Calculate stable time step DG method for advection-diffusion

Question

For stable time steps for the RKDG method for transport equations we require that

$$ \Delta t \le \frac{\Delta x CFL}{(2k + 1)|\lambda|}, $$ where $\lambda$ is the eigenvalue of our conservation law and $k = 0, 1, \dots$. For diffusion I believe we require that

$$ \Delta t \le \frac{\Delta x^{2}}{\nu}, $$ where $\nu$ is the diffffusion coefficient. To calculate a stable time step I am doing the following, $$ \Delta t \le \min \left\{\frac{\Delta x^{2}}{\nu},\frac{\Delta x CFL}{(2k + 1)|\lambda|}\right\}. $$ It works reasonably well for $k = 1$ up to 160 elements. For $k = 2$, it only produces stable time steps for up to 80 elements. The solution does not blow up but I do not get the correct rate of convergence. As such, I was curious if someone had a literature reference or could provide the correct expression on how to calculate stable time steps that would yield the correct rates of convergence. For the time being I would like to stick with explicit RK methods for simplicity as I'm still learning DG. As a side note, the CFL condition I'm choosing is quite small, i.e. $CFL = 0.05$ to $CFL = 0.01$.

Answer 1

Generally you should consider:

Convection:

$$ \Delta t_C \le CFL \cdot \alpha_{RK}(p) \cdot \frac{\Delta x}{(2k + 1)|\lambda|}. $$

Diffusion:

$$ \Delta t_D \le DFL \cdot \beta_{RK}(p) \cdot \frac{\Delta x^2}{(2k + 1)^2\nu}. $$

Finally:

$$\Delta t = \text{min}(\Delta t_C,\Delta t_D).$$

Here $\alpha$ and $\beta$ are scaling factors for different RK methods depending on the polynomial degree and the spatial operator.

Note that $CFL<=1$ and $DFL<=1$. Moreover these conditions do only hold for Cartesian meshes or to be more precise - for the one dimensional case. For unstructured meshes you also have to consider metric terms.