Showing papers on "L-stability published in 1976"

Numerical Solution of Large Systems of Stiff Ordinary Differential Equations in a Modular Simulation Framework

Abstract: This chapter discusses the numerical solution of large systems of stiff ordinary differential equations (o.d.e.s.) in a modular simulation framework. A stiff ordinary differential equation is one in which one component of the solution decays much faster than others. Many chemical engineering systems give rise to systems of stiff o.d.e.s. Such situations occur in chemical reactors where the rate constants for the reactions involved are widely separated, and in multistage systems, for example in a distillation column, because the time constants for the various components and plates may be very different and also as the time constant for the reboiler is large compared with the time constant of a plate. Several methods were selected for numerical testing: (1) four conventional methods, none of which are oriented to solving stiff systems including Euler, Adams–Moulton–Shell (Shell, 1958); Runge–Kutta–Merson (Merson, 1957); and Gear's nonstiff option (Gear, 1971a, b); (2) four explicit stiff techniques include Richards, Lanning, and Torrey (1965), Nigro (1969 ), Treanor (1966), Fowler and Warten (1967); (3) four implicit stiff techniques include Klopfenstein and Davis (1971), Sandberg and Schichman (1968), Brandon (1972, 1974c), and Gear's stiff option (Gear 1971a, b).

Generalized Padé and Chebyshev Approximations for Point Kinetics

Abstract: New methods of the Pade and Chebyshev type are applied to the solution of the point kinetics equations. A theoretical analysis is presented in an attempt to derive global approximations by opposition to the local ones that have been used up to now. It is shown that many of the techniques developed earlier for integrating stiff systems of ordinary differential equations are particular cases of this general formalism. Numerical studies are presented for various reactivity insertions. The results confirm the theoretical analysis and indicate the range of applicability of the methods presented. (auth)

On the numerical solutions of some Volterra equations on infinite intervals

Abstract: The paper discusses long time behavior and error bounds for discretized Volterra equations. A key property is positivity of the quadrature which is combined with monotonicity properties of the nonlinearities in the equations. It is shown how the positivity of discretization quadrature is linked with \(A\)-stability property of linear multistep methods for ordinary differential equations. Some of the results are new when applied to differential equations with monotone nonlinearities and \(A\)-stable discretizations.

Problem #11: generation of Runge-Kutta equations

Abstract: Generate a set of equations for an explicit k-th order, m stage, Runge-Kutta method for integrating an autonomous system of ordinary differential equations, k and m as large as possible. The number of conditions and variables for various k and m are given in Table 1. Tabulate the costs C(m,k).

On Liapunov Stability of Stiff Non-Linear Multistep Difference Equations,

Abstract: : The concepts of G-stability and (G,mu)-stability recently introduced by Dahlquist are useful for discussing Liapunov stability of solutions to systems of non-linear difference equations, generated by applying linear multistep formulas to monotone, dissipative, arbitrarily stiff systems of non-linear differential equations. In this paper, the theory of G-stability and (G,mu)-stability is reviewed and a construction is proposed which facilitates the finding of a quadratic Liapunov function. By this construction it is proved that, for the four-parameter family of all three-step formulas which are second-order accurate, A-stability is necessary and sufficient for G-stability. Some results on (G,mu)-stability are also obtained by this construction.