Picard’s iterative method for nonlinear multicomponent transport equations

Abstract: In this paper, we present a Picard’s iterative method for the solution of nonlinear multicomponent transport equations. The multicomponent transport equations are important for mixture models of the ionized and neutral particles in plasma simulations. Such mixtures deal with the so-called Stefan–Maxwell approaches for the multicomponent diffusion. The underlying nonlinearities are delicate and it is not necessary to be an analytical function of the dependent variables. The proposed solver method is based on Banach’s contraction fix-point principle that allows to solve such nonlinearities without making any use to Lagrange multipliers and constrained variations. Such an improvement allows to solve delicate nonlinear problems and we test the application to model with multicomponent transport equations.


Introduction
We are motivated to apply nonlinear multicomponent transport equations, which are given by delicate plasma processes.
ABOUT THE AUTHOR Juergen Geiser, researcher and lecturer at the Ruhr-University of Bochum, Germany, has been involved in teaching and research projects and has collaborated with engineering and physicist groups on numerical modeling of technical and physical models. The research activity refers to the mathematical modeling, numerics, and analysis of transport and flow problems in engineering applications, e.g. groundwater modeling and plasma modeling. He is a specialist in multiscale solvers and iterative solvers and most of the topics of the special issue. Moreover, Juergen Geiser is the author of scientific books and editor of various scientific journals and has thus able to manage the editorial activity. He is also a visiting professor at the Centrale Supelec in the laboratory: Mathematics in Interaction with Computer Sciences, Chatenay-Malabry, Cedex, France.

PUBLIC INTEREST STATEMENT
In this paper a multi component model is presented to simulate ionized and neutral particles in plasma simulations. An improved diffusion operator is presented based on the Stefan-Maxwell approach. Such nonlinear diffusion operators are solved with novel solver methods based on the Banach's contraction fixpoint principle. For higher accuracy, multilevel iterative methods are presented to solve the nonlinearities. An extension to exponential Picard's iterative methods is discussed, which allows to split into linear and nonlinear parts. The novel schemes are tested in blow-up, Bernoulli examples.
A real-life example based on a multicomponent diffusion model is solved with the new methods. The paper also discusses future work to the novel iterative solvers.
We deal with problems of normal pressure, room temperature plasma applications, which are used for medical and technical processes. Here, the increasing importance of plasma chemistry based on the multicomponent plasma is a key factor for such a trend, see for low pressure plasma (Senega & Brinkmann, 2006) and for atmospheric pressure regimes (Tanaka, 2004). Due to the fact of the influence of the mass transfer in the multicomponent mixture, also the standard conservation laws have to be improved, such improvements are well-known in fusion research, see the modeling of in high-ionized plasmas (Igitkhanov, 2011), but only few works are done for weak-ionized plasma for atmospheric pressure regimes.
We concentrate on a diffusion reaction model for a mixture of H, H 2 , H + 2 species. The model equation results in a delicate nonlinear multidiffusion equation. Such nonlinear equations are often solved with standard fix-point iteration schemes. Here, we propose a novel higher order Picard's iterative method, which allows to accelerate the nonlinear solver based on intermediate level. Such a treatment allows to save computational time and we obtained higher order accurate results.
The paper is outlined as follows.
In Section 2, we present our mathematical model. The Picard's iterative methods are discussed in Section 3. The numerical experiments are presented in Section 4. In the contents, that are given in Section 5, we summarize our results.

Mathematical model
In the following, a model is presented due to the motivation in Senega and Brinkmann (2006), which deals with a fluid dynamical description of a plasma model based on a small Knudsen number.
The Knudsen number is the ratio of the mean free path over the typical domain size L of the apparatus. For small Knudsen numbers Kn ≈ 0.01 − 1.0, we deal with a Navier-Stokes equation, where for large Knudsen numbers Kn ≥ 1.0, we deal with the Boltzmann equations.
We deal with the following plasma model of a mixture of H, H 2 , H + 2 .
We take into account the dissociation and ionization reactions, which are given as: where the electron temperature is given as T e = 17, 400 K and the gas temperature values remain constant T h = 600 K. The diffusion coefficients are given as in the following formula: where the parameters are: f ij is a correction factor of order unity, m ij = m i m j m i +m j is the reduced mass, m i mass of species i, m j mass of species j, p pressure, t i , T j the temperature of the corresponding species, and Ω (1,1) ij a collision integral (Hirschfelder, Curtiss, & Bird, 1966).
We assume the following binary diffusion parameters for our experiments: (1)

Iterative method
In the following, we discuss Picard's iterative method with different levels. Picard's iterative methods are known to solve delicate nonlinear problems (see Ramos, 2008Ramos, , 2009).
In the following, we discuss the basic ideas and we develop improved Picard's iterative methods, e.g. exponential schemes, for our special treatments. (9) The basic Picard's iterative method is given as: where the Picard's operator is given as: Picard's operator is applied for the two-level method as: Further, the operator is applied for the three-level method as: Proof We combine two two-level methods, given as: and where if we subtract Equation (23) with (24), we obtain: □

Multilevel iterative method based on Picard's iterative method
The multilevel iterative method is given as: Theorem 1 We have the following construction formula for the k-level iterative method based on Picard's iterative method: • Two-level method is given as: • i-level method with i ≥ 3 is given as: Proof The case i = 2 is the standard Picard method and clear.
For the case i ≥ 3 we have the following complete induction given as: We start from i = 3 (and k = i − 2) and have: The induction step is given as i → i + 1, where we have to multiply (u 1 − u 0 ) to the i-level method as: The same is done for the right hand side and we have proven the formula. □

Remark 2
The next level methods are given as: • Four-level method: • Five-level method:

Exponential Picard's iterative method
In the following, we discuss an extension of Picard's iterative method with a linear operator (exponential idea).
The exponential Picard's iterative method is given as: where the variation of constant formula is applied and we obtain: So the two-level method is given as: Further, the three-level method is given as: The multilevel iterative method is given as: Theorem 2 We have the following construction formula for the k-level iterative method based on Picard's iterative method: • Two-level method is given as: • k-level method with k ≥ 3 is given as: Proof The case k = 2 is the variation of constants with the standard Picard method and is clear.
For the case k ≥ 3, we have the following complete induction given as in the proof of Theorem 1. □

Remark 3
The next level methods are given as: • Four-level method: • Five-level method:

Numerical experiments
In the following experiments, we discuss the improvements of our novel Picard's iterative methods to the standard approaches. We start with test examples of blow-up and Bernoulli's equations, where we could compare to analytical solutions. A multicomponent model is discussed in the next steps and the benefits to the higher order multilevel methods are presented.

First numerical example: blow-up equation
As a nonlinear differential example, we chose the Bernoulli's equation: where the analytical solution can be derived as (see also Geiser, 2008): When we apply p = 2, we have For the numerical experiments, we have the following analytical solution: We apply the Picard's iterative method, which is given in the following as a two-level method. Here, we approximate in each sub-interval [t n , t n+1 ], n = 0, 1, … , N, the integral: For the numerical integration, we apply the different Simpson's rules of higher order.

Remark 4
The three-level method has the benefit of applying the numerical integration for the differences between k and k − 1 solutions of the right-hand side, such that we can accelerate the solver process, while we skip an additional numerical integration in each step.
Based on the reference solution, we deal with the following errors: where method is the different Picard's methods and k is the number of iterative steps.
We apply the Picard's iteration with the two-level method (see Figure 1 and Table 1).
Remark 5 Here, we see the improvement till K = 5, while here the accuracy is reached. The reduction of the errors is done with each iterative step.
We apply the Picard's iteration with the three-level method (see Figure 2 and Table 2).
Remark 6 Here, we see the improvement till K = 4, while here the accuracy is reached. The reduction of the errors is done with each iterative step.
We apply the blow-up experiment with = 0.999 and obtain the following solutions. The Picard's iteration with the two-Level method is given in Figure 3 for = 0.999.
The Picard's iteration with the three-level method is given in Figure 4 for = 0.999.

Remark 7
We discussed the multilevel Picard's method with different iterative steps to a blow-up problem. We saw that we obtain high accurate solutions with k ≈ 3 − 5 iterative steps and that a three-level method can obtain faster numerical results.

Second numerical example: Bernoulli's equation
As a nonlinear differential example, we chose the Bernoulli's equation: where the analytical solution can be derived as (see also Geiser, 2008): Using u(0) = 1, we find that c = 1 + 2 + 4 .
. The computation of the inverse matrices is given as: The numerical errors of the different schemes are given in Figure 6.
The numerical solution of the three-and four-level Picard's fix-point schemes with different iterative steps is given in Figure 7.
The numerical errors of the four-level Picard's fix-point schemes compared with a reference solution of fine time steps of a three-level method (see Figure 8).
The solutions of the numerical experiments are given in Figure 9.

Remark 10
We tested all different level properties, meaning three-level and four-level Picard's fixpoint schemes, with different iterative step sizes, meaning k = 3, 4, 5, 10. For all applications, we saw only marginal differences, such that a three-level method is sufficient to resolve the nonlinear problem. (97)

Conclusions and discussion
We present the coupled model for a multicomponent transport model, which can be applied for solving nonlinear diffusion equations, e.g. for multicomponent plasma transport models. The Picard's methods are flexible and we could derive multilevel methods, with exponential treatments. The benefit of the methods is to resolve the nonlinearity with more accuracy based on the intermediate