Partial differential equation stability

In the context of the numerical solution of the Boltzmann equation and the numerical solution of partial differential equations, stability ensures that roundoff and other errors in the calculation are not amplified to cause the calculation to blow up. An algorithm for solving an evolutionary (e. g., a time-dependent) partial differential equation is stable if the numerical solution at a fixed time remains bounded as the time stepsize goes to zero. An important theorem in mathematics, the so-called Lax equivalence theorem, states that an algorithm converges if it is consistent and stable. 

A marching-ahead solution to a parabolic partial differential equation is conceptually straightforward and directly analogous to the marching-ahead method we have used for solving ordinary differential equations. The difficulties associated with the numerical solution are the familiar ones of accuracy and stability. 

To study the response of the kiln to transient conditions and to different control schemes, the set of partial differential equations (39), (40), (44), and (48) were solved using a hybrid analog-digital computer, the EAI Hydac 2000. A description of the computer and of the methods used in the solution are given in the paper by Weekman et al. (1967). The kiln conditions used in the simulation to be discussed are given in Table V. The dynamic model was first used to study the effect of fast coke on kiln stability. 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

Dassl, solves stiff systems of differential-algebraic equations (DAE) using backward differentiation techniques [13,46]. The solution of coupled parabolic partial differential equations (PDE) by techniques like the method of lines is often formulated as a system of DAEs. It automatically controls integration errors and stability by varying time steps and method order. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

In order to build up and implement efficient numerical schemes for partial differential equations, it is necessary to have informations on the mathematical properties of the system of equations—this has been done in the previous sections—as well as on the stability and the convergence properties of the schemes this is the purpose of numerical analysis. 

M. Renardy, On the linear stability of hyperbolic partial differential equations and viscoelastic flows, Z. Angew. Math. Phys., 45 (1994) 854-865. 

There are many ways to combine the various finite differences that may be used for each of the terms of the mass balance equation, and there are as many ways to approximate a partial differential equation by a finite-difference scheme. The choice is limited in practice, however, for two reasons. First, we need the numerical calculation to be stable, and there is a condition to satisfy to achieve numerical stability. Second, we need to control the numerical errors that are made during the calculations. Replacing a partial difference term with any of the possible finite difference terms gives a tnmcation error. These tnmcation errors accumulate during the calculation of a numerical solution. The error contribution... 

Finite difference methods Methods used for the calculation of ntunerical solutions of systems of partial differential equations. The differential elements in the differential equations are replaced by corresponding finite differences, giving difference equations. Stability and accuracy conditions must be satisfied (Chapter 10, Section 10.3). 

In all these cases, the correct design must grow from the equations of mass, energy, and momentum balance to which we now turn in the next few sections. From these we proceed to the design problem (Sec. 9.5) and hence to elementary considerations of optimal design (Sec. 9.6). The stability and sensitivity of a tubular reactor is a vast and fascinating subject. Since the steady state equations are ordinary differential equations, the equations describing the transient behavior are partial differential equations. This... 

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. 

The well-posedness of the two-fluid model has been a source of controversy reflected by the large number of papers on this issue that can be found in the literature. This issue is linked with analysis of the characteristics, stability and wavelength phenomena in multi -phase flow equation systems. The controversy originates primarily from the fact that with the present level of knowledge, there is no general way to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem. The mathematical theory of well posedness for systems of partial differential equations describing dispersed chemical reacting flows needs to be examined. 

Equation (7) is a nonlinear partial differential equation. According to conditions of convergence and stability, applying against wind differential scheme, it may be discretized as... 

The fourth and final step in the stability analysis is the reduction of the linear system of partial differential equations to a system of ordinary linear differential equations, the solution of which, subject to the appropriate boundary conditions, yields the eigenfunction

associated complex wave velocity c. 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

In modds of tubular reactors, material and energy balance are expressed by partial differential equations in time and space variaUes. Althou detailed numerical studies have been made in order to duddate the transient behaviour of tubular reactors, analytical studies have largely been confined to the question of existence, multiplicity, and stability of the reactm steady-state profiles, since the dimination of transirait behaviour often reduces tbe balance equations to a system of ordinary differential equations. 

Friedrichs (1901-1982) the famous book on partial differential equations, including the fundamental CFL-condition as a numerical stability criterion for imsteady partial differential equations. Once the Nazis had taken power, Lewy was dismissed in 1933, so that he emigrated over France to the USA, where he was appointed professor of applied mathematics at the California Institute of Technology, Berkeley CA. Lewy was awarded the Leroy P. Steele Prize of the American Mathematical Society AMS in 1979, and the 1985 Wolf-Prize as a life-time achievement. 

Kopell and Howard also investigated the stability properties of these plane waves as solutions of the partial differential equations (10). The plane waves discussed in Theorem 1, coming from a Hopf bifurcation, have small amplitude close to the point of bifurcation, in which case applies... 

The general method for solving Eqs. (11) consists of transforming the partial differential equations with the help of Fourier-Laplace transformations into a set of linear algebraic equations that can be solved by the standard techniques of matrix algebra. The roots of the secular equation are the normal modes. They yield the laws for the decays in time of all perturbations and fluctuations which conserve the stability of the system. The power-series expansion in the reciprocal space variables of the normal modes permits identification of relaxation, migration, and diffusion contributions. The basic information provided by the normal modes is that the system escapes the perturbation by any means at its disposal, regardless of the particular physical or chemical reason for the decay. 

Although only approximate analytical solutions to this partial differential equation have been available for x(s,D,r,t), accurate numerical solutions are now possible using finite element methods first introduced by Claverie and coworkers [46] and recently generahzed to permit greater efficiency and stability [42,43] the algorithm SEDFIT [47] employs this procedure for obtaining the sedimentation coefficient distribution. 

Stability means the errors in the solution do not grow exponentially as the solution proceeds but damp out. Convergence means that the solution of the difference equation approaches the exact solution of the partial differential equation as At and Ax go to zero with M fixed. Using smaller sizes of At and Ax increases the accuracy in general but greatly increases the number of calculations required. Hence, a digital computer is often ideally suited for this type of calculation. 

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