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According to the Courant–Friedrichs–Lewy condition, in the 1 dimensional case we have:

$C=\frac{u\cdot\Delta_t}{\Delta_x} \leq C_{max} $

It is said that $C_{max}$ changes depending on the method used, but for explicit methods it should be $C_{max}=1$.

However, a problem that I did has a different solution. According to the solution, when this scheme: $v_j^{n+1}=\alpha_{-2}v^n_{j-2}+\alpha_{-1}v^n_{j-1}+\alpha_0v^n_j+\alpha_1v^n_{j+1}+\alpha_2v^n_{j+2}$ is applied to solve the advection equation, $u_t+au_x=0$ with fixed courant number $\mu=\frac{k}{h}$, then $x_j-at_n\in[x_j-2nh,x_j+2nh]\Leftrightarrow |a\mu|\leq 2$.

  1. I don't know why the solutions state that $|a\mu|\leq 2$ when the Wikipedia article seems to state that it should be $\leq1$ since this is a 1D explicit scheme?
  2. What is the general method for finding $C_{max}$?
Twilight Sparkle
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1 Answers1

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First, you should know that the CFL condition is a necessary condition of stability for solving PDEs with finite difference method.

For the advection equation in your problem, the general solution is $$u(x)=u(x-at),$$ which means the information propagates with a velocity $a$ in the domain. So the information can travel a distance $|a\Delta{t}|$ in one time iteration. To accurately solve the problem, the space discretization at $x$ should at least contain the interval $[x-|a\Delta{t}|,x+|a\Delta{t}|]$. And the CFL condition is derived from this idea, such that $|a\Delta{t}|\leq{C\Delta{x}}$ where $C\Delta{x}$ is the range of the information propagates and it depends on the space discretization scheme.

In your five point stencil case, the value at $v_j$ is from $v_{j-2}$ and $v_{j+2}$, so $C=2$. If you choose a three point stencil or a first order method(two point stencil, such as upwind method), generally, $C=1$.

For problem 2, one of the general methods for stability analysis in finite different method is called Von Neumann analysis. It can also provide such a same result for this problem.

enigne
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