Integration of Stiff Equations

Abstract

In the study of chemical kinetics, electrical circuit theory, and problems of missile guidance a type of differential equation arises which is exceedingly difficult to solve by ordinary numerical procedures. A very satisfactory method of solution-of these equations is obtained by making use of a forward interpolation process. This scheme has the unusual property of singling out and approximating a particular solution of the differential equation to the exclusion of the manifold of other solutions. This behavior may be explained by a simple geometrical interpretation of the significance of the forward interpolation process. The differential equations to which this method applies are called "stiff."A typical example of a stiff equation is the equation representing the rate of formation of free radicals in a complex chemical reaction. The free radicals are created and destroyed so rapidly compared to the time scale for the over-all reaction that to a first approximation the rate of production is equal to the rate of depletion. This is the notion of the pseudo-stationary state. In some cases such as the fast reactions occurring in flames or detonations, this approximation is not sufficiently accurate. The method described in the present paper provides a means for obtaining solutions to equations of this type to any degree of accuracy.The nunmerical procedure described here can easily be extended to sets of simultaneous first-order differential equations. In any particular region, the differential equations can be uncoupled by introducing suitable linear combinations of the original dependent variables. Some of the uncoupled equations may be "stiff" in which case they can be integrated by the methods discussed here; whereas other uncoupled equations may be integrated by the more usual procedures.

A comparison of high-order time integrators for thermal convection in rotating spherical shells

“…where C N L , C Q , C LS , and C MP are, respectively, the costs of the evaluation of the non-linear terms, the evaluation of the operator Q, the solution of the blockdiagonal linear systems (20) in the case of taking Q explicitly, and the matrix products by the block-tridiagonal matrices H m when Q is taken implicitly. The integer N GMR is the average number of iterations performed by the linear iterative solver (GMRES in our case).…”

“…However it will be seen that matrix-free methods based on Krylov techniques [23], GMRES [24] in our case, can be used efficiently, to solve the block-tridiagonal linear systems with the same memory requirements as in the Q-explicit case. The increase of the cost in solving the linear systems (20) may be offset by the increase of the time stepsize. The initial approximation for the solution of the linear system is obtained by extrapolation from the previous steps, as is usually done in the implementations of the BDF.…”

“…, . The IMEX-BDF formulae mentioned before are related to the backward differentiation formulae (BDF) [20,16]. These are collocation multistep methods, which obtain u n+1 from the previous approximations u n−j , j = 0, 1, .…”

“…They can be solved with the same memory requirements and number of operations as the Q-explicit method, because Q u and Q l must be evaluated in any case. The systems (20) are solved by using backward or forward block substitution, by inverting the same diagonal submatrices as in the Q-explicit method. In order to implement this possibility in a symmetric way, the two options are used alternately, that is, one step is performed with Q u implicit and Q l explicit, and vice-versa in the following step.…”

A semi-analytical approach to molecular dynamics

“…Bonded potentials thus differ by approximately three orders of magnitude from the remaining potentials, and render molecular dynamics stiff [10]. Stiffness spells trouble for numerical simulation as it typically requires unduly small time step sizes, particularly for explicit integrators: the presence of significant fast motions in MD can limit the time step to below one femtosecond.…”