An Implicit Flux-Corrected Transport Algorithm Used for Gas Discharge Calculations

Abstract

An implicit flux-corrected transport (FCT) and diffusion algorithm was developed and used in many gas discharge calculations. Such calculations require the use of a fine mesh where the electric field changes rapidly; that is, near electrodes or in a streamer front. If diffusion is included using an explicit method, then the von Neumann stability condition severely limits the time-step that can be used; however, this limitation does not apply to implicit methods. Further, for gas discharge calculations including space-charge effects, it is necessary to solve the continuity equations with no negative number densities nor point-by-point oscillation in the number density. This is because the electron number densities are finely balanced with the ion number densities to determine the space-charge distribution and hence the electric field which drives the motion of the particles. An efficient way to solve the particle transport equation, with the required properties, is to use FCT. The most accurate form of FCT developed by the author is implicit fourth-order FCT; hence, the method presented incorporates implicit diffusion into the implicit fourth-order FCT scheme to produce a robust algorithm that has been successfully used in many calculations.

1. Introduction

Gas discharge calculations often involve the solution of continuity equations, including transport and diffusion, for electrons, positive ions and negative ions to produce space-charge distributions that determine the electric field and hence the particle motion. The number densities are finely balanced to give a net charge and there must be no negative values or point-by-point oscillations in any of the number densities; the FCT method has these required properties.

FCT was first developed by Boris and Book [1], and extended and generalised by Zalesak [2], because no single method can produce strictly positive results with no point-by-point ripples. The FCT method has been developed and used over many years, and in 2013 J. P. Book celebrated the 40th anniversary of the first FCT paper [3].

The simplest first-order method that works is upwind differencing, which gives strictly positive results, and never produces point-by-point ripples, but generates serious numerical diffusion [4]. However, the results for just one time-step, using the upwind differencing method, are similar to those for higher-order methods. It is these properties that make the upwind differencing method the base method for FCT algorithms [2].

The upwind differencing method is equivalent to taking only the first term in a Taylor series expansion of the continuity equation in time. If the second term is included, the resulting second-order algorithm is the Lax Wendrof method. This method gives a more accurate solution, but generates ripples and negative values [4]. Similar results are produced by most higher-order methods.

Zalesak’s generalisation of FCT [2] involves developing an anti-diffusive flux using some higher-order methods which will remove the diffusion from the upwind differencing solution to produce the high-order solution everywhere, except where to do so would produce a ripple or a negative value.

The heart of the FCT method is the flux-corrector, which monitors the anti-diffusive flux and modifies it to prevent ripples and negative values. This may seem a drastic measure; however, the flux-corrector only acts at a few locations a few times during the calculation, and the upwind difference solution is very close to the higher-order solution for one time-step. Thus, the high-order solution is used most of the time. Two forms of flux-corrector were developed, one by Boris and Book [1] and the other by Zalesak [2]; in the author’s experience, both methods give equivalent results, and since Boris and Book’s method is simpler that is the one the author uses.

One of the most accurate FCT algorithms developed by the author is an implicit fourth-order scheme by Steinle and Morrow [5], but this scheme does not include diffusion. When diffusion is included in the calculations explicitly, then the von Neumann stability criterion severely limits the time-step possibility. The von Neumann criterion is $s\le 1/2$ with $s=\delta tD/\delta x$² where $\delta t$ is the time-step, $D$ is the diffusion coefficient and $\delta x$ the mesh spacing. Thus, for small mesh spacings, diffusion can dominate the time-step. However, this time-step limitation does not apply for an implicit solution of the diffusion equation.

Thus, implicit diffusion was incorporated into the implicit fourth-order FCT scheme to produce a robust algorithm that is stable, embodies the required FCT properties, and can include diffusion. Many researchers are reproducing and extending the original work produced using this method, so it is appropriate to finally publish the full algorithm.

The implicit scheme has been used for a variety of applications. For example, it was used to describe streamer propagation in air over a 5 cm gap using a moving mesh and ~10⁷ time-steps [6]. These calculations were extended to a gap of 50 cm using a similar number of time-steps to explain the stepped nature of lightning and the positive streamer that rises from the earth to meet a lightning leader from the clouds [7]. These calculations are a tribute to the stability and accuracy of the methods, as errors did not accumulate to destroy the results. Later, the method was used in spherical and cylindrical coordinates to explain positive glow corona [8]. A recent application was to explain the mechanism of Ball and Bead Lightning using spherical coordinates [9,10,11,12], and also the rapid radial expansion of a lightning channel in cylindrical coordinates [13]. In both cases, the electrons had disappeared and continuity equations for positive and negative ions were solved, including ion diffusion. The most recent application was to examine a reduced order model for a streamer discharge in air [14].

The prototype partial differential equation to be solved using finite difference methods for electron and ion motion is

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial \left(w\rho \right)}{\partial x}} - {\displaystyle \frac{\partial }{\partial x}}(D{\displaystyle \frac{\partial \rho }{\partial x}})=0$$

(1)

where $\rho$ is the particle density, $w$ is the drift velocity and $D$ the diffusion coefficient. For the method described here, the time-step used is not limited by the diffusion, but is limited to a Courant number $c \le 1$, where $c=\delta tw/\delta x$.

2. Numerical Methods

The implicit fourth-order space- and time-centred scheme developed by Steinle and Morrow [5] was modified to include implicit diffusion to give the following equation for the high-order scheme of an FCT algorithm:

$$\begin{array}{c}12\delta x_j{\rho }_j^{*} + 3[\delta x_{j+1/2}{c}_{j+1/2}({\rho }_{j+1}^{*} + {\rho }_j^{*}) - \delta x_{j-1/2}{c}_{j-1/2}({\rho }_j^{*} + {\rho }_{j-1}^{*})] + \delta x_{j+\frac{1}{2}}[2 \hfill \\ + c_{j+\frac{1}{2}}^2 - 6s_{j+\frac{1}{2}}]({\rho }_{j+1}^{*} - {\rho }_j^{*}) - \delta x_{j-\frac{1}{2}}[2 + c_{j-\frac{1}{2}}^2 - 6s_{j-\frac{1}{2}}]({\rho }_j^{*} - {\rho }_{j-1}^{*}) = 12\delta x_j{\rho }_j -\hfill \\ 3[\delta x_{j+1/2}{c}_{j+1/2}({\rho }_{j+1} + {\rho }_j) - \delta x_{j-1/2}{c}_{j-1/2}({\rho }_j + {\rho }_{j-1})]+\delta x_{j+1/2}[2 + c_{j+1/2}^2 + \hfill \\ 6s_{j+1/2}\left]\right({\rho }_{j+1} - {\rho }_j\left) - \delta x_{j-1/2}[2 + c_{j-1/2}^2 + 6s_{j-1/2}\left]\right({\rho }_j - {\rho }_{j-1}\right)\hfill \end{array}$$

(2)

Here, ${\rho }_j$ is the number density at mesh point $x_{j }$ at time $n$, and ${\rho }_j^{*}$ is the solution to the high-order scheme at time level $n+1$.

$c_{j+1/2}=\delta t{w}_{j+1/2}/\delta x_{j+1/2}$ is the local Courant number, and $w_{j+1/2}$ is the drift velocity at the interface between mesh points $j$ and $(j+1)$. The local diffusion number is

$$s_{j+1/2} = \delta t{D}_{j+1/2}/{\left(\delta x_{j+1/2}\right)}^2$$

(3)

where $D_{j+1/2}=(D_{j+1}+D_j$)/2.

The equation is solved on a non-uniform mesh defined in the set ($x_j|j=1,m$), so that two interleaved mesh spacings are specified as $\delta x_{j+1/2} = x_{j+1} - { x}_{j }$ and $\delta x_j = \frac{1}{2}(\delta x_{j+1/2} + \delta x_{j-1/2})$.

Where possible, c and s are time-centred by making an estimate of the electric field half a time-step ahead ($\delta t/2$). This can be achieved by calculating the solution at $\delta t/2$ using the current electric field at time-step n, estimating the electric field and then performing a full time-step using the time-centred values; other methods are possible. This procedure makes the calculation more accurate, but is time consuming and may not be necessary if the solution is not changing rapidly.

From a solution of Equation (2), the high-order flux for use in the FCT method can be derived:

$$\begin{array}{c}{\mathsf{\Phi }}_{j+1/2}^H = \delta x_{j+1/2}\{[2 + c_{j+1/2}^2]({\rho }_{j+1}^{*} - {\rho }_j^{*} - {\rho }_{j+1} + {\rho }_j) + 3c_{j+1/2}({\rho }_{j+1}^{*} + {\rho }_j^{*} +\\ {\rho }_{j+1} + {\rho }_j) - 6s_{j+1/2}({\rho }_{j+1}^{*} - {\rho }_j^{*} + {\rho }_{j+1} - {\rho }_j)\}/12\end{array}$$

(4)

The low-order scheme used is composed of fully implicit diffusion coupled with upwind differences for the convective term, as follows:

$$\begin{array}{c}2\delta x_j{\rho }_{j }^{\#} - 2\delta x_{j+1/2}{s}_{j+1/2}\left({\rho }_{j+1}^{\#} - {\rho }_j^{\#}\right) + 2\delta x_{j-1/2}{s}_{j-1/2}\left({\rho }_j^{\#} - {\rho }_{j-1}^{\#}\right)=2\delta x_j{\rho }_j - \hfill \\ \delta x_{j+1/2}\left[\left(c_{j+1/2} + \left|c_{j+1/2}\right|\right){\rho }_j + \left(c_{j+1/2} - \left|c_{j+1/2}\right|\right){\rho }_{j+1}\right] + \delta x_{j-1/2}[(c_{j-1/2} \hfill \\ + |c_{j-1/2}|){\rho }_{j-1} + (c_{j-1/2} - |c_{j-1/2}|){\rho }_j]\hfill \end{array}$$

(5)

where ${\rho }_{j }^{\#}$ is the transported and diffused solution of the low-order equation at time level n + 1.

From the solution of the low-order Equation (5), a low-order flux can be derived, namely

$$\begin{array}{c}{\mathsf{\Phi }}_{j+1/2}^L = {\displaystyle \frac{\delta x_{j+1/2}}{2}}[\left(c_{j+1/2} + \left|c_{j+1/2}\right|\right){\rho }_j + \left(c_{j+1/2} - \left|c_{j+1/2}\right|\right){\rho }_{j+1} - \\ 2s_{j+1/2}\left({\rho }_{j+1}^{\#} - {\rho }_j^{\#}\right)]\end{array}$$

(6)

The anti-diffusive flux required to transform the low-order solution into the high-order solution can then be found as

$${\mathsf{\Phi }}_{j+1/2}={\mathsf{\Phi }}_{j+1/2}^H - {\mathsf{\Phi }}_{j+1/2}^L$$

(7)

Then, either Boris and Book’s flux limiter [1] or Zalesak’s flux limiter [2] is applied to these anti-diffusive fluxes to determine what fraction ‘F’ of the anti-diffusive flux can be used without generating ripples or negative values. The fraction F can vary from 0 to 1, where F = 0 means the lower-order solution is used, F = 1 means the high-order solution is used, and an intermediate value, $0\le F\le 1$, means some combination of higher- and lower-order solution is used.

Thus, we obtain a corrected flux ${\mathsf{\Phi }}_{j+1/2}^{^}$ where

$${\mathsf{\Phi }}_{j+1/2}^{^} = F{\mathsf{\Phi }}_{j+1/2}$$

(8)

A new solution is computed at time level $n+1$, as follows:

$${\rho }_j^{n+1} = {\rho }_j^{\#} + ({\mathsf{\Phi }}_{j-1/2}^{^} - {\mathsf{\Phi }}_{j+1/2}^{^})/\delta x_j$$

(9)

For $c\le 1,$ Equations (2) and (5) can be solved efficiently using the Thomas algorithm [15].

Note that source terms are not included in these equations because they do not add any particle transportation or diffusion; they are treated as an additive term at each time step in the calculation.

3. Numerical Tests

Several basic numerical tests were performed to ensure that the method behaves as expected.

3.1. A Square-Wave Dynamic Test

A square-wave of density was used as an input on a uniform mesh with c = 0.8 and s = 10; this means that there is a lot of diffusion and one would expect that using the FCT method was unnecessary. Figure 1 shows the result of this test as the dashed curve where only the high-order method was used; it is clear that the large amount of diffusion did not stop the solution oscillating. When the full FCT method is used, the solid curve is obtained with no oscillations, giving the expected result for the highly diffused square-wave.

3.2. Gaussian Wave Dynamic Test

For an exact test solution in the dynamic case, an initial Gaussian distribution was used:

$$\rho (x, t=0)=A({t_0)}^{-1/2}exp(-{\displaystyle \frac{x^2}{4D{t}_0}})$$

(10)

where $D=5 \times { 10}^5$ ${\mathrm{c}\mathrm{m}}^2{\mathrm{s}}^{-1}$, $t_0=1.81 \times { 10}^{-9}$ $\mathrm{s}$, and A is adjusted to give a maximum amplitude of 10. After a time $t$, the distribution becomes [16]

$$\rho (x, t)=A(t+{t_0)}^{-1/2}exp(-{\displaystyle \frac{x^2}{4D(t+t_0)}})$$

(11)

The results of running this test for 201 time-steps is shown in Figure 2, where the solid curve is the exact solution and the circles are the results of applying the FCT algorithm. Clearly, the fit is excellent and there is no tendency to flatten the curve.

3.3. Steady State Test

For electrons moving into an anode and being absorbed, the appropriate boundary condition is that their density goes to zero [17]. The case where electrons are emitted from a cathode into a uniform electric field, propagating at a uniform velocity across the gap, and then being absorbed by the anode, is a classic singular perturbation problem [18]. This problem has the following exact solution for the steady state [19]:

$$\rho (x)={\displaystyle \frac{A[1-exp(wx/D\left)\right]}{1-exp(wd/D)}}$$

(12)

where A is determined by the flux at the emitting boundary and d is the electrode separation.

The results of the steady state test are shown in Figure 3, where the FCT solution after 3334 time-steps is compared with the exact solution given by Equation (12), with c = 0.8, s = 6.0, and w = 2 $\times {10}^7$ cm/s. The maximum deviation from the exact solution is 0.13%. Fewer time-steps could have been used to achieve this result; however, the result presented shows the extreme stability of the method over many time-steps.

3.4. Diffusion-Free Results

When diffusion is negligible, the algorithm reverts to the fourth-order implicit algorithm developed by Steinle and Morrow [5]. This gives very accurate results with no numerical diffusion or ripples, as demonstrated in Figure 4 for a square wave of electrons moving from left to right, with c = 0.8 and s = 0, and with w = 2 × ${10}^7$ cm/s.

4. Two-Dimensional Calculations

The implicit algorithm has been used to make two-dimensional calculations using the time-split method of Boris and Book [1]. With the time-split method, the one-dimensional code is used along parallel lines of one axis for half a time-step, followed by a parallel line of calculations along the other axis for half a time-step. Tests with square objects show that the object can be moved around an area circling several times with no more distortion than shown in Figure 4 for the case with no diffusion. This method was used by Morrow and Blackburn [20] to complete two-dimensional calculations in a void filled with air; the calculation was carried out in cylindrical geometry with azimuthal symmetry and with no undue complications or complexity.

5. Conclusions

The method presented is an extremely accurate and stable one for computing transport and diffusion under conditions where diffusion dominates, as well as conditions where diffusion has a negligible contribution. The problems of numerical diffusion or oscillatory behaviour due to steep gradients are all overcome by using flux-correction principles. The method involves the use of a single algorithm which can be applied throughout any region; thus, there is no necessity to change algorithms within the domain, which can present many difficulties. Further, the algorithm can be applied to two-dimensional calculations using the time-splitting method.