coefficients of f n ij. 5. Scale the Fourier coefficients of f n ij (multiplied it by i), and then use inverse FF to get the point values of the x-derivative f x. 6. Compute the v-derivative by the ninth order upwinding-biased finite difference formula. 7. Analytically differentiate m(v) on the right hand side of (1.4). 8. Find the time step t by the CFL condition and solve the method of lines ODE to get f n+1 ij by the classical fourth order Runge-Kutta method. All the computations in this paper were carried out on SUN ULRA-30 workstations with a f77 -fast -r8" compile option. We use the FF subroutine in the IMSL library for the spectral method in x. 3 Numerical Results he main numerical results are presented in this section. We have performed many more numerical tests, including many tests for calibrating purpose to make sure that what we present are not numerical artifacts. However we will show only a selected group of representative results. In all the figures we plot the maximum amplitude, over x, oftheelectric field je n i j as a function of time t n, in a logarithm scale. 5

6 Frame Jun 2000 Frame Jun bump equilibria Figure 3.1: Example 3.1, M= Maxwellians and scaled Maxwellians Example 3.1. We solve the nonlinear problem (equations (1.4), (1.5)) with the following initial data: f(0;x;v)=0:01 cos(x)m(v): (3.1) his corresponds to a 2ß period in x with a small () amplitude perturbation. For comparison, we also compute the linear problem (equation (1.6), (1.7) ) with the same initial data (3.1). he results are plotted in Fig We can observe that the electric field exponentially decays in both the linear and the nonlinear cases in a similar fashion, until machine zero is reached. Example 3.2. We nowchange the initial condition to effectively increase the x-period, by taking a = 2 in (1.12). For a small amplitude perturbation f(0;x;v)= 0:0001 cos(x)m(2v); (3.2) the electric field is still observed to exponentially decay both for the linear and for the nonlinear cases, Fig he tail in the nonlinear case after machine zero is reached is a numerical artifact which goes away with grid refinements. However, when we increase the magnitude of the perturbation f(0;x;v)=0:5cos(x)m(2v) (3.3) then the electric field does not seem to decay at all for the nonlinear case, see Fig We have performed many more numerical experiments with a continuum of amplitudes for Example 3.1 (a=1) and Example 3.2 (a=2). It seems that numerical evidence supports the following plausible conclusions: 6

7 Frame Jun 2000 Frame Jul log(e-max) Figure 3.2: Example 3.2 with a small amplitude perturbation (3.2), M=1024. Frame Jun 2000 Frame Jun Figure 3.3: Example 3.2 with a larger amplitude perturbation (3.3), M=

8 Frame Jun 2000 Frame Jun Figure 3.4: Example 3.3, M=1024. ffl For fixed x-period 2aß, when the amplitude increases, the electric field in the nonlinear problem changes from an exponential decay similar to the linear problem to no-decay. ffl For fixed amplitude in the initial perturbation, when the x-period increases, the electric field in the nonlinear problem decays slower. Also it becomes no-decay with much smaller amplitude in the initial perturbation. he following examples further verify these observations. Example 3.3. We now change the form of the initial perturbation to f(0;x;v) =0:01 cos(x) exp(sin(x))m(v): (3.4) Clearly the electric field decays exponentially both in the linear and in the nonlinear cases, Fig It seems that the form of the initial perturbation has less effect on the decay of the electric field than the amplitude or the x-period. Example 3.4. We nowchange the initial condition to effectively increase the x-period, by taking a = ß in (1.12). For a small amplitude perturbation f(0;x;v) = 0:0001 cos(x) exp(sin(x))m(ßv); (3.5) the electric field is still observed to exponentially decay both for the linear and for the nonlinear cases, Fig he numerical noises can be reduced by refining the mesh in v, Fig However, when we increase the magnitude of the perturbation f(0;x;v)= 0:01 cos(x) exp(sin(x))m(ßv) (3.6) then the electric field does not seem to decay for the nonlinear case, Fig. 3.7, while (of course) it exponentially decays in the linear case. We notice that there are some 8

9 Frame Jun 2000 Frame Jun 2000 Figure 3.5: Example 3.4 with a small amplitude perturbation (3.5), M=1024. Frame Jun 2000 Frame Jun Figure 3.6: Example 3.4 with a small amplitude perturbation (3.5), M=

10 Frame Jun 2000 Frame Jun 2000 Figure 3.7: Example 3.4 with a large amplitude perturbation (3.6), M=1024. Frame Jun 2000 Frame Jul 2000 Figure 3.8: Example 3.4 with a large amplitude perturbation (3.6), M=

11 Frame Jul 2000 Frame Jul 2000 Figure 3.9: Example 3.4 with a large amplitude perturbation (3.6), M=4096. numerical noises in the linear case in Fig hese noises are greatly reduced and eventually disappear when we perform grid refinements twice, see Fig. 3.8 and Fig Example 3.5. In order to show the limitation of our numerical approach, we further increase the effective x-period by taking a = 2ß in (1.12). An initial condition f(0;x;v) = 0:01 cos(x) exp(sin(x))m(2ßv) (3.7) gives the results in Fig. 3.10, where one could not clearly observe the exponential decay even in the linear case, although it does seem that the nonlinear case has more non-monotone behavior for large t. For this example, even for the very small perturbation f(0;x;v)= 0: cos(x) exp(sin(x))m(2ßv) (3.8) we still could not clearly observe the exponential decay of the electric field, either in the linear or in the nonlinear case, see Fig In order to verify that this not numerical, we refine the mesh and get essentially the same picture, see Fig An algebraically decaying equilibrium In this section, we show an example on an algebraically decaying equilibrium. It seems that the results are qualitatively similar to those obtained with the Maxwellians for this example. Example 3.6. We use the initial data (3.1) with m(v) replaced by an algebraically decaying equilibrium: 8 m 3 (v) = 3ß(1 + (v) 2 ) : (3.9) 3 In Fig. 3.13, Fig and Fig we show the time histories of the maximum of the electric fields for the linear and nonlinear problems. It seems that in both 11

12 Frame Jun 2000 Frame Jun 2000 EMAX E Figure 3.10: Example 3.5 with a large amplitude perturbation (3.7), M=1024. Frame Jun 2000 Frame Jun E E E E-06 1E-06 9E-07 8E-07 EMAX 7E-07 6E-07 5E-07 4E-07 3E-07 2E-07 1E E E E E-06 1E-06 9E-07 8E-07 EMAX 7E-07 6E-07 5E-07 4E-07 3E-07 2E-07 1E Figure 3.11: Example 3.5 with a very small amplitude perturbation (3.8), M=

13 Frame Jul 2000 EMAX 1.4E E E E-06 1E-06 9E-07 8E-07 7E-07 6E-07 Frame Jul 2000 EMAX 1.4E E E E-06 1E-06 9E-07 8E-07 7E-07 6E-07 5E-07 4E-07 3E-07 2E-07 1E-07 5E-07 4E-07 3E-07 2E-07 1E Figure 3.12: Example 3.5 with a very small amplitude perturbation (3.8), M=2048. cases the maximum of the electric fields decay exponentially, similar to the result of the Maxwellian case in Example 3.1. We observe some numerical noises but these are pushed" to larger time for more refined meshes, indicating that they are indeed numerical artifacts bump equilibria In this section, we show some examples on 2-bump equilibria. It seems that the results are qualitatively similar to those obtained with the 1-bump equilibria in some cases but different in some others. Example 3.7. We use the initial data (3.1) with m(v) replaced by a 2-bump equilibria m 1 (v) = p 5))2 2ß (exp( (v)2 )+0:1 exp( ((v )): (3.10) 2 2 In Fig. 3.16, we show the 2-bump equilibria (3.10) and the time history of the maximum of electric field for the nonlinear problem. It seems to decay exponentially, similar to the Maxwellian case in Example 3.1. Example 3.8. We now change the initial data to be (3.1) with m(v) replaced by another 2-bump equilibria m 2 (v) = 10a 11 p 2))2 2ß (exp( (av)2 )+0:1 exp( (a(v )); a = ß: (3.11) 2 2 In Fig. 3.17, we show the 2-bump equilibria (3.11) and the time history of the maximum of electric field for the nonlinear problem. It seems to decay, but not always exponentially. his is different from the result of the Maxwellian case in Example

14 Frame Jul 2000 Frame Jun Figure 3.13: Example 3.6, M=1024. Frame Jul 2000 Frame Jun Figure 3.14: Example 3.6, M=

15 Frame Jul 2000 Frame Jul Figure 3.15: Example 3.6, M=4096. Frame Jun 2000 Frame Jun m(v) v (a) 2-bump equilibria (3.10). (b) Electric field, nonlinear problem. Figure 3.16: Example 3.7, M=

16 Frame Jun Frame Jun 2000 m(v) v (a) 2-bump equilibria (3.11). (b) Electric field, nonlinear problem. Figure 3.17: Example 3.8, M= Concluding Remarks We have used a high order and carefully calibrated numerical method to solve the spatially periodic Vlasov-Poisson system to study the so-called Landau damping phenomenon, namely an exponential decay of the maximum of the electric field with time. It seems that for the nonlinear Vlasov-Poisson system, Landau damping exists for analytical perturbations with small amplitude to either a Maxwellian, or to some polynomially decaying equilibria, even to some equilibria with double bumps. his demonstrates that Landau damping is robust. he longer the spatial period, the slower the decay becomes. For some long period cases our numerical method is not powerful enough to detect whether the electric field decays or not. References [1] C. Cercignani, I. Gamba, J. Jerome and C.-W. Shu, Device benchmark comparisons via kinetic, hydrodynamic, and high-field models, Comput. Meth. Appl. Mech. Engin., Vol. 181, 2000, pp [2] C.Z. Cheng and J. Knorr, he integration of the Vlasov Equation in configuration space, J. Comput. Phys., Vol. 22, 1976, pp [3] J.P. Holloway and J.J. Dorning, Undamped plasma waves, Phys. Rev. A, Vol. 44, 1991, pp [4] R.J. Gagne and M.M. Shoucri, A splitting scheme for the numerical solution of a one-dimensional Vlasov equation, J. Comput. Phys., Vol. 24, 1977, pp [5] Y. Guo and W. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., Vol. XLVII, 1995, pp [6] Y. Guo and W. Strauss, Nonlinear instability of double-humped equilibria, J. Ann. Inst. Henri Poincare, Vol. 12, 1995, pp

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