Journal of Mathematics

Volume 2019, Article ID 8649308, 16 pages

https://doi.org/10.1155/2019/8649308

## A Leap-Frog Finite Difference Method for Strongly Coupled System from Sweat Transport in Porous Textile Media

^{1}College of Mathematics and Statistics, Shenzhen University, China^{2}Shenzhen Key Laboratory of Advanced Machine Learning and Applications, China^{3}Guangdong Key Laboratory of Intelligent Information Processing, China

Correspondence should be addressed to Qian Zhang; nc.ude.uzs@qgnahzam

Received 25 March 2019; Accepted 15 May 2019; Published 12 June 2019

Academic Editor: Tepper L. Gill

Copyright © 2019 Qian Zhang and Chao Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we present an uncoupled leap-frog finite difference method for the system of equations arising from sweat transport through porous textile media. Based on physical mechanisms, the sweat transport can be viewed as the multicomponent flow that coupled the heat and moisture transfer, such that the system is nonlinear and strongly coupled. The leap-frog method is proposed to solve this system, with the second order accuracy in both spatial and temporal directions. We prove the existence and uniqueness of the solution to the system with optimal error estimates in the discrete norm. Numerical simulations are presented and analyzed, respectively.

#### 1. Introduction

Single/multicomponent flow in porous textile media attracted considerable attention in the last several decades. See [1–4] for the single-component models and [5–9] for the multicomponent models. In this paper, we study the multicomponent sweat transport coupled with vapor and heat in porous textile media. In [10], Ye et al. proposed a quasi-steady-state single-component model which consists of a steady-state air equation and dynamic state equations for other components. Under certain conditions, the multicomponent model reduces to a new single-component model, and the physical process can be viewed as sweat transport (vapor and heat flow) governed by the conservation of mass and energy: where is the porosity of the media, is the vapor concentration, is the temperature, is the thermal conductivity, is the latent heat of evaporation/condensation, and is the molecular weight of water. The effective volumetric heat capacity is defined by where is the molar heat capacity and is the volumetric heat capacity of fiber.

By Darcy’s law, the gas velocity is defined aswhere is the permeability and is the dynamic viscosity, which usually is density-dependent for the compressible flow. Here we choose a linear form of , where is a certain constant.

By the Hertz-Knudsen equation [11], the phase change rate is defined aswhere is a positive constant, the saturation pressure is determined from experimental measurements [12], and the pressure is given by , where is the universal gas constant.

With nondimensionalization, the sweat transport process (1)-(2) can be described by the following system:where , , , and is a smooth and increasing function satisfying .

Since the right boundary is exposed to environment and the left boundary is connected to the body, we consider commonly used Robin type boundary conditionsand the initial conditionsPhysically, parameters , , , , , and are nonnegative constants [1, 2, 6]. We define initial condition parameters with and being positive constants.

Due to the strong nonlinearity and the coupling of the system, both theoretical and numerical analyses of the system are difficult. Numerical analysis for some related systems of parabolic/elliptic equations can be found in [13–20]. Existence and uniqueness of a classical solution for a steady-state model was given in [10]. Existence of a weak solution for the corresponding dynamic models was given in [21, 22]. Positivity of temperature and nonnegativity of vapor density were also proved here. Recently, a finite difference method second-order in space and first-order in time for the system (6)-(12) was presented in [23], where the backward semi-implicit Euler scheme is applied in the temporal direction and central finite difference approximations are used in the spatial direction. In [23], authors presented optimal error estimates under the assumption that the step size and are smaller than a positive constant.

In this paper, we propose an uncoupled leap-frog finite difference method for the system (6)-(12) with second-order accuracy in both spatial and temporal directions. We prove the existence and uniqueness of a solution to the finite difference system with optimal error estimates in the discrete norm, under the condition that the mesh size and are smaller than a positive constant which depends solely upon the physical parameters involved in the equations. Due to the strong nonlinearity and the coupling of equations, the method presented in [23] does not apply to the leap-frog scheme directly. One of the difficulties is to show convergence of the numerical solution without restriction on the grid ratio. In this paper, we assume that the solution to the system (6)-(12) satisfies that for some positive constants , , , and .

The manuscript is organized as follows: in Section 2, we present an uncoupled leap-frog finite difference method for the nonlinear sweat transport system. In Section 3, we prove the existence and uniqueness of the solution to the sweat transport system with the optimal error estimate in the discrete norm. Numerical results will be presented in Section 4 to support our theoretical results.

#### 2. The Leap-Frog Finite Difference Scheme

For convenience of calculations, we add the equation (6) times into the equation (7); thus the governing system (6)-(7) can be modified as Due to the practical interest in a long time period, say 8–24 hours, we present an uncoupled leap-frog finite difference scheme in the temporal direction and the central finite difference (volume) scheme in the spatial direction for the above system with the initial/boundary conditions (8)-(12).

Let be a positive number, let be a uniform partition in , and let be a uniform partition in , where and are the step size in the spatial and temporal directions, respectively. Denote in the spatial cell and let be a mesh function defined on , where . Some notations are introduced below: from which The discrete system is defined byand the discrete initial conditionswhere

The computational procedure of the uncoupled leap-frog scheme at each time step is listed below:

*Step 1. *The vapor concentration can be calculated by solving the tridiagonal linear systems defined in (18)-(20).

*Step 2. *With the updated vapor concentration , we can get and correspondingly.

*Step 3. *Finally, the temperature can be obtained by solving the tridiagonal linear system (21)-(23).

#### 3. The Leap-Frog Scheme and the Optimal Error Estimate

In this section, we will show the existence and uniqueness of the solution to the system (18)-(26) with optimal error estimates in the discrete norm. Let and be two mesh functions on . We define the inner product and norms by

Let be the solution of the system (6)-(12) and , . The error functions are defined byWe state our main result in the theorem below:

Theorem 1. *Suppose that the solution of the system (6)-(12) is in , satisfying (13). Then there exist positive constants and , independent of and , such that, when , the finite difference scheme (18)-(26) is uniquely solvable and*

To prove the theorem, we make a stronger assumption that there exists , independent of , such that the inequality,holds for . We prove the assumption and the theorem by induction method. By the initial condition (26), this is true for . In the next subsection we will show that this is also true for . In this part, we let be a generic positive constant, which is associated with the physical parameters , , , , , , , the parameters involved in initial and boundary conditions and the solution of the system (6)-(12). is independent of time step , mesh size , , and constant .

##### 3.1. The Leap-Frog Scheme and Preliminaries

For convenience of calculations, we further introduce some notations. Let ; thus the sweat transport system (6)-(7) can be reduced towith the initial and boundary conditionsThe discrete leap-frog system (18)-(23) is modified as

Let and . We denote by and the corresponding finite difference solution andWe getand the initial conditionswhere Subtracting the system (36) from the system (38), we get the error equations

andand by (40), we can directly derive the inequality To prove our main theorem, the following formula will be often used:In the following lemma, we present discrete Sobolev interpolation formulas and the proof can be found in [24].

Lemma 2. *Let and be two mesh functions. Then for any positive constant ,*

Lemma 3.

*Proof. *From (30) for , we haveWhen , with the inverse inequality we haveWhen , by taking in Lemma 2,The first part of (54) is obtained and the second part and the inequality (55) can be proved similarly.

In addition, by Lemma 3, there exist constants and such that, when ,and

##### 3.2. The Existence and Uniqueness

Since the coefficient matrix in the system (18)-(20) is strictly diagonally dominant, thus the system (18)-(20) has a unique solution . Here we will discuss the boundedness of .

Multiplying (41)-(43) by , , and , respectively, we getWith (44), (51), (59), (60), and (40), we see that and by using (60) again, we have and with (52), Substituting the last three equations into (61) results in where we have noted . Moreover, by the assumption of the induction,Since we have the fact that , thus When , we can get the inequality asSince are independent of , by (13) when and are small enough,

Now we try to prove our main theorem. By noting (44), (60), (40), and Lemma 3,We can see that, when , the assumption of induction and (68) show that and when , by (68), which means there exists an independent of , such thatMultiplying the error equation (41) by leads tothat is,We can see that, when , and when , with Lemma 2, where is independent of . Then there exists , when ,