Stability, numerical

Trichloroethylene is not attacked by dilute acids or alkalis, but when heated with sodium hydroxide under pressure it yields sodium gly-collate. In the presence of light and oxygen dichloroethanoyl chloride is formed, which can react with any moisture present to give small amounts of highly corrosive HCl. Numerous stabilizers have been patented. 

The heightened appreciation of resonance problems, in particular, has been quite recent [63, 62], and contrasts the more systematic error associated with numerical stability that grows systematically with the discretization size. Ironically, resonance artifacts are worse in the modern impulse multiple-timestep methods, formulated to be symplectic and reversible the earlier extrapolative variants were abandoned due to energy drifts. 

In order to compare the efficiency of the SISM with the standard LFV method, we compared computational performance for the same level of accuracy. To study the error accumulation and numerical stability we monitored the error in total energy, AE, defined as... 

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

Equation (8.29) provides no guarantee of stability. It is a necessary condition for stability that is imposed by the discretization scheme. Practical experience indicates that it is usually a sufficient condition as well, but exceptions exist when reaction rates (or heat-generation rates) become very high, as in regions near thermal runaway. There is a second, physical stability criterion that prevents excessively large changes in concentration or temperature. For example. An, the calculated change in the concentration of a component that is consumed by the reaction, must be smaller than a itself Thus, there are two stability conditions imposed on Az numerical stability and physical stability. Violations of either stability criterion are usually easy to detect. The calculation blows up. Example 8.8 shows what happens when the numerical stability limit is violated. 

When transient problems are considered, the time derivative appearing in Eq. (32) also has to be approximated numerically. Thus, besides a spatial discretization, which has been discussed in the previous paragraphs, transient problems require a temporal discretization. Similar to the discretization of the convective terms, the temporal discretization has a major influence on the accuracy of the numerical results and numerical stability. When Eq. (32) is integrated over the control volumes and source terms are neglected, an equation of the following form results ... 

This section addresses some of the problems with NLP optimization software. The primary determinant of solution reliability with LP solvers is numerical stability and accuracy. If the linear algebra subsystem of an LP solver is strong in these... 

Thus, due to limitations on the available computer memory, DNS of homogeneous turbulent reacting flows has been limited to Sc 1 (i.e., gas-phase reactions). Moreover, because explicit ODE solvers (e.g., Runge-Kutta) are usually employed for time stepping, numerical stability puts an upper limit on reaction rate k. Although more complex... 

Curtis, W. K., R. O. Fox, and K. Halasi (1992). Numerical stability analysis of a class of functional differential equations. SIAM Journal of Applied Mathematics 52,... 

As linear regression is a very fundamental operation, several methods have been developed in order to improve the numerical stability of the calculation. It is beyond the objective of this book to discuss these issues in any detail. We do feel, however, that the reader has to be aware of the potential problems and should be able to avoid them as much as possible. 

For increased numerical stability, we introduce the immittance matrix U [rj) by the usual relation... 

The main limitation of FDTD is the need for computer memory and runtime, dominantly. The Courant criterion limits the time step of FDTD in order to ensure numerical stability. 

Aside from the numerical stability consideration, one can actually rationahze the cp (r)-formulation. Starting from the following identity. 

Relativistic variational principles are usually formulated as prescriptions for reaching a saddle point on the energy hypersurface in the space of variational parameters. The results of the variational calculations depend upon the orientation of the saddle in the space of the nonlinear parameters. The structure of the energy hypersurface may be very complicated and reaching the correct saddle point may be difficult [14,15]. If each component of the wavefunction is associated with an independent set of nonlinear parameters, then changing the representation of the Dirac equation results in a transformation of the energy hypersurface. As a consequence, the numerical stability of the variational procedure depends on the chosen representation. 

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