Commit 30ead698 authored by Holger Class's avatar Holger Class Committed by Martin Schneider

time discretization

parent 9a28187b
......@@ -201,10 +201,49 @@ in the Doxygen documentation at
\url{http://www.dumux.org/doxygen-stable/html-\DumuxVersion/modules.php}.
The documentation includes a detailed description for every model.
\subsubsection{Temporal discretization}
We discretize time with an explicit or implicit Euler
method.
% TODO: make section with more details on temporal discretization
\subsubsection{Time discretization}
Our systems of partial differential equations are discretized in space and in time.
Let us consider the general case of a balance equation of the following form
\begin{equation}\label{eq:generalbalance}
\frac{\partial m(u)}{\partial t} + \nabla\cdot\mathbf{f}(u, \nabla u) + q(u) = 0,
\end{equation}
seeking an unknown quantity $u$ in terms of storage $m$, flux $\mathbf{f}$ and source $q$.
All available Dumux models can be written mathematically in form of \eqref{eq:generalbalance}
with possibly vector-valued quantities $u$, $m$, $q$ and a tensor-valued flux $\mathbf{f}$.
For the sake of simplicity, we assume scalar quantities $u$, $m$, $q$ and a vector-valued
flux $\mathbf{f}$ in the notation below.
For discretizing \eqref{eq:generalbalance} we need to choose an
approximation for the temporal derivative $\partial m(u)/\partial t$.
While many elaborate methods for this approximation exist,
we focus on the simplest one of a first order difference quotient
\begin{equation}\label{eq:euler}
\frac{\partial m(u_{k/k+1})}{\partial t}
\approx \frac{m(u_{k+1}) - m(u_k)}{\Delta t_{k+1}}
\end{equation}
for approximating the solution $u$ at time $t_k$ (forward) or $t_{k+1}$ (backward).
The question of whether to choose the forward or the backward quotient leads to the
explicit and implicit Euler method, respectively.
In case of the former, inserting \eqref{eq:euler} in \eqref{eq:generalbalance}
at time $t_k$ leads to
\begin{equation}\label{eq:expliciteuler}
\frac{m(u_{k+1}) - m(u_k)}{\Delta t_{k+1}} + \nabla\cdot\mathbf{f}(u_k, \nabla u_k) + q(u_k) = 0,
\end{equation}
whereas the implicit Euler method is described as
\begin{equation}\label{eq:impliciteuler}
\frac{m(u_{k+1}) - m(u_k)}{\Delta t_{k+1}}
+ \nabla\cdot\mathbf{f}(u_{k+1}, \nabla u_{k+1}) + q(u_{k+1}) = 0.
\end{equation}
Once the solution $u_k$ at time $t_k$ is known, it is straightforward
to determine $m(u_{k+1})$ from \eqref{eq:expliciteuler},
while attempting to do the same based on \eqref{eq:impliciteuler}
involves the solution of a nonlinear system.
On the other hand, the explicit method \eqref{eq:expliciteuler} is stable only
if the time step size $\Delta t_{k+1}$ is below a certain limit that depends
on the specific balance equation, whereas the implicit method \eqref{eq:impliciteuler}
is unconditionally stable.
\subsubsection{Algorithms to solve equations}
The governing equations of each model can be solved monolithically or sequentially.
......
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