# stability of a numercial scheme for a hyperbolic system?

This is related to my question here https://math.stackexchange.com/questions/4447383/lax-wendroff-scheme-stability-analysis-for-a-linear-system-of-conservation-laws .

Consider the numerical scheme given by the finite difference equation: \begin{align*} Q_{i,j,k}^{n+1}&= \Big(I- \frac{\Delta t^2}{\Delta x^2}A^2-\frac{\Delta t^2}{\Delta y^2}B^2-\frac{\Delta t^2}{\Delta z^2}C^2\Big)Q_{i,j,k}^{n} \\ & \ +(\frac{\Delta t^2}{2\Delta x^2}A^2- \frac{\Delta t}{2\Delta x}A) Q_{i+1, j, k}^n- (\frac{\Delta t^2}{2\Delta x^2}A^2- \frac{\Delta t}{2\Delta x}A)Q_{i-1, j, k}^n \\ &\ +(\frac{\Delta t^2}{2\Delta y^2}B^2 - \frac{\Delta t}{2\Delta y}B) Q_{i, j+1, k}^n- ( \frac{\Delta t^2}{2\Delta y^2}B^2- \frac{\Delta t}{2\Delta y}B)Q_{i, j-1, k}^n \\ &\ + (\frac{\Delta t^2}{2\Delta z^2}C^2 - \frac{\Delta t}{2\Delta z}C)Q_{i, j, k+1}^n- (\frac{\Delta t^2}{2\Delta z^2}C^2- \frac{\Delta t}{2\Delta z}C)Q_{i, j, k-1}^n \\ &\quad + \frac{\Delta t^2}{8\Delta x\Delta y}(AB + BA)(Q_{i+1, j+1, k}^n- Q_{i-1, j+1, k}^n)- (Q_{i+1, j-1, k}^n- Q_{i-1, j-1, k}^n) \\ & \quad + \frac{\Delta t^2}{8\Delta x\Delta z}(AC + CA)(Q_{i+1, j, k+1}^n- Q_{i-1, j, k+1}^n)- (Q_{i+1, j, k-1}^n- Q_{i-1, j, k-1}^n) \\ &\quad + \frac{\Delta t^2}{8\Delta z\Delta y}(CB + BC)(Q_{i, j+1, k+1}^n- Q_{i, j+1, k-1}^n)- (Q_{i, j-1, k+1}^n- Q_{i, j-1, k-1}^n) \end{align*} Where $$Q_{i,j,k}^{n+1}\in \mathbb{R}^4$$ is a vector of $$4$$ components, at the updated one-step time $$t_{n+1}= (n+1)\Delta t$$
$$Q_{i,j,k}^{n+1}$$ is given as linear combination of values of a $$19$$-points stencil in the space, and $$A, B, C$$ are constant $$4\times 4$$ matrices, making this system linear.

Is this numerical scheme stable ? for what (sufficient) condition ?
Is it convergent ?
I appreciate any hints or directions I should look for.

• What have you already tried? Have you tried to read any of the books that explain how to assess the stability of schemes? May 19, 2022 at 17:21
• My reference was Leveque's book "Finite Volume methods for hyperbolic problems", I assume this scheme is consistent so we only need to prove it is stable to conclude convergence (why does Lax equivalence theorem apply here?) May 20, 2022 at 2:02
• So far I tried some Von Neumann analysis math.stackexchange.com/questions/4453633/… May 20, 2022 at 2:05