Hello, everyone:

Is there any general method to deal source terms in PDE, such as du/dt+u*du/dx=f(u) where f(u) is very large at the inlet of a piper or f(u) is the chemical reaction terms.

Thanks.

Hello,

Sure it has! But it's not quite general... You always have a lot of choices depending on your problem.

How are you solving this(these) equation(s)? Are you coding yourself the solver? Do you use a software?

Assuming you're using a finit volume approach, as you discretize your equations for the hole domain, you'll have a linear system like this:

[A]*[u]=[B]

Where [A] you have the coeficients that multiply u, [u] you have the solution and in be [B] you have the independent terms.

Depending in the dependency between f(u) and u, you'll have to linearize that before solving the system - i.e. if you have f(u) = u^2, you'll have to treat like f(u)= u*u', where you know u' and are looking for u.

And then probably you'll have a part of the source term that will be in [B] and a other part in [A], as the source term is not entirely dependent from u nor independent.

I hope I could answer your question.

As it's my first post, and I'm kind of new in CFD, I apologize for any mistakes in my explanation.

Best regards, Victor

Your equation is nonlinear so the first thing to try is a Newton-type method. To do this, write the whole system of PDE as a single statement of the form

F(u) = 0

where in this case, F(u) = u_t + u u_x - f(u). Then take the functional derivative

F'(u)(du) = (du)_t + u (du)_x + (du) u_x - f'(u)(du)

which is linear in du. Now take an initial guess u and solve the linear system

F'(u)(du) = -F(u)

for du. Then update u = u + du and repeat until converged.

If assembling the Jacobian matrix F'(u) is very expensive, then Jacobian-free Newton-Krylov methods (only a preconditioner for the Krylov iteration is actually formed) can be used. Also, various quasi-Newton methods can also work well although they generally don't have quadratic convergence. If the nonlinearity is very strong, a continuation may be needed.