Stability explicit numerical solution

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. 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

Numerical solution of the ODEs describing initial value problems is possible by explicit and implicit techniques, which are described in the Sections 11.1.1 and 11.1.2, respectively. It is worth noting that the techniques are formulated for the solution of a single equation (Equation 11.1), but they can be used for solving multiple ODEs as well. Theoretical background of these methods as well as their stabilities are described elsewhere [1,2] and will not be discussed here. 

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. 

Next, we examine the stability of the numerical solution of this problem obtained from using the explicit Euler method. Momentarily we ignore the truncation and roundoff errors. Applying Eq. (5.60), we obtain the recurrence equation... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Methods applying reverse differences in time are called implicit. Generally these implicit methods, as e.g. the Crank-Nicholson method, show high numerical stability. On the other side, there are explicit methods, and the methods of iterative solution algorithms. Besides the strong attenuation (numeric dispersion) there is another problem with the finite differences method, and that is the oscillation. 

It must be stressed that the hypotheses of Corollary 5.2 give sufficient, but not necessary, conditions for the existence of a positive periodic solution possessing strong stability properties. Furthermore, since the singlepopulation periodic solutions Eft) and 2(0 are not explicitly computable, as the corresponding rest points were in Chapter 1, it does not seem possible to obtain explicit formulas for A,2and A21. However, these crucial Floquet exponents can be easily approximated numerically. One must... 

The performance of numerical methods for chemical continuity equations is generally characterized in terms of accuracy, stability, degree of mass conservation, and computational efficiency. The simplest of such methods is provided by the forward Euler or fully explicit scheme, by which the solution y" 1 at time tn y is given by... 

In [211] the authors obtained a new embedded 4(3) pair explicit four-stage fourth-order Runge-Kutta-Nystrom (RKN) method to integrate second-order differential equations with oscillating solutions. The proposed method has high phase-lag order with small principal local truncation error coefficient. The authors given the stability analysis of the proposed method. Numerical comparisons of this new obtained method to problems with oscillating and/or periodical behavior of the solution show the efficiency of the method. 

We start by summarizing the behavior of our prototype DDE, eq. (10.1). The explicit solution is presented in eq. (10.10), but it is not immediately obvious what this function looks like. Numerical analysis combined with the algebraic approach to linear stability analysis described in the previous section yields the following results. 

Since sin pzk/2 > 0, the inequality always holds if At > 0, thus guaranteeing stability. (Of course, mesh sizes must be kept small in order to reduce truncation errors and to ensure convergence to solutions of the PDE.) Unlike the conditionally stable explicit scheme studied earlier, this implicit scheme, which requires only tridiagonal matrix inversion, is unconditionally stable. We have tacitly assumed a positive time step At > 0 in arriving at this stability, which is the usual case. But in Chapter 21, we will introduce reverse time integration where we have At < 0. For such applications, the stability requirements are altered, and the nature of the numerical truncation errors changes. 

Numerical integration techniques are necessary in modeling and simulation of batch and bio processing. In this chapter we described error and stability criteria for numerical techniques. Various numerical techniques for solution of stiff and non-stiff problems are discussed. These methods include one-step and multi-step explicit methods for non-stiff and implicit methods for stiff systems, and orthogonal collocation method for ordinary as well as partial differential equations. These methods are an integral part of some of the packages like MATLAB. However, it is important to know the theory so that appropriate method for simulation can be chosen. 

---