I have to solve a 2D vorticity transport equation (very similar to viscous Burgers 2D). I had to change the equation for a non-dimensional form.
Vorticity transport Eq.:
$$\zeta_t+u \cdot \zeta_x + v \cdot \zeta_y=\frac{1}{\text{Re}}(\zeta_{xx}+\zeta_{yy})$$
I have to use the Lax-Wendroff 2 steps scheme. I've made a discretization:
Lax-Wendroff 2 Steps:
1º STEP: $$\zeta^{n+1/2}_{i,j} = S^n_{12} - 2\Omega(u_{i,j}\Delta I^n_{i+1/2,j} + v_{i,j}\Delta J^n_{i,j+1/2}) + \Theta(S^n_{ 32} - S^n_{12})$$ 2º STEP: $$\zeta^{n+1}_{i,j} = \zeta^{n}_{i,j} - \Omega(u_{i,j}\Delta I^{n+1/2}_{i+1/2,j} + v_{i,j}\Delta J^{n+1/2}_{i,j+1/2}) + \Theta(S^n - 4\zeta^{n}_{i,j})$$ where, $$S^n_{12} = \zeta^n_{i+1/2,j} + \zeta^n_{i-1/2,j} + \zeta^n_{i,j+1/2} + \zeta^n_{i,j-1/2}$$ $$S^n_{32} = \zeta^n_{i+3/2,j} + \zeta^n_{i-3/2,j} + \zeta^n_{i,j+3/2} + \zeta^n_{i,j-3/2}$$ $$\Delta J^n_{i+1/2,j} = \zeta^n_{i+1/2,j} - \zeta^n_{i-1/2,j}$$ $$\Delta J^n_{i+3/2,j} = \zeta^n_{i,j+1/2} - \zeta^n_{i,j-1/2}$$ $$\Omega = \frac{\Delta t}{\Delta x} $$ $$\Theta = \frac{\Delta t}{\Delta x^2\,\text{Re}} $$
First question: I'm not sure if this discretization is correct. Second problem: I don't know how to analyze the stability condition. Can I use the same stability condition for a Lax-Wendroff 1-step scheme?
Results: VORTICITY
