Explicit solution instability

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. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

The same equations, albeit with damping and coherent external driving field, were studied by Drummond et al. [104] as a particular case of sub/second-harmonic generation. They proved that below a critical pump intensity, the system can reach a stable state (field of constant amplitude). However, beyond the critical intensity, the steady state is unstable. They predicted the existence of various instabilities as well as both first- and second-order phase transition-like behavior. For certain sets of parameters they found an amplitude self-modula-tion of the second harmonic and of the fundamental field in the cavity as well as new bifurcation solutions. Mandel and Erneux [105] constructed explicitly and analytically new time-periodic solutions and proved their stability in the vicinity of the transition points. 

Recall that both accuracy and stability are considerations in an explicit method. If the time step is too large, specifically if the coefficient of D20 in the difference formula given in cell D21 is less than zero, the the solution will be unstable. Try giving a too large time step and observe the solution. The instability will be unmistakenly obvious. 

Because an explicit finite-difference procedure is being used to solve the momentum and energy equations, the solution can become unstable, i.e., as the solution proceeds it can diverge increasingly from the actual solution. The analysis of the conditions under which such an instability will develop that was given in Chapter 4 for the case of forced convection in a duct essentially applies here and shows that in order to avoid instability, AZ should be selected so that ... 

The explicit and implicit methods have their advantages and disadvantages, and one method is not necessarily better Ilian the other one. Next you will see that the explicit method is easy to implement but imposes a limit on the allowable time step to avoid instabilities in the solution, and the iinplicit method requires the nodal temperatures to be solved simultaneously for each time step but imposes no limit on the magnitude of the time step. We limit the discussion to. bne- and two-dimensional cases to keep the complexities at a manageable leyel, but the analysis can feadily be extended to threc-dimen.sional ca.ses and other coordinate systems. 

If one has found an instability, other procedures are, however, needed in order to construct explicitly a solution of the new type. Obviously there is a large choice of methods for that purpose. There is, however,... 

If the film surfaces are given a small wavy perturbation, the equations of motion now become Equations 5.3 and 5.4, with p replaced by 4>. Clearly d> can be eliminated from these equations in the same manner as was p in Equation 5.5. Hence the basic differential equations (Equations 5.9 and 5.10) for the velocity components, and naturally also the solutions to these equations, such as Equation 5.17, show no explicit dependence on van der Waals and electrical double-layer forces. As we shall see shortly, these forces do make their appearance in the boundary conditions. Note that the use of Equations 5.3 and 5.4 involves an assumption that the film is at rest before perturbation, i.e., flow due to film drainage (see Chapter 7) is neglected. This assumption amounts to requiring that the instability develop rapidly in comparison with changes in film thickness due to drainage. 

The numerous complexities of the physical domain represented by all of the components and associated detailed aspects of a system that affect the stability of the system must be (1) realistically included into the mathematical models, (2) accurately resolved by the numerical solution methods, and (3) shown to not have introduced artifacts into the calculations. The numerically enhanced mathematical stability of implicit methods, the potential numerical instability of explicit methods, and the dissipative and dispersive characteristics of implicit and explicit methods require careful investigations. Jensen (1992) has given examples of some of these effects. 

All the methods presented so far, e.g. the Euler and the Runge-Kutta methods, are examples of explicit methods, as the numerical solution aty +i has an explicit formula. Explicit methods, however, have problems with stability, and there are certain stability constraints that prevent the explicit methods from taking very large time steps. Stability analysis can be used to show that the explicit Euler method is conditionally stable, i.e. the step size has to be chosen sufficiently small to ensure stability. This conditional stability, i.e. the existence of a critical step size beyond which numerical instabilities manifest, is typical for all explicit methods. In contrast, the implicit methods have much better stability properties. Let us introduce the implicit backward Euler method. 

Now there is no unstable behavior regarding the size of h (Figure 2.2). Note that in the explicit Euler s method we were approximating the solution by a polynomial, and there exists no pol3momial that can approximate the exponential term as x tends to 00, hence, the instability. By using the implicit method, we have expressed the solution in the form of a rational function, which can go to zero, as t tends to 00. 

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