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I need to find the CFL Condition of the following Finite Difference Scheme.

$$ u_t + cu_x = 0\\ u_{j+1,m} = Au_{j,m+3} + Bu_{j,m+2} + Cu_{j,m+1} + Du_{j,m} + Eu_{j,m-1} $$ So far I have the following work:

$$ \left(\frac{u_{j+1,m}-Du_{j,m}}{\Delta t}\right) + c\left(\frac{Au_{j,m+3}-Eu_{j,m-1}}{4\Delta x}\right) = 0\\ u_{j+1,m} = \frac{c\Delta t}{4\Delta x}\left(Eu_{j,m-1}-Au_{j,m+3}\right)+Du_{j,m}\\ u_{j+1,m} = \frac{\sigma}{4}\left(Eu_{j,m-1}-Au_{j,m+3}\right)+Du_{j,m}\\ 0 \leq \left|\frac{c\Delta t}{4\Delta x}\right| \leq 1\\ 0 \leq \left|\frac{c\Delta t}{\Delta x}\right| \leq 4 $$ Numerical Domain of Dependence

I have drawn out the Domain of dependence in the image above and solved the CFL condition to be $0 \leq \left|\sigma = \frac{c\Delta t}{\Delta x}\right| \leq 4$, but I do not think this answer is correct. What did I do incorrectly? Is the CFL condition supposed to be a value of 4 on both sides of the domain of dependence from $m$ even though it goes up to $m+3$ but only down to $m-1$?

EDIT: Is the following the correct way to solve this?

$$ x_{m-1} \leq x_m-ct_{j+1} \leq x_{m+3}\\ -\Delta x\leq -c\Delta t \leq 3\Delta x\\ \boxed{-1 \leq \frac{-c\Delta t}{\Delta x} \leq 3} $$

where the boxed solution is the CFL condition.

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  • $\begingroup$ @EditPiAf If taking the derivatives of the scheme with respect to t and x and plugging them into the given equation is not correct, then what should my next steps be from here? I am not sure how to proceed to find the actual solution to the CFL condition. $\endgroup$ Dec 8, 2020 at 18:42

1 Answer 1

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The first 3 equations of OP's attempt modify the expression of the scheme $$ u_{j+1,m} = Au_{j,m+3} + Bu_{j,m+2} + Cu_{j,m+1} + Du_{j,m} + Eu_{j,m-1} \, . \tag{1} $$ A simple comparison with the above time-stepping formula shows that they aren't equivalent. One shouldn't "take the derivatives of the scheme with respect to $t$ and $x$" as proposed in comments to OP.

You may have a read at Sec. 1.6 p. 34 of the book by J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations (SIAM). You are expected to prove something similar to the Theorem 1.6.1 therein:

Theorem. For the explicit difference scheme (1) for the advection equation $u_t+cu_x=0$ with $\Delta t/\Delta x = \lambda$ held constant, a necessary condition for stability is the Courant-Friedrichs-Lewy (CFL) condition $$ \sigma = |c\lambda| \leq \sigma_\max . $$ And the goal is to find the optimal value of the critical Courant number $\sigma_\max$ such that the above property is true. Note that the Theorem 1.6.1 of the book corresponds to the case $A=B=0$, for which $\sigma_\max = 1$. Comparing the numerical and physical domains of dependence looks like a good strategy to solve it. Keep in mind that $c$ has arbitrary sign (a priori), and that (LeVeque, 2002, p. 69)

CFL Condition: A numerical method can be convergent only if its numerical domain of dependence contains the true domain of dependence of the PDE, at least in the limit as $\Delta t$ and $\Delta x$ go to zero.

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