# CFL Condition of general Finite Difference Scheme

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:

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

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.

• @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. Dec 8, 2020 at 18:42

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.
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.