Question about Discontinous Galerkin Method
 

Hi,

I use the DG to solve the 1d shallow water.
I want to know how calculate the state value in the Riemann solver ( Roe for example ). If I use the formula in attachments i have minor oscillation but if i use Uh=U0 the result is good.

What is the right ?

Thanks in advance

What is written there in the equation is just the solution representation, i.e. the basis functions times your DOF. For the input into the flux functions, you should evaluate U on the boundaries, then calculate the fluxes from that.
Are you using a modal or a nodal basis? If you are using a modal basis, u0 would be (I'm guessing here) the value of your first Legendre polynomial, i.e. the mean value. That will give you a first order approximation of your fluxes, and thus result in a globally first order method. The correct way to do is indeed to evaluate the basis fully at the location it is needed.

I recommend you do a convergence study for a simple model problem (Burgers, linear advection etc) first to make sure you got everything right.

Thanks for your response,

I use here a legendre polynomial basis, I think that are modal approach ( if it lagrange basis here is nodal ... right ? ). In the case when I use the fully basis, osciallations are generated in the solution in same problems ( Haudraulic jump for example ).

Here, what I want know is how evaluate the physical state to calculate the flux at each edge.

Thanks,

You do that evaluation the full basis / full expansion at the location where you need it. If you are getting oscillations (for a smooth problem), then you will need to check your code. If you are talking about non-smooth problems, then oscillations are to be expected if you don't use any form of limiting.