Von Neumann Stability Analysis - Illustration of The Method

Illustration of The Method

The von Neumann method is based on the decomposition of the errors into Fourier series. To illustrate the procedure, consider the one-dimensional heat equation

$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$

defined on the spatial interval, which can be discretized as

$\quad (1) \qquad u_j^{n + 1} = u_j^{n} + r \left(u_{j + 1}^n - 2 u_j^n + u_{j - 1}^n \right)$

where

and the solution of the discrete equation approximates the analytical solution of the PDE on the grid.

Define the round-off error as

$\epsilon_j^n = N_j^n - u_j^n$

where is the solution of the discretized equation (1) that would be computed in the absence of round-off error, and is the numerical solution obtained in finite precision arithmetic. Since the exact solution must satisfy the discretized equation exactly, the error must also satisfy the discretized equation. Thus

$\quad (2) \qquad \epsilon_j^{n + 1} = \epsilon_j^n + r \left(\epsilon_{j + 1}^n - 2 \epsilon_j^n + \epsilon_{j - 1}^n \right)$

is a recurrence relation for the error. Equations (1) and (2) show that both the error and the numerical solution have the same growth or decay behavior with respect to time. For linear differential equations with periodic boundary condition, the spatial variation of error may be expanded in a finite Fourier series, in the interval, as

$\quad (3) \qquad \epsilon(x) = \sum_{m=1}^{M} A_m e^{ik_m x}$

where the wavenumber with and . The time dependence of the error is included by assuming that the amplitude of error is a function of time. Since the error tends to grow or decay exponentially with time, it is reasonable to assume that the amplitude varies exponentially with time; hence

$\quad (4) \qquad \epsilon(x,t) = \sum_{m=1}^{M} e^{at} e^{ik_m x}$

where is a constant.

Since the difference equation for error is linear (the behavior of each term of the series is the same as series itself), it is enough to consider the growth of error of a typical term:

$\quad (5) \qquad \epsilon_m(x,t) = e^{at} e^{ik_m x}$

The stability characteristics can be studied using just this form for the error with no loss in generality. To find out how error varies in steps of time, substitute equation (5) into equation (2), after noting that

$\begin{align} \epsilon_j^n & = e^{at} e^{ik_m x} \\ \epsilon_j^{n+1} & = e^{a(t+\Delta t)} e^{ik_m x} \\ \epsilon_{j+1}^n & = e^{at} e^{ik_m (x+\Delta x)} \\ \epsilon_{j-1}^n & = e^{at} e^{ik_m (x-\Delta x)}, \end{align}$

to yield (after simplification)

$\quad (6) \qquad e^{a\Delta t} = 1 + \frac{\alpha \Delta t}{\Delta x^2} \left(e^{ik_m \Delta x} + e^{-ik_m \Delta x} - 2\right).$

Using the identities

$\qquad \cos(k_m \Delta x) = \frac{e^{ik_m \Delta x} + e^{-ik_m \Delta x}}{2} \qquad \text{and} \qquad \sin^2\frac{k_m \Delta x}{2} = \frac{1 - \cos(k_m \Delta x)}{2}$

equation (6) may be written as

$\quad (7) \qquad e^{a\Delta t} = 1 - \frac{4\alpha \Delta t}{\Delta x^2} \sin^2 (k_m \Delta x/2)$

Define the amplification factor

$G \equiv \frac{\epsilon_j^{n+1}}{\epsilon_j^n}$

The necessary and sufficient condition for the error to remain bounded is that, However,

$\quad (8) \qquad G = \frac{e^{a(t+\Delta t)} e^{ik_m x}}{e^{at} e^{ik_m x}} = e^{a\Delta t}$

Thus, from equations (7) and (8), the condition for stability is given by

$\quad (9) \qquad \left\vert 1 - \frac{4\alpha \Delta t}{\Delta x^2} \sin^2 (k_m \Delta x/2) \right\vert \leq 1$

For the above condition to hold at all, we have

$\quad (10) \qquad \frac{\alpha \Delta t}{\Delta x^2} \leq \frac{1}{2}$

Equation (10) gives the stability requirement for the FTCS scheme as applied to one-dimensional heat equation. It says that for a given, the allowed value of must be small enough to satisfy equation (10).

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