Von Neumann stability analysis

The Von Neumann stability analysis (according to John von Neumann ), sometimes also L2 stability analysis, is the standard method for investigating the stability of numerical methods for solving time-dependent partial differential equations .

The process was developed by John von Neumann in Los Alamos as part of the Manhattan Project . The method was kept under lock and key during the war and was not published until 1947 by John Crank and Phyllis Nicolson . In 1968 Heinz-Otto Kreiss demonstrated further central properties of the analysis method.

The linear, one-dimensional case

Given a linear, partial differential equation with constant coefficients of the form on an interval ${\displaystyle [0,L]}$

${\displaystyle u_{t}+Au_{x}=0,}$

Initial data as well as a numerical method for the solution. The condition that the method is stable in the L2 standard then means that the error produced by the numerical method remains limited for given step sizes and for . The first step of the Von Neumann stability analysis is to periodically continue the solution to the complete real numbers. ${\displaystyle u_{0}=u(0)}$ ${\displaystyle \Delta x}$ ${\displaystyle \Delta t}$ ${\displaystyle t\to \infty }$

The periodic error at the time at the discretization point can now be converted into a Fourier series ${\displaystyle E_{i}^{n}}$ ${\displaystyle t_{n}}$ ${\displaystyle x_{i}}$

${\displaystyle E_{i}^{n}=\sum _{-N}^{N}c_{j}^{n}e^{ij\Delta xI}}$

be developed. Here denotes the imaginary unit . The numerical method then defines an evolution of the coefficients of the Fourier series in time using a so-called amplification matrix . The L2 stability condition is then reduced to the fact that the numerical method is stable precisely when the spectral radius of the amplification matrix is ​​less than or equal to one in absolute terms. ${\displaystyle I}$ ${\displaystyle G}$ ${\displaystyle \rho (G)}$

example

The simplest case is the linear advection equation

${\displaystyle u_{t}+au_{x}=0,}$

where is a real number. One of the simplest imaginable numerical methods for solving such equations is the explicit Euler method for time integration coupled with central differences on an equidistant grid in space. So the second term becomes using ${\displaystyle a}$ ${\displaystyle u^{n+1}=u^{n}-(\Delta t)au_{x}}$

${\displaystyle au_{x}\approx {\frac {a}{2\Delta x}}(u_{i+1}^{n}-u_{i-1}^{n})}$

approximated. Overall, the procedure results

${\displaystyle {\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}+{\frac {a}{2\Delta x}}(u_{i+1}^{n}-u_{i-1}^{n})=0,}$

which also defines the development of the errors and also of every single term of the Fourier series expansion. If we consider the j th summand, inserting it into the above formula and dividing by with gives : ${\displaystyle e^{ij\Delta xI}}$ ${\displaystyle \phi =j\Delta x}$

${\displaystyle c_{j}^{n+1}-c_{j}^{n}+{\frac {a\Delta t}{2\Delta x}}c_{j}^{n}(e^{I\phi }-e^{-I\phi })=0.}$

The amplification matrix is ​​now given by

${\displaystyle G(\phi )={\frac {c_{j}^{n+1}}{c_{j}^{n}}}=1-I{\frac {a\Delta t}{\Delta x}}\sin \phi .}$

The method is stable if for all , which is not the case here, since the method is thus unstable regardless of the choice of step sizes. The employees of the Manhattan Project observed this behavior, which led von Neumann to develop the stability analysis. The upwind procedure is used in space ${\displaystyle L_{2}}$ ${\displaystyle |G|\leq 1}$ ${\displaystyle \phi }$ ${\displaystyle |G|^{2}=1+\left({\frac {a\Delta t}{\Delta x}}\right)^{2}\sin ^{2}\phi .}$

${\displaystyle au_{x}\approx {\frac {a}{\Delta x}}(u_{i}^{n}-u_{i-1}^{n})}$

is used, the Courant-Friedrichs-Lewy condition results

${\displaystyle {\frac {a\Delta t}{\Delta x}}<1}$,

so conditional stability.

Other equations

In the non-linear case or in the case of variable coefficients, the method can be applied by linearizing and freezing the coefficients; however, in the general case the analysis only supplies one necessary condition for stability, in special cases also a sufficient one. Furthermore, the condition on the spectral radius is now:

${\displaystyle \rho (G)\leq 1+c\Delta t}$.

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

Web links

• Martin Neumann: Explanation of the analysis in a script Computational Physics I: Fundamentals of the University of Vienna