Solutions of ODEs

Hindmarsh, A.C., "Preliminary Documentation of GEARBI Solution of ODE Systems with Block Iterative Treatment of the Jacobian," Lawrence Livermore Laboratory Report UCID-30149, December 1976. 

When applied to the solution of odes, the BI method (Chap. 4) uses a backward difference for the derivative on the left-hand side of (8.9) and the argument of the function on the right-hand side is the future, unknown, concentration vector. In our notation, at the point along the row of concentrations, this is... 

The two methods are BDF and extrapolation. Both methods are used for the numerical solution of odes and are described in Chap. 4. The extension to the solution of pdes is most easily understood if the pde is semidiscretised that is, if we only discretise the right-hand side of the diffusion equation, thus producing a set of odes. This is the Method of Lines or MOL. Once we have such a set, as seen in (8.9), the methods for systems of odes can be applied, after adding boundary conditions. 

In order to understand the problem of finding TS with three or more DOFs, it is useful to address the question of dimensionalities, in configuration and phase space. In classical, Hamiltonian dynamics, transition states are grounded on the idea that certain surfaces (more precisely, certain manifolds) act as barriers in phase space. It is possible to devise barriers in phase space, since in phase space, in contrast to configuration space, two trajectories never cross [uniqueness of solutions of ODEs, see Eq. (4)]. In order to construct a barrier in phase space, the first step is to construct a manifold if that is made of a set of trajectories [8]. 

In numerical solutions of ODE-IVP, the solution y at the point t" is represented by y". The simplest method is Euler s method, which is obtained by writing a difference expression for the derivative ... 

We have seen in Chapter 3 that finite difference equations also arise in Power Series solutions of ODEs by the Method of Frobenius the recurrence relations obtained there are in fact finite-difference equations. In Chapters 7 and 8, we show how finite-difference equations also arise naturally in the numerical solutions of differential equations. 

Applications of Laplace Transforms for Solutions of ODE 369 The dynamic material balance for constant volumetric flow is... 

The following general algorithm applies to the solution of ODEs [5-7]... 

Gear86] Gear C. W. (1986) Maintaining solution invariants in the numerical solution of ODEs. SIAM J. Sci. Stat. Comput. 7(3) 734-743. 

Sha86] Shampine L. F. (1986) Conservation laws and the numerical solution of ODEs. Comp, and Maths, with Appls., Part B. 12. 

When it comes to the numerical solution of ODEs, everyone begins with Euler s method because it is easy to understand and simple to program. Even though its low accuracy keeps it from being widely used for solving ODEs, it gives us an idea of the basic concept of numerical solution for a differential equation simply and clearly. Let us consider a first-order differential equation ... 

Using Euler s explicit method, calculate the PRE involved in the solution of ODE ... 

In case of start-up flow from the rest state, a numerical solution of ODE set (11.4) and (11.5a)-(11.5c) should be obtained using the following initial conditions ... 

Although the PCAS and PCAF methods are similar in form, these two methods are fundamentally different. The objective function of PCAF contains the production rates of species, and the matrix F can be calculated from the right-hand side of ODE (2.9). The objective function of PCAS contains the concentrations of species [the solution of ODE (2.9)], and the matrix S has to be obtained from the solution of the sensitivity differential equations (5.7) and is therefore computationally more time consuming. Put another way, PCAS investigates the effect of parameter changes on the solution of the kinetic system of ODEs, whilst PCAF examines the effect of parameter changes on the right-hand sides of the kinetic system of ODEs (2.9). 

Application of the QSSA is successful if the solution of ODE (7.69) is almost identical to the solution of the coupled DAEs (7.70 and 7.71). What is considered as almost identical may depend on the actual problem and the accuracy required, but in reaction kinetic modelling, a 1 % error for all species at any time is usually considered acceptable. It was emphasised in Sect. 7.2 that the aim of chemical kinetic simulations is the accurate calculation of the concentrations of important species or those of important reaction features. Therefore, the statement above can be refined so that the application of the QSSA is successful if the solutions of Eqs. (7.69) and (7.70-7.71) are almost identical considering the concentrations of important species and the important features. 

---