Subsections

• 3.2.1 Initial Conditions

3.2 The Diffusion Equation

In this section, the discretization of parabolic time-variant problems is described. In its simplest representation, the right-hand side of the diffusion equation is time-invariant. Solving a diffusion problem the diffusion equation becomes time-variant. It describes the out-diffusion of matter, driven by its own concentration gradient. The diffusion flux ${\mathrm{\bf J}}$ can be written as

$${\mathrm{\bf J}}=-D \;\operatorname{grad}c$$ (3.45)

where $D$ denotes the diffusion coefficient and $c$ is the concentration of the diffusing material.
Additionally, the conservation of material must be fulfilled

$$\operatorname{div}{\mathrm{\bf J}} = -\frac{\partial c}{\partial t}.$$ (3.46)

After insertion of (3.46) in (3.45), the diffusion equation can be reformulated as

$$\operatorname{div}\operatorname{grad}c = \frac{1}{D} \;\frac{\partial c}{\partial t}.$$ (3.47)

As described in the previous section, this equation can be discretized as

$$\sum_{\forall \;j\;:\;\exists\; \mathrm{edge} \;< p_ip_j >} \frac... ..._{j} - c_{i}) = \frac{1}{D} \;\left(\frac{\partial}{\partial t}c_i\right) \;V_i$$ (3.48)

The time derivative in this formula can be discretized by several methods. By the backward Euler method, the time derivative is approximated by [66]

$$\frac{\partial}{\partial t}c(t+\Delta t)\approx \frac{c(t+\Delta t)-c(t)}{\Delta t}=\frac{c^{k+1}-c^{k}}{\Delta t},$$ (3.49)

with $\Delta t$ the sampling interval. The discrete notation by backward Euler time discretization follows by

$$\sum_{\forall \;j\;:\;\exists\; \mathrm{edge} \;< p_ip_j >} \frac... ...+1} - c_i^{k+1}\right) = \frac{1}{D} \;\frac{c_i^{k+1}-c_i^{k}}{\Delta t}\;V_i.$$ (3.50)

By separating the unknowns to the left-hand side, this expression becomes

$$\frac{1}{\Delta t} \;c_i^{k+1}\;V_i + D \left( - \sum_{\forall \;... ...\frac{A_{ij}}{d_{ij}} \;c_i^{k+1} \right) = \frac{1}{\Delta t} \;c_i^{k} \;V_i.$$ (3.51)

In matrix notation, (3.51) can be written as

$${\mathrm{\bf K}} \cdot {\mathrm{\bf x}} = {\mathrm{\bf B}},$$ (3.52)

with

$${\mathrm{\bf K}}=\frac{1}{\Delta t} \;{\mathrm{\bf M}} + D \;{\mathrm{\bf S}},\qquad$$ (3.53)

$$\qquad\qquad\qquad\qquad x_{i} =c_i^{k+1} \qquad\forall \; i,$$ (3.54)

$$m_{ij} = 0 \qquad\forall \;i,j \;:\; i\neq j,$$ (3.55)

$$m_{ii} =V_i \qquad\forall \; i,$$ (3.56)

$$s_{ij} = -\frac{A_{ij}}{d_{ij}} \qquad\forall \;i,j \; :\;\exists\;\mathrm{edge}\;< p_i p_j >,\qquad\qquad$$ (3.57)

$$s_{ii} =\sum_{\forall\;j\;:\;\exists\;\mathrm{edge} \; < p_ip_j >} \frac... ...ij}}{d_{ij}}=-\sum_{\forall\;j\;:\;\exists\;\mathrm{edge} \; < p_ip_j >} s_{ij} \qquad\forall \; i,$$ (3.58)

$$b_{i} = \frac{1}{\Delta t} \;V_i \;c_i^{k} \qquad\forall \;i.$$ (3.59)

The necessary boundary conditions for this parabolic equation can be handled as shown in the previous section. Also the requirements for an M-matrix (see Section 2.3) are satisfied. The conditions for ${\mathrm{\bf S}}$ can be handled as in the previous section and ${\mathrm{\bf M}}$ consists of positive diagonal entries only. Additionally, an initial condition is required.

3.2.1 Initial Conditions

The initial concentration distribution at initial time $\tau$ is defined as

$$c({\mathrm{\bf x}},\tau)=C({\mathrm{\bf x}}).$$ (3.60)

The discrete system is satisfied by the discrete formulation

$$c_i(\tau)=c_i^0=C_{i} p_i .$$ (3.61)

The concentration distributions $c_i^1, c_i^2, \ldots, c_i^{k+1}$ can be computed by an sequential evaluation of .