Von Neumann stability analysis
In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to check the stability of finite difference schemes as applied to linear partial differential equations.[1] The analysis is based on the Fourier decomposition of numerical error and was developed at Los Alamos National Laboratory after having been briefly described in a 1947 article by British researchers John Crank and Phyllis Nicolson.[2] This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time.Later, the method was given a more rigorous treatment in an article[3] co-authored by John von Neumann.A finite difference scheme is stable if the errors made at one time step of the calculation do not cause the errors to be magnified 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 numerical scheme is said to be unstable.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 the equation under consideration is nonlinear.In certain cases, von Neumann stability is necessary and sufficient for stability in the sense of Lax–Richtmyer (as used in the Lax equivalence theorem): The PDE and the finite difference scheme models are linear; the PDE is constant-coefficient with periodic boundary conditions and has only two independent variables; and the scheme uses no more than two time levels.[4] Von Neumann stability is necessary in a much wider variety of cases.It is often used in place of a more detailed stability analysis to provide a good guess at the restrictions (if any) on the step sizes used in the scheme because of its relative simplicity.The von Neumann method is based on the decomposition of the errors into Fourier series.To illustrate the procedure, consider the one-dimensional heat equationof the discrete equation approximates the analytical solutionis the solution of the discretized equation (1) that would be computed in the absence of round-off error, andis the numerical solution obtained in finite precision arithmetic.must satisfy the discretized equation exactly, the errorsatisfies the equation, too (this is only true in machine precision).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 with respect toOften the assumption is made that the error grows or decays exponentially with time, but this is not necessary for the stability analysis.If the boundary condition is not periodic, then we may use the finite Fourier integral with respect to: 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: if a Fourier series is used or if a Fourier integral is used.The stability characteristics can be studied using just this form for the error with no loss in generality.equation (6) may be written as Define the amplification factor The necessary and sufficient condition for the error to remain bounded is thatThus, from equations (7) and (8), the condition for stability is given by Note that the termThe highest value the sinusoidal term can take is 1 and for that particular choice if the upper threshold condition is satisfied, then so will be for all grid points, thus we have Equation (11) gives the stability requirement for the FTCS scheme as applied to one-dimensional heat equation.Similar analysis shows that a FTCS scheme for linear advection is unconditionally unstable.