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Calculate stable time step DG method for advection-diffusion

Computational Science Asked on August 4, 2021

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 $CFL = 0.01$.

One Answer

Generally you should consider:

Convection:

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

Diffusion:

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

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.

Correct answer by ConvexHull on August 4, 2021

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