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Non-stationary linear transport model

The weak form of diffusion-type differential equation leads to
$\displaystyle \mbox{\boldmath$K$}$   $\displaystyle \mbox{\boldmath$r$}$$\displaystyle +$   $\displaystyle \mbox{\boldmath$C$}$$\displaystyle \frac{{\rm d}\mbox{\boldmath$r$}}{{\rm d} t} = \mbox{\boldmath$F$},$     (2.8)

where the matrix $ K$ is a general non-symmetric conductivity matrix, $ C$ is a general capacity matrix and the vector $ F$ contains contributions from external and internal sources. The vector of unknowns, $ r$, can hold nodal values of temperature, humidity, or concentration fields, for example.

Time discretization is based on a generalized trapezoidal rule. Let us assume that the solution is known at time $ t$ and the time increment is $ \Delta t$. The parameter $ \alpha\in\langle 0, 1\rangle$ defines a type of integration scheme; $ \alpha=0$ results in an explicit (forward) method, $ \alpha=0.5$ refers to the Crank-Nicolson method, and $ \alpha=1$ means an implicit (backward) method. The appromation of solution vector and its time derivative yield

$\displaystyle \tau$ $\displaystyle =$ $\displaystyle t+\alpha\Delta t = (t+\Delta t) - (1-\alpha)\Delta t,$ (2.9)
$\displaystyle \mbox{\boldmath$r$}$$\displaystyle _{\tau}$ $\displaystyle =$ $\displaystyle (1-\alpha)$$\displaystyle \mbox{\boldmath$r$}$$\displaystyle _t+\alpha$$\displaystyle \mbox{\boldmath$r$}$$\displaystyle _{t+\Delta t},$ (2.10)
$\displaystyle \frac{{\rm d}\mbox{\boldmath$r$}}{{\rm d} t}$ $\displaystyle =$ $\displaystyle \frac{1}{\Delta t}\left(\mbox{\boldmath$r$}_{t+\Delta t}-\mbox{\boldmath$r$}_t\right).$ (2.11)
$\displaystyle \mbox{\boldmath$F$}$$\displaystyle _{\tau}$ $\displaystyle =$ $\displaystyle (1-\alpha)$$\displaystyle \mbox{\boldmath$F$}$$\displaystyle _t+\alpha$$\displaystyle \mbox{\boldmath$F$}$$\displaystyle _{t+\Delta t},$ (2.12)

Let us assume that Eq. (2.8) should be satisfied at time $ \tau$. Inserting into Eq. (2.8) leads to

$\displaystyle \left[\alpha \mbox{\boldmath$K$} + \frac{1}{\Delta t} \mbox{\bold... ... + (1-\alpha) \mbox{\boldmath$F$}_{t} + \alpha \mbox{\boldmath$F$}_{t+\Delta t}$     (2.13)

where the conductivity matrix $ K$ contains also a contribution from convection, since it depends on $ r$$ _{t+\Delta t}$
$\displaystyle \mbox{\boldmath$K$}$$\displaystyle = \int_{\Omega}$$\displaystyle \mbox{\boldmath$B$}$$\displaystyle ^T \lambda$   $\displaystyle \mbox{\boldmath$B$}$$\displaystyle \mathrm{d}\Omega + \underbrace{\int_{{\Gamma}_{\overline{c}}} \mbox{\boldmath$N$}^T h \mbox{\boldmath$N$} \mathrm{d}\Gamma}_{\textrm{Convection}}$     (2.14)

The vectors $ F$$ _t$ or $ F$$ _{t+\Delta t}$ contain all known contributions
$\displaystyle \mbox{\boldmath$F$}$$\displaystyle _t = - \underbrace{\int_{\Gamma_{\overline{q}}} \mbox{\boldmath$N... ..._{\Omega}\mbox{\boldmath$N$}^T\overline{Q}_t\mathrm{d}\Omega}_{\textrm{Source}}$     (2.15)

next up previous contents
Next: Non-stationary nonlinear transport model Up: Solution procedures Previous: Arc-length method   Contents
Borek Patzak 2017-12-30