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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 $CFL = 0.01$.

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  • $\begingroup$ What rate of convergence are you talking about here? $\endgroup$ – wwfe Jul 10 at 14:25
  • $\begingroup$ To measure the spatial accuracy of the DG method. I get order 2 for P1, so thats fine. But using P2 basis functions I do not get a rate of convergence of order 3 for very refined meshes. I have dome some testing and how I calculate my DT makes a huge difference. The only problem is the way I'm calculating it I feel is not correct as if it was, I would be getting the correct rate of convergence. As a note, I start with 10 elements and keep doubling. $\endgroup$ – Simon Jul 10 at 17:13
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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.

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  • $\begingroup$ Thank you for the informative reply. For the case of SSP RK methods (I'm using a 5th-order linear SSP), what would beta and alpha be? For the diffusion term I am using the LDG with alternative fluxes (to achieve a compact stencil). For the convection term I am using the Lax-Friedrichs scheme for the numerical flux. $\endgroup$ – Simon Jul 11 at 0:12
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    $\begingroup$ This depends on the spatial operator (DG), the RK method and the physical system you are considering. $\endgroup$ – ConvexHull Jul 11 at 0:14
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    $\begingroup$ To be more precise: You have to calculate the discrete Eigenvalues of your system. $\endgroup$ – ConvexHull Jul 11 at 0:14
  • $\begingroup$ In terms of the spatial DG operator, I am merely using the original 1D formulation by Cockburn and Shu, i.e. the modal approach. In terms of the physical problem, I am solving the linear advection-diffusion for simplicity. I hope I'm making sense.. $\endgroup$ – Simon Jul 11 at 0:19
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    $\begingroup$ The correct way would be to calculate the characteristic polynomial of your discrete system and plot a stability diagram in a complex plain. $\endgroup$ – ConvexHull Jul 11 at 0:23

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