1
$\begingroup$

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.

$\endgroup$
3
  • 1
    $\begingroup$ What have you already tried? Have you tried to read any of the books that explain how to assess the stability of schemes? $\endgroup$ May 19 at 17:21
  • $\begingroup$ 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?) $\endgroup$
    – NotaChoice
    May 20 at 2:02
  • $\begingroup$ So far I tried some Von Neumann analysis math.stackexchange.com/questions/4453633/… $\endgroup$
    – NotaChoice
    May 20 at 2:05

1 Answer 1

5
$\begingroup$

It is worth making some additional points. What you set out is just one version of the Lax-Wendroff method. That scheme is unique in one space dimension but has several free parameters in two or three dimensions. The two-dimensional case is thoroughly investigated in T.B. Lung, P.L. Roe, Toward a reduction of mesh imprinting, Int J. Num Meth. Fluids, vol 76, p 450, 2014. If you roll your sleeves up you will work out the 3D case quite easily. (If you have access to symbolic manipulation software)

The other thing is that Lax Wendroff, by itself, is no longer the gold standard for wave equations. Some of the citations in that paper may point you in good directions.

$\endgroup$

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.