5

Pros: With trigonometric basis functions your problem size is $N \text{log}(N)$ instead of $N^2$. Stabilization techniques are easy to implement and cheap: Filtering in the modal space. Zero padding in the modal space. No aliasing due to the Galerkin ansatz. Energy/Entropy stable disctretizations, e.g. via a skew symmetric implementation, are quite easy. ...


3

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 ...


3

I believe Numeca is developing a Flux Reconstruction (related to DG) extension of their solver. I am sure other companies are doing so as well. However, commercial codes have a somewhat different set of requirements than academic codes. Commercial codes target speed and robustness more than accuracy, and both of these are harder to get with DG. I think that ...


3

I recommend you first read answer here to find out why $\tilde{E}(\mathbf{k}) \neq \frac{1}{2} \tilde{\mathbf{u}}(\mathbf{k}) \cdot \tilde{\mathbf{u}}(\mathbf{k})$. In order to find velocity profile from Fourier transform ($\mathscr{F}$) of kinetic energy ($E(\mathbf{r}) = \frac{1}{2} \mathbf{u}(\mathbf{r}) \cdot \mathbf{u}(\mathbf{r})$), you could obtain it ...


2

You go the grouping in the term $\nabla u^+-\nabla u^-$ mostly right but not quite. To see this, remember that the functions $\phi_i^{\pm}$ are discontinuous, and that consequently $u_0^+$ is not necessarily equal to $u_3^-$ -- if they were, you'd end up with a function that is continuous at that point. Instead, you need one degree of freedom per adjacent ...


2

The paper of Cockburn and Shu [1] explains this. If the solution is $$ u_h(x) = u_i + u_{xi} \phi(x), \qquad \phi(x) = \frac{x - x_i}{\Delta x/2} $$ Then the limiter is $$ u_{xi} = minmod(u_{xi}, u_i - u_{i-1}, u_{i+1}-u_i) $$ Does your solution representation look as in the first equation above ? Also, the solution with normalised Legendre basis will look ...


2

You can use discontinuous Galerkin methods also for $P_0$ elements. It's true that the gradient in the cell interior is zero, so your formulation will exclusively consist of the jump terms at cell interfaces.


1

To expand on Wolfgang Bangerth's answer, I think P0 DG schemes reduce to two-point cell-centered finite volume schemes. I don't know if DG convergence analysis always includes $p = 0$, but the resulting finite volume schemes can be shown to converge under appropriate "mesh orthogonality" conditions. https://math.unice.fr/~minjeaud/Donnees/...


1

Per the regulations of Stack Exchange. I have included the "answer" to my question. So I managed to fix my problem. First, off I was missing a factor of 0.5 when calculating the surface integral. Additionally, I am using the following normalization $\phi(x) = \sqrt{2m + 1} L(\xi)$.


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