Von Neumann stability analysis

The von Neumann stability analysis (after John von Neumann ), sometimes called L2 - stability analysis is the standard method for testing the stability of numerical methods for the solution of time-dependent partial differential equations.

The procedure was developed by John von Neumann in Los Alamos during the Manhattan Project. During the war the method was kept under wraps and not published until 1947 by John Crank and Phyllis Nicolson. 1968 proved Heinz -Otto Kreiss other key characteristics of the analytical method.

The linear, one-dimensional case

Is given on an interval a linear partial differential equation with constant coefficients of the form

Initial data and a numerical method for the solution. The condition that the method is stable in the L2 norm, then states that the produced by the numerical method error for given increments and for limited stays. The first step of the Von Neumann stability analysis is now to the solution periodically to continue to complete the real numbers.

The periodic error in the time discretization is now in a Fourier series

Be developed. Here denotes the imaginary unit. The numerical method then defines an evolution of the coefficients of the Fourier series in time by means of a so-called Amplifikationsmatrix. The L2 - stability condition is then reduced to the fact that the numerical method is stable if the spectral radius of Amplifikationsmatrix is smaller in absolute value equal to one.

Example

The simplest case is the linear advection equation

Where is a real number. One of the simplest imaginable numerical methods for the solution of such equations is the explicit Euler method coupled to time integration with central differences on an equidistant grid in space. The second term is thus by

Approximated. Overall, the method yields

Which defines the evolution of the error and each term of the Fourier series development. Consider the jth addend, the result insertion in the above formula and dividing by comprising:

The Amplifikationsmatrix is now given by

The process is stable if all what is not the case here, since this method is independent of the choice of the step size unstable. This behavior observed employees of the Manhattan Project, which led to the development of von Neumann stability analysis. If the room the upwind schemes

Used, we obtain the Courant -Friedrichs - Lewy condition

So conditional stability.

Other equations

In the nonlinear case, or in the case of variable coefficients, the process can be applied by linearization and freezing the coefficients, however, the analysis in the general case, only a necessary condition for stability, in special cases also sufficient. Furthermore, the condition on the spectral radius is:

A general method for the complete stability analysis of nonlinear equations is not known.