Partial differential equation unstability

The partial differential equations used to model the dynamic behavior of physicochemical processes often exhibit complicated, non-recurrent dynamic behavior. Simple simulation is often not capable of correlating and interpreting such results. We present two illustrative cases in which the computation of unstable, saddle-type solutions and their stable and unstable manifolds is critical to the understanding of the system dynamics. Implementation characteristics of algorithms that perform such computations are also discussed. 

An algorithm for a partial differential equation is said to be stable if the truncation error introduced in a step is not amplified in the latter calculation steps. Unstable algorithms cannot be used in solving a diffusion equation because the errors would explode and overwhelm the values of C. 

The partial differential equations of the previous section are numerically solved. In order to minimize the number of spatial pivots required for a given accuracy, a central differences scheme was chosen. For certain flow problems the results may become unstable and the algorithm then allows for a smooth transition to one-sided differences with the penalty of an increased local error which has to be compensated by a larger number of pivots. 

Still be very sensitive to a particular variable. On the other hand, an unstable condition is such that the least perturbation will lead to a finite change and such a condition may be regarded as infinitely sensitive to any operating variable. Sensitivity can be fully explored in terms of steady state solutions. A complete discussion of stability really requires the study of the transient equations. For the stirred tank this was possible since we had only to deal with ordinary differential equations for the tubular reactor the full treatment of the partial differential equations is beyond our scope here. Nevertheless, just as much could be learned about the stability of a stirred tank from the heat generation and removal diagram, so here something may be learned about stability from features of the steady state solution. 

Nevertheless, one feature of the one dimensional model containing dispersion terms is of considerable interest. These terms increase the order of the partial differential equations and, under certain conditions, lead to nonuniqueness of the steady state profile through the reactor (109). For certain ranges of operating conditions and parameter values, three or more steady state profiles can be obtained for the same feed conditions. The two outlying steady-state profiles will be stable (at least to small perturbations), whereas the intermediate profile will be unstable. The profile generated as a solution to equations (12.7.38) and (12.7.47) will depend on the initial guesses for T and C involved in the trial-and-error solution. 

Linear stability theory can show definitively that a system is unstable, but it gives no information about the ultimate fate of the process as the disturbance grows. Furthermore, linear stability theory can show only conditional stability. There are two ways to attack the problem of finite disturbances. One is direct numerical simulation of the full set of nonlinear partial differential equations. This approach has become increasingly popular as computer power has grown, but a fundamental difficulty of distinguishing physical from numerical instability is always present. The other, employed less now than in the past, is to expand the nonlinear equations in... 

If the chemical system has a single unstable steady state and hence shows oscillatory behavior in the absence of starch, complex formation can stabilize the homogeneous steady state and make possible the appearance of Turing structures at parameters which would yield oscillatory kinetics in the complex-free system. Observe that in the above partial differential equation system (12) the effective ratio of diffusion coefficients is (1 -f K )c, which can be much greater than unity even if c < 1. Consequently, the presence of a species that forms an appropriate complex with the activator can allow Turing structures to form for, in principle, any ratio of the activator and inhibitor diffusion coefficients. 

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