Neumann criterion

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

Using this looser criterion of self-reproduction - it is looser since, unlike von Neumann s requirement, it does not force the self-reproducing structure to be capable of universal construction - Langton discovered a relatively simple self-reproducing structure embedded within a two-dimensional CA that we will refer to as Langton s Loop figure 11.4 shows a few snapshots of its 151 time-step reproduction cycle. 

Nauk (UkrainRSR) 1966(7), 871-74 CA 65, 19919 (1966) "Criterion of Uni-dimensional Instability of Gas Detonations (The criterion was derived by using Zel dovich-Von Neumann model, which represents a detonation wave in an ideal gas as a stationary complex consisting of a shock wave and the front of an instantaneously occurring reaction with a characteristic induction time that follows the shock wave at a definite distance. The results showed that the criterion assumes the form dependent... 

When p = 0.5, the method is the Crank-Nicholson implicit method. The expansion point should be taken at (i+l/2,j). The truncation error is of the order (Ax)2 plus order (Ay)2. No stability criterion comes out of the von Neumann analysis, but difficulties can come about if diagonal dominance is not kept for the tridiagonal algorithm. 

According to the specified criterion, the detonation exists only if at the Neumann state, dT /T divided by the coefficient of dM/M on the left-hand side of equation (29) exceeds dA/A divided by the coefficient of dM/M on the left-hand side of equation (28). For sufficiently strong leading shocks, the value of M at the Neumann state is such that this criterion to reduce approximately to... 

The stability criterion of the technique is extracted by means of the ordinary von Neumann method as... 

Concerning the stability of the previous schemes, von Neumann analysis leads to the Courant criterion... 

Finally, for the stability criterion of the algorithm, von Neumann s method yields... 

Study the finite difference literature on parabolic equations and transient modeling, and summarize the von Neumann stability criterion for explicit and implicit schemes. What are its strengths and limitations What new stability tests are available to study nonlinearities and heterogeneities Comment on group velocity and wave-based stability analysis. 

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