0
$\begingroup$

I'm working with a 2D Navier Stokes PDE in the unstabilized version - the equation is a linear equation of the type $\frac{∂u}{∂t} = F(u,t)$.

In order to perform time discretization with FDM (finite difference method), with theta method, this equation turns into

$$\frac{u^{n+1}-u^{n}}{Δt} = θF(u^{n+1},t^{n+1}) + (θ-1)F(u^{n},t^{n})$$

Now well, by using SUPG (Streamline Upwind Petrov Galerkin) stabilization this PDE is a second order nonlinear equation.

QUESTION: How does the theta method look in this case?

Improve this question
$\endgroup$
  • $\begingroup$ What precisely is your question? You've written down how the semi-discrete equation looks like. Are you asking how this equation needs to be stabilized if you add SUPG? $\endgroup$ – Wolfgang Bangerth May 11 '20 at 19:46
  • $\begingroup$ the question is how it looks the semi discret form for a secod order non lineal equation. $\endgroup$ – Rubi C.g. May 12 '20 at 19:35
  • $\begingroup$ "the equation is a linear equation of the type..." - Why should we think this equation is linear? $\endgroup$ – Maxim Umansky May 14 '20 at 4:42
1
$\begingroup$

It becomes a root finding problem. Find $u^{n+1}$ such that

$$ u^{n+1} + \Delta t \theta F(u^{n+1},t^{n+1}) = u^n + \Delta t (\theta - 1) F(u^{n},t^{n}).$$

Now, you can multiply by a test function, form your trilinear form $w(\cdot,\cdot,\cdot)$ to obtain the weak problem; Find $u^{n+1}$ in an appropriate space $V$ such that

$$w(u^{n+1},u^{n+1},v) = (u^n + \Delta t (\theta - 1) F(u^{n},t^{n}),v)$$

for all $v\in V$. It is just left to add SUPG term $s(u^{n+1},v)$ to get your final problem; Find $u^{n+1}$ in an appropriate space $V$ such that

$$w(u^{n+1},u^{n+1},v) + s(u^{n+1},v) = (u^n + \Delta t (\theta - 1) F(u^{n},t^{n}),v)$$

for all $v\in V$. You are missing the pressure unknowns, I do not know if it is deliberate and you are working with compressible Navier-Stokes equations or you made a mistake but either way you can find the proper SUPG term to use by looking at the literature.

Since you are solving a non-linear root finding problem, you will have to linearize it somehow. Two popular methods are Newton linearization and Picard linearization.

Improve this answer
$\endgroup$
  • $\begingroup$ but this si the same task as for linear equations, it is the same for second order? $\endgroup$ – Rubi C.g. May 12 '20 at 19:34
  • $\begingroup$ Abstractly, there is no difference. That is correct. However, to solve the problem related to $$w(u^{n+1},u^{n+1},v) + s(u^{n+1},v) = (u^n + \Delta t (\theta - 1) F(u^{n},t^{n}),v)$$, you will have to do some linearization. One example is the so called lagging method; $$w(u^{n},u^{n+1},v) + s(u^{n+1},v) = (u^n + \Delta t (\theta - 1) F(u^{n},t^{n}),v)$$. This effectively linearizes the Navier-Stokes equations. It is a first order convergent nonlinear iteration technique. The main problem is for the first few time steps the approximate solution can be far from the actual solution. $\endgroup$ – Abdullah Ali Sivas May 12 '20 at 21:35
  • $\begingroup$ thank you so much! $\endgroup$ – Rubi C.g. May 13 '20 at 16:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.