Fixed-Point and STILS Method to Solve a Coupled System of Transport Equations

Abstract

In this paper, a coupled system of two transport equations is studied. The techniques are a fixed-point and Space-Time Integrated Least Square (STILS) method. The nonstationary advective transport equation is transformed to a “stationary” one by integrating space and time. Using a variational formulation and an adequate Poincare inequality, we prove the existence and the uniqueness of the solution. The transport equation with a nonlinear feedback is solved using a fixed-point method.

1. Introduction

This work is motivated by the crystal dissolution and precipitation model in saturated porous medium [1]. In [1], the authors present a macroscopic model describing ions transport by fluid flow in a porous medium undergoing dissolution and precipitation reactions. Such models received much attention during the past years (e.g., [2–4]).

All these papers deal with the upscaled formulation of the phenomena. A rigorous justification, starting from a well-posed microscopic (pore-scale) model and applying a suitable upscaling, has been given for important classes of problems. For instance, [5] presents homogenization as a method for upscaling and contains an overview with particular emphasis on porous media flow including chemical reactions. In this respect, we also mention [6], where the reaction rates and isotherms are linear (see also [7] or [8]), where nonlinear cases as well as multivalued interface conditions are analyzed.

In [9], the authors study the pore-scale analogue of the model proposed in [1], which is built on Stokes flow in the pores, transport of dissolved ions by convection and diffusion, and dissolution-precipitation reactions on the surface of the porous skeleton. They use regularisation techniques and a fixed-point argument to obtain existence of a weak solution in general domains. The results obtained are a rigorous justification of the macroscopic model in [1].

In this paper, we give a mathematical analysis of the macroscopic model in [1] using the fixed-point theorem and the STILS method to solve the transport equations.

The least squares method is widely used to solve partial differential equations. We can refer to [10, 11] for application on elasticity and fluid mechanics problems. Some general mathematical results have been obtained for this method in the case of first-order time-dependent conservations laws. With the space-time objects below, the STILS method transforms a nonstationary problem into a “stationary” problem by integrating space and time. This “stationary” problem is of advective form. For instance, in [12], an equivalence between the advective formulation and that of anisotropic diffusion is established.

The STILS method is originated to [13, 14]. In [13, 14], a least squares method is used to solve a 2D stationary first-order conservation equation with regularity assumptions on the advection velocity. A comparison between the least squares solution and the renormalized solution in the sense of [15] for some equations is given in [16]. The STILS method leads to some numerical schemes which are much simpler than the usual ones (like the streamline diffusion method, the characteristic method, and the discontinuous finite element method with flux limiter). Some numerical examples are presented in [17–19].

In this paper, we make use of both the fixed-point theorem and the so called STILS method to solve nonlinear transport equations in the model described below.

This paper is organized as follows. Section 2 provides a brief presentation of the model proposed in [1]. Assumptions and preliminary results useful to the resolution are also presented. Section 3 is devoted to an existence and uniqueness theorem. The fixed-point theorem and the STILS method are both used to deal with the above mentioned model.

2. Model Equations and Problem Description

2.1. Model Equations

It is useful to recall the model equations suggested in [1] without going into the details. We assume having two species and , for example, ions, say being a cation and an anion. In addition, there may be a crystalline solid present at the porous skeleton. and may precipitate at the surface of the porous skeleton to form , and conversely, the crystalline solid may dissolve. The stoichiometry of the reaction is supposed to be as follows:

and denote positive numbers. Let , be the molar concentration of in the solution relative to the water volume, and let be the molar concentration of relative to the mass of the porous skeleton. The particle is attached to the surface of the porous skeleton and thus is immobile. The conservation of the corresponding total masses leads to the partial differential equations where the water content is supposed to be constant and not affected by the reaction (1), is the bulk density, is the diffusion/dispersion tensor, and is the specific discharge vector. If we define then equations (2) and (3) imply that the quantity verifies

Another equation for results from a description of the precipitation and dissolution processes. Following the detailed discussion in [1], we have where and are, respectively, the dissolution and precipitation rates and the reaction velocity. is a nonlinear smooth nonnegative function depending on and . A typical example is leading to

Summarizing the discussion done in about precipitation-dissolution reaction, we have for the crystalline solid the equation or equivalently where

which means for and for .

is the set-valued Heaviside function defined by

We make now some assumptions which lead to the model we study in this paper. Equation (4) gives

Then, in this model, we consider equations (2), (5), and (9).

If the dispersive transport is negligible compared to the advective transport, it is reasonable to tend to zero. This assumption cancels the corresponding terms in (2) and (5). We assume also that anywhere; it means in (9). The velocity of the solute transport in (2) is the gradient of the hydraulic potential. Setting the model equations have the following form:

In this paper, we study the coupled system (14)–(16) in addition to the appropriate boundary conditions.

2.2. Problem Description

2.2.1. Notations

Let , be a domain with a Lipschitz boundary satisfying the cone property.

If is given, set . Let where is the outer normal to at . Let

Let be the classical Hilbert space of order , and

We also define the following Hilbert space : equipped with the graph norm:

Let

The problem consists in finding satisfying the following partial differential equation system

Equations (14) and (16) give system . In systems and , initial conditions and inflow boundary conditions are given for and . The technique is to solve first and after . With the solutions and and equation (16), we obtain .

2.2.2. Preliminary Results

The STILS method will be used to solve the problem like in [12, 20]. The method leads to a variational formulation problem, and we use the classical Lax-Milgram theorem to find the solution. So we give the following lemma to prove the bilinear map coercivity.

Lemma 1 (curved Poincaré inequality). There exists a constant such that

The proof is in [20].

Let be a polynomial with respect to with bounded coefficients defined in .

Let be defined by

is also a transport problem with a nonlinear feedback. We use both STILS method and fixed-point theorem to solve the problem. So we need to be bounded in ; hence, we give the following lemma.

Lemma 2. Let be a positive real number and a positive function such that Let be the sequence of functions defined by Then, we have

Proof. Assume that there exists such that We know that the velocity field is bounded in its time component and has a constant dot product (equal to 1, too) with the vector . This ensures (see Proposition 7 in [12]) that is filled by the characteristics in the following ways:
There exists an such that for almost each point of , there exists an integral curve that connect to the space-time inflow boundary: On this integral curve (characteristic), system (29) gives Hence, thanks to (28). Moreover, since is filled by characteristics, we have Finally, we conclude that for any , . This ends the proof of the lemma.

3. Existence and Uniqueness Theorem

Theorem 3. Let and . System (24) has an unique solution .

Proof. The proof will be done in two steps.
Step one. In the system , we assume that the divergence of the velocity is zero; then, we can use the result in [12] or in [21, 22] cited in [12] to prove the existence of the boundary trace on of a function in .
So, using the extension of in system , this later is equivalent to searching in such that with . Now, like in [12], we define the following convex quadratic form : The Gateau derivative of is Hence, is the solution of (37) if and only if satisfies Using the Lax-Milgram theorem, we have the solution of the variational problem (40). Lemma 1 is used to prove the bilinear application coercivity.
Step two. This last step is devoted to the problem
We set which is a polynomial function with respect to with coefficients depending on . Then, becomes This is a transport equation with a nonlinear feedback due to the chemical reactions (dissolution-precipitation). To solve it, we first split (42) into the following systems: In order to solve the nonlinear system (43), we use the fixed-point theory. For and satisfying (28), let be the sequence of functions defined by Using Lemma 2, we prove that is bounded in .
Put solution of Let us show that is a Cauchy sequence in and that its limit is the solution of (42).
Let and be integers such that Equations (47) and (48) give By a variational formulation, (49) give We choose then Hence, Since the function is regular enough, we have So using the Cauchy-Schwarz inequality in the right hand side of (52), we get Hence, Let ; we have Using the curved Poincaré inequality of Lemma 1, we get So, if , the sequence is a Cauchy sequence in which is a Hilbert space then converges to .
Let us now prove that the limit of is the solution of (42). For this purpose, we write Since tends to zero in and , it remains to be proved that tends to zero. We have We know that is bounded and tends to in so tends to zero. Then, is the solution of the first equation of (42). The boundary and initial conditions are obtained by the continuity of the trace mapping.

4. Conclusion

The existence and uniqueness solutions of a coupled transport equations have been proved by a fixed-point and STILS method. Free divergence of the transport velocity is considered in this work, but the method used can be extended to no free divergence case. The techniques used in this paper can be applied in other coupled nonstationary systems. The spatial operator in dimension, for example, can be replaced by an operator in dimension, where the component is the time derivative. The obtained result can be improved adding a concrete example like solving a salt-wedge intrusion model or a coupled system arising in epidemiology. All those things and the numerical simulation will be done in the future work.

Data Availability

There are no data supporting this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this work.

Acknowledgments

This research work was supported by UFR SAT of Gaston Berger University. I appreciate the several discussions with colleagues that help to improve the quality of this work.