Von Neumann Analysis of Difference Schemes

In this section, we summarize the basics of Von Neumann analysis provided in [8]. Consider the ($N$+1)D real-valued grid function $U_{{\bf m}}(n)$, defined for integer $n$ and for ${\bf m} = [m_{1},\hdots,m_{N}] \in \mathbb{Z}^{N}$, the set of all integer $N$-tuples. Such a grid function will be used, in a finite difference scheme, as an approximation to the continuous solution $u({\bf x},t)$ to some problem, at the location ${\bf x} = {\bf m}\Delta$, and at time $t=nT$, where $\Delta$ is the grid spacing, and $T$ is the time step. Here, and henceforth in this appendix, we have assumed that the grid spacing is uniform in all the spatial coordinates, and that the spatial domain is unbounded. We define the space step/time step ratio to be

$$v_{0} \triangleq \frac{\Delta}{T}$$

The spatial Fourier transform of $U_{{\bf m}}(n)$ is defined by

$$\hat{U}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n) = \frac{1}{(2\p... ...\Delta{\bf m}\cdot\mbox{{\scriptsize\boldmath$\beta$}}}U_{{\bf m}}(n)\Delta^{N}$$

and is a periodic function of $\beta$ $=[\beta_{1},\hdots,\beta_{N}]^{T}$, a vector of spatial wavenumbers. The transform can be inverted by

$$U_{{\bf m}}(n) = \frac{1}{(2\pi)^{N/2}}\int_{[-\pi/\Delta, \pi/\D... ...{\mbox{{\scriptsize\boldmath$\beta$}}}(n)d\beta_{1} d\beta_{2}\hdots d\beta_{N}$$

where $\beta$ $\in[-\pi/\Delta, \pi/\Delta]^{N}$ refers to the space enclosed by the intervals $-\pi/\Delta\leq\beta_{j}\leq\pi/\Delta$, for $j=1,\hdots, N$. If, for a given grid spacing $\Delta$, we define the discrete spatial $L_{2}$ norm of $U_{{\bf m}}(n)$ by

$$\Vert U(n)\Vert _{2} = \left(\sum_{{\bf m}\in\mathbb{Z}^{N}}U_{{\bf m}}^{2}(n)\Delta^{N}\right)^{1/2}$$

and the corresponding spectral $L_{2}$ norm of $\hat{U}_{\mbox{\boldmath $\beta$}}(n)$ by

$$\Vert\hat{U}(n)\Vert _{2} = \left(\int_{[-\pi/\Delta, \pi/\Delta]... ...ldmath$\beta$}}}(n)\vert^{2}d\beta_{1} d\beta_{2}\hdots d\beta_{N}\right)^{1/2}$$

then if $U_{{\bf m}}(n)$ and $\hat{U}_{\mbox{\boldmath $\beta$}}(n)$ are in their respective $L_{2}$ spaces, Parseval's relation gives

$$\Vert U(n)\Vert _{2} = \Vert\hat{U}(n)\Vert _{2}$$