Comment on "Improvements for drift-diffusion plasma fluid models with explicit time integration"
aa r X i v : . [ phy s i c s . c o m p - ph ] J u l Comment on ”Improvements for drift-diffusionplasma fluid models with explicit time integration”
Jiayong Zou
Southwest Electric Power Design Institute Co., Ltd., China Electric PowerEngineering Consulting Group, Chengdu 610021, ChinaE-mail: zoujiayong [email protected]
Abstract.
Recently, J. Teunissen reported a fully explicit method, namely thecurrent-limit approach, which claimed to overcome the dielectric relaxation timerestriction for the drift-diffusion plasma fluid model. In this comment, we point outthat the current-limit approach is not mathematically consistent, and discuss aboutthe possible reason why the inconsistency was not visibly noticed.
Keywords : fluid model, plasma, consistency, current-limit approacha short note highlighted in red is added on 31 July, 2020.Recently, this note was accepted by Plasma Sources Sci. Technol., together withthe response by J. Teunissen. In the response, it was claimed that the consistency shouldbe discussed when both △ t and △ x approach 0. When these two parameters approach0, the current-limiting approach turns off, thus the method is consistent. We in factdisagree with this argument.Considering a simple problem, ∂n∂t + ∂n∂x = 0 , x ∈ [0 , π ] (1)with n ( x,
0) = 1000000 sin( x ) and n (0 , t ) = n (2 π, t ). Clearly, the exact solution is n ( x, t ) = 1000000 sin( x − t ).Following the current-limiting method, we solve it using the following scheme: ∂n∂t + \ f i +1 / − \ f i − / ∆ x = 0 , (2) \ f i +1 / = ( t f i +1 / > t f i +1 / f i +1 / ≤ t (3)where f / is an upwind flux. The above scheme should work if the current-limitingapproach was correct. Does anyone think the above scheme will give correct results?We are not willing to submit a comment to the response which is a waste ofreviewers’ time, so we leave an additional note in this comment. omment on ”Improvements for drift-diffusion plasma fluid models with explicit time integration” x → t →
0, namely, the truncation error should vanish.Without loss of generality, we omit the diffusion term in the fluid model. Thetransport equations in the fluid model may be written as ∂n∂t + ∂f∂x = s ( n ) , (4)and the semi-discretized form of Eq. (4) is, dn i dt + f i +1 / − f i − / ∆ x = s ( n i ) . (5)Then, Eq. (5) is consistent if it always converges to Eq. (4) as ∆ x →
0, namely dn i dt + f i +1 / − f i − / ∆ x = s ( n i ) + O (∆ x p ) , with p > . (6)According to the current-limit approach, if the flux f i + > f max where f max = ε Ee ∆ t (see Eqs. (17) and (18) of [1]), the flux f i + is limited to be d f i + = f max , and Eq. (5)reads dn i dt + \ f i +1 / − f i − / ∆ x = s ( n i ) . (7)Comparing Eqs. (7) and (5), when the current-limit approach is turned on, anotherproblem is solved dn i dt + f i +1 / − f i − / ∆ x = s ( n i ) + f i +1 / − f max ∆ x , (8)namely, an additional term f i +1 / − f max ∆ x is added to the source term. Moreover, whenboth f i +1 / and f i − / are greater than f max , both of them will be limited, therefore, Eq.(5) becomes dn i dt + f max − f max ∆ x = s ( n i ) ⇒ dn i dt = s ( n i ) , (9) omment on ”Improvements for drift-diffusion plasma fluid models with explicit time integration” Table 1.
Numerical convergence rate for Eq. (10) solved with Eq. (11)∆ t error ( | exp(1) − y h (1) | ) numerical convergence rate0.04 0 . . . . . which is obviously different from the initial problem (Eq. (4)), namely, the convectionterm is directly omitted in the computation. In either case, the current-limit approachdoes not converge to Eq. (4) which implies that the scheme is not consistent.In [1], a convergence order in time was numerically observed (Fig. 3 and Fig. 5 in[1]). We present the following example to show this is possible in a numerical experimenteven for an inconsistent approach: dydt = y, y (0) = 1 . (10)We solve Eq. (10) with the following scheme until t = 1, y n +1 − y n ∆ t = 1 . × y n , y (0) = 1 . (11)Results in Tab. 1 show a numerically observed first order convergence. However, Eq.(11) is not consistent with Eq. (10), but is consistent with another different problemsimilar to Eq. (10): dydt = y + 0 . y, y (0) = 1 . (12)Now we discuss on the possible reason for why there were no visible differencesbetween the current-limit approach and the explicit scheme shown in [1]. In the fluidmodel, f = nv and s ( n ) = αn | v | = α | f | , with α not small, and typically α ≫
1. Whenboth f i +1 / and f i − / are limited, f i − / − f i +1 / may cancel to a large degree; when f i − / < f max < f i +1 / (i.e., only f i +1 / is limited), because f i +1 / and f i − / are thefluxes of a same cell, they are generally close to each other, therefore, f max is close to f i +1 / . In either case, the additional term may be much smaller than s ( n i ). Therefore,the inconsistency may not be visibly noticed. This coincides with the observation in theexample of Eq. (10) and Eq. (11).Finally, we wish to emphasize that in this comment we focus on the mathematicalcharacteristics of the current-limit approach itself, not on a possible way to get a visuallysimilar result. We feel that a mathematically correct scheme is preferred for reliablesimulations. References [1] Jannis Teunissen. Improvements for drift-diffusion plasma fluid models with explicit timeintegration.
Plasma Sources Science and Technology , 29: 015010, 2020. omment on ”Improvements for drift-diffusion plasma fluid models with explicit time integration” [2] Peter L. G. Ventzek, Timothy J. Sommerer, Robert J. Hoekstra, and Mark J. Kushner. Two-dimensional hybrid model of inductively coupled plasma sources for etching. Applied PhysicsLetters , 63(5):605-607, 1993.[3] Bo Lin, Chijie Zhuang, Zhenning Cai, Rong Zeng, and Weizhu Bao. An efficient and accurate MPI-based parallel simulator for streamer discharges in three dimensions.
Journal of ComputationalPhysics , 401:109026, 2020.[4] Andrea Villa, Luca Barbieri, Marco Gondola, and Roberto Malgesini. An asymptotic preservingscheme for the streamer simulation.
Journal of Computational Physics , 242:86-102, 2013.[5] Andrea Villa, Luca Barbieri, Marco Gondola, Andres R. Leon-Garzon, and Roberto Malgesini.An efficient algorithm for corona simulation with complex chemical models.
Journal ofComputational Physics , 337:233-251, 2017.[6] Arnold D.N. Stability, Consistency, and Convergence of Numerical Discretizations. In: EngquistB. (eds)