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Convective Term in the Douglas-Rachford ADI Scheme

July 20, 2015, 7:17 am
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1. Introduction

For the mathematical simulation of thermal field distribution in the ground, it is necessary to account for convective heat transfer (heat transfer by means of mass transfer). Convective heat transfer is caused by the filtration of water into the ground resulting, for example, from precipitation. The temperature distribution is described by the partial differential equation of heat conduction where, in the case of convection, there is a so-called convective term. Since the computational domain is arbitrary, the heat equation is solved numerically by using finite-difference methods (FDM). The Douglas – Rachford ADI scheme is one of these methods. In this paper, we focus on modification of the scheme to account for convection in the computational domain.

2. Unconditionally stable scheme for one-dimensional heat equation with convective term

Let’s consider a one-dimensional heat equation with convective term (1):

(1)   Image may be NSFW.
Clik here to view. \begin{equation*} u'_{t}=(ku'_{x})'_{x}+vu'_{x}+f(x,t),\;\;\; x\in{[0,a]},\; t\in{[0,T]} \end{equation*}

where Image may be NSFW.
Clik here to view. k=k(x,t) is the conduction coefficient, Image may be NSFW.
Clik here to view. v=v(x,t) is the fluid velocity, Image may be NSFW.
Clik here to view. f(x,t) is the heat source or sink density function and Image may be NSFW.
Clik here to view. u=u(x,t) is the sought temperature value at the point Image may be NSFW.
Clik here to view. x at time Image may be NSFW.
Clik here to view. t. A boundary condition of the first type is specified for equation (1)

(2)   Image may be NSFW.
Clik here to view. \begin{equation*} u(0,t)=\mu_{0}(t),\;\;\;u(a,t)=\mu_{1}(t), \end{equation*}

as well as the initial temperature distribution:

(3)   Image may be NSFW.
Clik here to view. \begin{equation*} u(x,0)=u_{0}(x). \end{equation*}

For the numerical solution of problem (1) – (3), we employ the FDM. We denote the spatial mesh with uniform step by Image may be NSFW.
Clik here to view. \omega_{h}=\{x_{i}=ih|i=\overline{0,N},\; h=a/N\} , and the time mesh by Image may be NSFW.
Clik here to view. \omega_{\tau}=\{t_{j}=j\tau|j=\overline{0,M},\; \tau=T/M\}. If we substitute the convective term derivative in equation (1) with central difference, then we obtain the finite-difference equation:

(4)   Image may be NSFW.
Clik here to view. \begin{equation*} \frac{y_{i}^{j+1}-y_{i}^{j}}{\tau}=\frac{k_{i,j+1}^{+0.5}\Lambda^{+}y_{i}^{j+1}-k_{i,j+1}^{-0.5}\Lambda^{-}y_{i}^{j+1}}{h}+v_{i,j+1}\Lambda^{0}y_{i}^{j+1}+f_{i,j}, \end{equation*}

Image may be NSFW.
Clik here to view. y_{0}^{j}=\mu_{0}(t_{j}), \; y_{N}^{j}=\mu_{1}(t_{j}), \; y_{i}^{0}=u_{0}(x_{i}), \; i=\overline{0,N}, j=\overline{0,M},

where

Image may be NSFW.
Clik here to view. k_{i,j+1}^{\pm0.5}=(k(x_{i},t_{j+1})+k(x_{i\pm1},t_{j+1}))/2, \; v_{i,j+1}=v(x_{i},t_{j+1}), \; f_{i,j}=f(x_{i},t_{j})

and Image may be NSFW.
Clik here to view. \Lambda^{-}y_{i}^{j+1}=\frac{y_{i}^{j+1}-y_{i-1}^{j+1}}{h}, \; \Lambda^{+}y_{i}^{j+1}=\frac{y_{i+1}^{j+1}-y_{i}^{j+1}}{h}}, \; \Lambda^{0}y_{i}^{j+1}=\frac{y_{i+1}^{j+1}-y_{i-1}^{j+1}}{2h}}

approximating the differential equation (1) with accuracy Image may be NSFW.
Clik here to view. O(\tau+h^{2}) [1]. Unfortunately, the above difference scheme is conditionally stable [2]. Therefore, instead of the difference scheme (4), the following unconditionally stable scheme [2] with the same order of accuracy is used:

(5)   Image may be NSFW.
Clik here to view. \begin{equation*} \frac{y_{i}^{j+1}-y_{i}^{j}}{\tau}=\mu_{i,j+1}\frac{k_{i,j+1}^{+0.5}\Lambda^{+}y_{i}^{j+1}-k_{i,j+1}^{-0.5}\Lambda^{-}y_{i}^{j+1}}{h}+ \end{equation*}

Image may be NSFW.
Clik here to view. +\tilde{v}_{i,j+1}^{+}k_{i,j +1}^{+0.5}\Lambda^{+}y_{i}^{j+1}+\tilde{v}_{i,j+1}^{-}k_{i,j +1}^{-0.5}\Lambda^{-}y_{i}^{j+1}+f_{i,j},

where

Image may be NSFW.
Clik here to view. \mu_{i,j+1}=\frac{1}{1+0.5h|\tilde{v}_{i,j+1}|}, \; \tilde{v}_{i,j+1}=\frac{v_{i,j+1}}{k_{i,j+1}}, \; \tilde{v}_{i,j+1}^{-}=(\tilde{v}_{i,j+1}-|\tilde{v}_{i,j+1}|)/2,

Image may be NSFW.
Clik here to view. \tilde{v}_{i,j+1}^{+}=(\tilde{v}_{i,j+1}+|\tilde{v}_{i,j+1}|)/2.

3. Convective term in the Douglas-Rachford Scheme

Let us modify the Douglas-Rachford scheme to calculate the thermal field distribution in the ground affected by filtration. Similar to the one-dimensional equation (see section 2), the Douglas-Rachford finite difference scheme is as follows:

(6)   Image may be NSFW.
Clik here to view. \begin{equation*} C_{eff}(T_{i,j,s}^{n})\frac{T_{i,j,s}^{*}-T_{i,j,s}^{n}}{t_{n+1}-t_{n}}=\frac{2(\mu_{+x}^{n}\Lambda_{x}^{+}-\mu_{-x}^{n}\Lambda_{x}^{-})}{x_{i+1}-x_{i-1}}T_{i,j,s}^{*}+ \end{equation*}

Image may be NSFW.
Clik here to view. +\frac{2(\Lambda_{y}^{+}-\Lambda_{y}^{-})T_{i,j,s}^{n}}{y_{j+1}-y_{j-1}}+\frac{2(\Lambda_{z}^{+}-\Lambda_{z}^{-})T_{i,j,s}^{n}}{z_{s+1}-z_{s-1}}+\Omega_{x}^{n}T_{i,j,s}^{*}

(7)   Image may be NSFW.
Clik here to view. \begin{equation*} C_{eff}(T_{i,j,s}^{n})\frac{T_{i,j,s}^{**}-T_{i,j,s}^{*}}{t_{n+1}-t_{n}}=\frac{2(\mu_{+y}^{n}\Lambda_{y}^{+}-\mu_{-y}^{n}\Lambda_{y}^{-})}{y_{j+1}-y_{j-1}}T_{i,j,s}^{**}- \end{equation*}

Image may be NSFW.
Clik here to view. -\frac{2(\Lambda_{y}^{+}-\Lambda_{y}^{-})T_{i,j,s}^{n}}{y_{j+1}-y_{j-1}}+\Omega_{x}^{n}T_{i,j,s}^{*}

(8)   Image may be NSFW.
Clik here to view. \begin{equation*} C_{eff}(T_{i,j,s}^{n})\frac{T_{i,j,s}^{n+1}-T_{i,j,s}^{**}}{t_{n+1}-t_{n}}=\frac{2(\mu_{+z}^{n}\Lambda_{z}^{+}-\mu_{-y}^{n}\Lambda_{z}^{-})}{z_{s+1}-z_{s-1}}T_{i,j,s}^{n+1}- \end{equation*}

Image may be NSFW.
Clik here to view. -\frac{2(\Lambda_{z}^{+}-\Lambda_{z}^{-})T_{i,j,s}^{n}}{z_{s+1}-z_{s-1}}+\Omega_{z}^{n}T_{i,j,s}^{*}

where

(9)   Image may be NSFW.
Clik here to view. \begin{equation*} \Lambda_{x}^{+}T_{i,j,s}=\frac{k(T_{i,j,s}^{n})+k(T_{i+1,j,s}^{n})}{2(x_{i+1}-x_{i})}(T_{i+1,j,s}-T_{i,j,s}), \end{equation*}

Image may be NSFW.
Clik here to view. \Lambda_{x}^{-}T_{i,j,s}=\frac{k(T_{i-1,j,s}^{n})+k(T_{i+1,j,s}^{n})}{2(x_{i}-x_{i-1})}(T_{i,j,s}-T_{i-1,j,s});

(10)   Image may be NSFW.
Clik here to view. \begin{equation*} \Lambda_{y}^{+}T_{i,j,s}=\frac{k(T_{i,j,s}^{n})+k(T_{i,j+1,s}^{n})}{2(y_{j+1}-y_{j})}(T_{i,j+1,s}-T_{i,j,s}), \end{equation*}

Image may be NSFW.
Clik here to view. \Lambda_{y}^{-}T_{i,j,s}=\frac{k(T_{i,j-1,s}^{n})+k(T_{i,j+1,s}^{n})}{2(y_{j}-y_{j-1})}(T_{i,j,s}-T_{i,j-1,s});

(11)   Image may be NSFW.
Clik here to view. \begin{equation*} \Lambda_{z}^{+}T_{i,j,s}=\frac{k(T_{i,j,s}^{n})+k(T_{i,j,s+1}^{n})}{2(z_{s+1}-z_{s})}(T_{i,j,s+1}-T_{i,j,s}), \end{equation*}

Image may be NSFW.
Clik here to view. \Lambda_{z}^{-}T_{i,j,s}=\frac{k(T_{i,j,s-1}^{n})+k(T_{i,j,s+1}^{n})}{2(z_{s}-z_{s-1})}(T_{i,j,s}-T_{i,j,s-1});

and

(12)   Image may be NSFW.
Clik here to view. \begin{equation*} \mu_{+\alpha}^{n}=\frac{1}{1+0.5(\alpha_{i+1}-\alpha_{i})|\tilde{v}_{i,j,s}^{\alpha}(n)|}, \end{equation*}

Image may be NSFW.
Clik here to view. \mu_{-\alpha}^{n}=\frac{1}{1+0.5(\alpha_{i}-\alpha_{i-1})|\tilde{v}_{i,j,s}^{\alpha}(n)|}, \; \tilde{v}_{i,j,s}^{\alpha}(n)=\frac{v_{i,j,s}^{\alpha}(n)}{k(T_{i,j,s}^{n})}

(13)   Image may be NSFW.
Clik here to view. \begin{equation*} \Omega_{\alpha}^{n}T_{i,j,s}=C_{w}(\tilde{v}_{i,j,s}^{+\alpha}(n)\Lambda_{\alpha}^{+}+\tilde{v}_{i,j,s}^{-\alpha}(n)\Lambda_{\alpha}^{-})T_{i,j,s} \end{equation*}

Image may be NSFW.
Clik here to view. \tilde{v}_{i,j,s}^{\pm\alpha}(n)=\frac{(\tilde{v}_{i,j,s}^{\alpha}(n)\pm|\tilde{v}_{i,j,s}^{\alpha}(n)|)}{2}, \; \alpha\in\{x,y,z\}

In formulas (6) – (12) the Image may be NSFW.
Clik here to view. C_{eff}(T_{i,j,s}^{n}) is the effective heat capacity, Image may be NSFW.
Clik here to view. k(T_{i,j,s}^{n}) is the effective conductivity of ground, Image may be NSFW.
Clik here to view. \vec{v}_{i,j,s}(n)=(v_{i,j,s}^{x}(n),v_{i,j,s}^{y}(n),v_{i,j,s}^{z}(n))^{T} is the filtration velocity vector, Image may be NSFW.
Clik here to view. T_{i,j,s}^{n} is the approximate temperature value at the node Image may be NSFW.
Clik here to view. (i,j,s) of the spatial mesh Image may be NSFW.
Clik here to view. \omega_{h}=\{(x_{i},y_{j},z_{s})|i=\overline{0,N_{x}}, j=\overline{0,N_{y}}, s=\overline{0,N_{z}}\} at the time Image may be NSFW.
Clik here to view. t_{n}\in{\omega_{\tau}}=\{t_{n}|n=\overline{0, N_{t}-1}\}. Image may be NSFW.
Clik here to view. C_{w}  in formula (13) defines the volumetric heat capacity of the water.

4. Benckmarks

When using the approach described in section 3 for convective heat transfer, there is no need to change the stability criterion of a difference scheme.
Here is an example of a problem where we compare the computational time using the ADI method with a high filtration velocity setting against the computational time without filtration.

Consider the following computational domain (see Fig. 1): 3 production boreholes and 9 different soils (layers). The geometric dimensions of the computational domain are: length – 60 m, width – 20 m, height – 100 m. A sufficiently high filtration velocity is manually specified for the soil layer A (see Fig. 1) Image may be NSFW.
Clik here to view. \vec{v}(x,y,z;T)=(0;10^{-5}\times(1-w(x,y,z;T));0)^{T} m\s, where Image may be NSFW.
Clik here to view. w(x,y,z;T) is the ice content at point Image may be NSFW.
Clik here to view. (x,y,z) and temperature Image may be NSFW.
Clik here to view. T=T(x,y,z).

The computation for 100 days (computational domain contained more than 1.3 million nodes) was performed on a single-threaded solver employing the ADI method and took 27 minutes, while the computation without filtration took 24 minutes.

Image may be NSFW.
Clik here to view. 3D model of computational domain

Fig. 1: 3D model of computational domain

The results with and without filtration are shown in Figures 2 – 5.

Image may be NSFW.
Clik here to view. Temperature isolines around borehole after 100 days		 Image may be NSFW.
Clik here to view. Temperature isolines around borehole with filtration
Fig. 2: Temperature isolines around center borehole after 100 days (plane section at Image may be NSFW.
Clik here to view. y=70)		 Fig. 3: Temperature isolines around center borehole with filtration after 100 days (plane section at Image may be NSFW.
Clik here to view. y=70)

Note that the solver based on the explicit finite-difference scheme (central difference for the approximation of the convective term was used), is more than 10 times slower in solving the problem with filtration vector Image may be NSFW.
Clik here to view. \vec{v}(x,y,z;T) specified in the computational domain. On the other hand, the time step when running such a solver is much smaller than when using the above solver, so the accuracy of the results is higher.

Image may be NSFW.
Clik here to view. Computation of heat transfer problem in the soil around the well		 Image may be NSFW.
Clik here to view. Computation of heat transfer problem in the soil around the well with filtration
Fig. 4: Temperature distribution after 100 days (plane section at Image may be NSFW.
Clik here to view. x=0)		 Fig. 5: Temperature distribution with filtration after 100 days (plane section at Image may be NSFW.
Clik here to view. x=0)

5. References

1. Samarskii A.A., The theory of difference schemes / A.A. Samarskii.– Moscow: Nauka, 1971. –552 p.
2. Polevikov V.K., Numerical methods of mathematical physics: the course of lectures / V.K. Polevikov. – Minsk: BSU, 2003. – 104 p.

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