From 604d511f9a06682b3f6a7f2ff172bf2c68cee274 Mon Sep 17 00:00:00 2001 From: Marco De Lucia Date: Thu, 3 Aug 2023 13:08:08 +0200 Subject: [PATCH] Improved doc about CFL for 2D FTCS scheme --- doc/ADI_scheme.org | 33 +++++++++++++++++++++++++++++++-- doc/ADI_scheme.pdf | Bin 239175 -> 242091 bytes 2 files changed, 31 insertions(+), 2 deletions(-) diff --git a/doc/ADI_scheme.org b/doc/ADI_scheme.org index 2de70da..9408089 100644 --- a/doc/ADI_scheme.org +++ b/doc/ADI_scheme.org @@ -487,10 +487,39 @@ C_{i,j}^{t+1} = & C^t_{i,j} +\\ \end{aligned} \end{equation} -The Courant-Friedrichs-Lewy stability criterion for this scheme reads: +The Courant-Friedrichs-Lewy stability criterion (cfr Lee, 2017) for +this scheme reads: + +#+NAME: eqn:CFL2DFTCS_Lee +\begin{equation} +\Delta t \leq \frac{1}{2 \max(\alpha_{i,j})} \cdot \frac{1}{\frac{1}{\Delta x^2} + \frac{1}{\Delta y^2}} +\end{equation} + + +Note that other derivations for the CFL condition are found in +literature. For example, the sources cited by [[https://en.wikipedia.org/wiki/FTCS_scheme][Wikipedia solution]] give: + +#+NAME: eqn:CFL2DFTCS_wiki +\begin{equation} +\displaystyle \Delta t\leq {\frac {1}{4 \max(\alpha) \left({\frac {1}{\Delta x^{2}}}+{\frac {1}{\Delta y^{2}}}\right)}} +\end{equation} + +We can produce a more restrictive condition than equation +[[eqn:CFL2DFTCS_Lee]] by considering the min of the $\Delta x$ and $\Delta +y$: + #+NAME: eqn:CFL2DFTCS \begin{equation} -\Delta t \leq \frac{(\Delta x^2, \Delta y^2)}{2 \max(\alpha_{i,j})} +\Delta t \leq \frac{\min(\Delta x, \Delta y)^2}{4 \max(\alpha_{i,j})} +\end{equation} + +In practice for the implementation it is advantageous to specify an +optional parameter $C$, $C \in [0, 1]$ so that the user can restrict +the "inner time stepping": + +#+NAME: eqn:CFL2DFTCS_impl +\begin{equation} +\Delta t \leq C \cdot \frac{\min(\Delta x, \Delta y)^2}{4 \max(\alpha_{i,j})} \end{equation} ** Boundary conditions diff --git a/doc/ADI_scheme.pdf 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