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Von Neumann stability analysis
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In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to verify the stability of finite difference schemes as applied to linear partial differential equations. The analysis is based on the Fourier decomposition of numerical error and was developed at Los Alamos National Laboratory during the World War II. It was briefly described first in 1947 article by Crank and Nicolson.
Later, it was also published in an article co-authored by von Neumann.
stability of numerical schemes is closely associated with numerical error.
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Encyclopedia
In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to verify the stability of finite difference schemes as applied to linear partial differential equations. The analysis is based on the Fourier decomposition of numerical error and was developed at Los Alamos National Laboratory during the World War II. It was briefly described first in 1947 article by Crank and Nicolson.
Later, it was also published in an article co-authored by von Neumann.
Numerical stability
The stability of numerical schemes is closely associated with numerical error. A finite difference numerical scheme is stable if the errors made at one time step of the calculation do not cause the errors to increase (remain bounded) as the computations are continued. A neutrally stable scheme is one in which errors remain constant as the computations are carried forward. If the errors decay and eventually damp out, the numerical scheme is said to be stable. If, on the contrary, the errors grow with time the solution diverges and thus the numerical is said to be unstable. The stability of numerical schemes can be investigated by performing von Neumann stability analysis. For time-dependent problems, stability guarantees that the numerical method produces a bounded solution whenever the solution of the exact differential equation is bounded. Stability, in general, can be difficult to investigate, especially when equation under consideration is nonlinear.
Illustration of the Method
von Neumann is applicable to discretized linear partial differential equations under the assumption of periodic boundary conditions. It is based on the decomposition of the errors into Fourier series. To illustrate the procedure, consider the one-dimensional heat equation
defined in the domain and its FTCS discretization
where
.
Define the error as
where is the exact value. Since exact solution must satisfy the discretized equation (1), it is easy to show that the error also satisfy the discretized equation. Thus
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
where the wave number with . 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
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:
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 thatto yield (after simplification)
Using the identities
equation (6) may be written as
Define the amplification factor
The necessary and sufficient condition for the error to remain bounded is that,
However,
Thus, from equations (7) and (8), the conditions for stability is given by
For the above condition to hold,
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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