In the course of weak formulation, if one encounters a term of integral of derivative of flow variable, how would one treat it?
I mean a term like this integral(g*h*db/dx) or integral(g*eta*db/dx) or integral(g*b*d(eta)/dx). where h, b, eta are all flow variables.
In this case, integration by parts would still bring us back to the same term. So what is the way around that?
This term is arising from the source term in q equation i.e. g*h*db/dx or g*eta*db/dx .
I don't understand pretty well. What is the dependence of g, h, db/dx? I suppose is on x, what are the integral limits? Would you mind to write the equation you have?
g is constant but h and b depend on x. But I have already sorted it out. Thanks.
If you perform a partial integration, do not forget to introduce the boundary terms for both CG and DG, and additionnally the interface contribution for DG; otherwise you will have a non-consistent formulation.
Everyone, please help!!
DG formulation for Shallow water equations
How can one correctly represent the nonlinear terms like Hu^2 or q^2/H or q*u in a Discontinuous Galerkin discretization? Because if I expand the conserved variables namely, the free surface z and the discharge q in series of their various basis functions. Then representation of their ratio q/z or product q*z becomes ratio or product of two series which I am not sure is correct. How does one work around this, please?
Thanks
Hi Chinedu,
there is unfortunately no "perfect" answer. The standard way is to use regular quadrature, and enhance the quadrature precision until the results become independent of that precision. You should consider that even rational functions can be approximated by polynomials (Taylor expansion). For some cases, you can use specific quadrature rules with appropriate weight functions (eg. based on zeros of Jacobi polynomials for polynomial weights).
For most cases, including shallow water, Euler and Navier-Stokes, our experience is that 2p+1 (with p the polynomial order, and hence the precision required to correctly integrate the mass term, or otherwise stated all terms for a linear equation) is a decent choice. For more non-linear problems, such as turbulence modeling etc, we have used a slightly higher precisions (2p+3).
Sorry for the apparently obvious and not so elegant suggestion,
Koen
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