FDM on nonlinear PDEs

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?

• 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? Commented May 11, 2020 at 19:46
• the question is how it looks the semi discret form for a secod order non lineal equation. Commented May 12, 2020 at 19:35
• "the equation is a linear equation of the type..." - Why should we think this equation is linear? Commented May 14, 2020 at 4:42

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.

• but this si the same task as for linear equations, it is the same for second order? Commented May 12, 2020 at 19:34
• 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. Commented May 12, 2020 at 21:35