# Proof of CFL condition for RKDG scheme

The cfl condition for linear advection equation $$u_t + a u_x = 0$$ using a DG method of degree $k$ polynomials, upwind flux and an RK scheme of $k+1$ stage/accuracy is stated to be $\frac{1}{2k+1}$ for $L^2$ stability. However I have not been able to find a paper which proves this though it is mentioned in some papers that the proof exists for k=0,1,2. Please point out any existing literature on this.

## 1 Answer

As far is I know, a proof for the general case has not been presented so far. The most detailed analysis has probably been performed by Krivodonova & Qin 2013 in this work. Quoting from the conclusion:

We have derived a closed form expressions for the eigenvalues of the DG spatial discretization applied to the onedimensional linear advection equation with periodic boundary conditions and the upwind flux. [...] We have also shown that $(p + 1)(p + 2)$ is a guaranteed bound on the size of the eigenvalues which can be used to compute the CFL condition for large p. However, we have also proved that the growth rate of the largest eigenvalue is less than $(p + 1)^2$. We conjecture that a more accurate rate is proportional to $(p + 1)^{1.75}$. This is in contrast with the currently assumed quadratic rate for the DGM [13] and various spectral methods [12].

• Thank you for the reference which I was not aware of. They study semi-discrete DG scheme which is useful to deduce cfl number for euler scheme. These cfl numbers decrease like 1/p^2 from the eigenvalue analysis. Multi-stage RK schemes have better cfl numbers according to literature but that is what I am looking for a proof. Feb 3, 2016 at 14:15