Navier-Stokes treating convection with IMEX scheme

NGSolve
1

I’m looking to solve navier-Stokes equation with an IMEX method which gives a convective term on the form

c(u_{n+1}, u_n, v) = \int_{\Omega} \nabla u_{n+1} u_n \cdot v d\Omega

as this gives unconditional stability. I have tried to do this with a bilinear form which includes a gridfunction un. Below is is the setup of the spaces and convective part of the bilinear form:

V = VectorH1(mesh,order=k, dirichlet=“wall|cyl|inlet”)
Q = H1(mesh,order=k-1)
X = VQ
(u,p), (v,q) = X.TnT()
gfu = GridFunction(X)
velocity, pressure = gfu.components
uin = CoefficientFunction((1.54y(0.41-y)/(0.410.41),0))
velocity.Set(uin, definedon=mesh.Boundaries(“inlet”))
un = gfu.components[0]
conv = BilinearForm(X)
c = InnerProduct(Grad(u) * un, v) dx
conv += c

Assembly of conv and updating of un is done while time stepping. This method seems to work for quite a few time steps, until much of the data suddenly become nan values. Any help with figuring out how to do implement this scheme properly would be much appreciated!

2

Hi,

if you want to trat the convection semi-implicitly, you need to at the c form to the BilinearForm with all the other implicit terms. As a consequence, you will need to reassemble the matrix and inverse in every time-step.

Best, Henry