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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?

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  • $\begingroup$ 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 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 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 at 4:42
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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.

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  • $\begingroup$ but this si the same task as for linear equations, it is the same for second order? $\endgroup$ – Rubi C.g. May 12 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 at 21:35
  • $\begingroup$ thank you so much! $\endgroup$ – Rubi C.g. May 13 at 16:52

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