Research Article | Open Access
Qian Zhang, Chao Huang, "A Leap-Frog Finite Difference Method for Strongly Coupled System from Sweat Transport in Porous Textile Media", Journal of Mathematics, vol. 2019, Article ID 8649308, 16 pages, 2019. https://doi.org/10.1155/2019/8649308
A Leap-Frog Finite Difference Method for Strongly Coupled System from Sweat Transport in Porous Textile Media
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.
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 , 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 , the phase change rate is defined aswhere is a positive constant, the saturation pressure is determined from experimental measurements , and the pressure is given by , where is the universal gas constant.
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 . 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 , 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 , 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  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 2. With the updated vapor concentration , we can get and correspondingly.
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
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
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 .
Lemma 2. Let and be two mesh functions. Then for any positive constant ,
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
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,