Showing papers on "L-stability published in 1981"
TL;DR: Methods of orders 5 and 6 are derived which require one evaluation of the Jacobian and oneLU decomposition per step and are A-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations.
Abstract: This paper deals with the solution of nonlinear stiff ordinary differential equations. The methods derived here are of Rosenbrock-type. This has the advantage that they areA-stable (or stiffly stable) and nevertheless do not require the solution of nonlinear systems of equations. We derive methods of orders 5 and 6 which require one evaluation of the Jacobian and oneLU decomposition per step. We have written programs for these methods which use Richardson extrapolation for the step size control and give numerical results.
89 citations
TL;DR: A critical evaluation of the test set is made, and changes are recommended that make the set a more useful tool.
Abstract: In 1975 Enright, Hull, and Lindberg published a set of problems for the testing of codes for stiff ordinary differential equation {ODE) problems. It has been widely used since then. In this paper a critical evaluation of the test set is made, and changes are recommended that make the set a more useful tool. Comments about the difficulties of testing software in this area are made which may lead to more thoughtful use of the test set.
50 citations
TL;DR: In this article, a numerical integration of systems of ordinary differential equations that arise on semidiscrete spatial differencing or finite element projection for evolution problems characterized by partial differential equations is examined.
16 citations
TL;DR: In this article, the authors consider a stiff system of nonlinear ordinary differential equations for which we know some but not all of the initial conditions and present an algorithm which determines the unknown initial values in such a way that the solution does not have an initial transient.
Abstract: We consider a stiff system of nonlinear ordinary differential equations for which we know some but not all of the initial conditions. This paper presents an algorithm which determines the unknown initial values in such a way that the solution does not have an initial transient. The algorithm was motivated by a problem from tonospheric physics in which two of the initial conditions are unknown. By physical considerations, the correct solution should not have an initial transient. This requirement allows us to uniquely specify the two unknown initial values.
14 citations
TL;DR: In this paper, the authors developed an efficient procedure to determine the unknown initial value in such a way that the solution of the system does not have an initial transient, which is meaningful only if the Jacobian matrix has exactly one stiff eigenvalue.
Abstract: Consider a stiff system of ordinary differential equations for which one of the initial values is unknown. We develop an efficient procedure to determine the unknown initial value in such a way that the solution of the system does not have an initial transient. The procedure works and is meaningful only if the Jacobian matrix of the system has exactly one stiff eigenvalue. We apply the procedure to the solution of a problem in the physics of the upper ionosphere.
10 citations
TL;DR: The quadrature methods are based upon a substitution of an explicit A-stable first approximation into a generalized convolution formula as discussed by the authors, and they are A -stable, explicit, and of arbitrarily high order.
Abstract: The quadrature methods are based upon a substitution of an explicit A-stable first approximation into a generalized convolution formula. They are A -stable, explicit, and of arbitrarily high order. The generalized convolution formula is derived and its order-raising properties examined. Two families of explicit A-stable first approximations are developed, generalizing results of Lawson and N0rsett. Various aspects of the numerical implementation are discussed. Numerical results supplement the paper and exemplify the various merits and weaknesses of the quadrature methods.
10 citations
TL;DR: Safe Point Methods (SPM) as mentioned in this paper are a class of methods which utilize standard explicit numerical methods, but make at each step a correction which eliminates the effect of the dominant eigenvalue of a simply separably stiff system.
Abstract: A stiff system of ODE’s is defined to be separably stiff if the stiffness is due to a small number of real negative eigenvalues well separated from the rest, and to be simply separably stiff if there is just one such eigenvalue. Although the majority of stiff systems encountered in practice are not separable, they do arise, and special techniques for them could have a place in packages which call on a variety of methods; they can also have an application in real-time problems. This paper develops a class of methods called Safe Point Methods (SPM), which utilize standard explicit numerical methods, but make at each step a correction which eliminates the effect of the dominant eigenvalue of a simply separably stiff system. The corrections are made by proceeding to (and, for some techniques, returning from) a so-called safe point, where the local solution does not have a large derivative. To find such points, it is necessary to find the dominant eigenvalue, and it is recommended that this be done by the powe...
7 citations
TL;DR: Most of the spatial discretization schemes of transport equations in slab geometry which have been developed recently are particular applications of a general finite element oriented formalism developed by this author and his collaborators for the numerical integration of systems of stiff ordinary differential equations.
6 citations
TL;DR: In this article, a general method of formulating Lagrangian and anti-Lagrangian equations for networks consisting of non-linear RLCM (M = memristor) multiports with hybrid representation is presented.
Abstract: A general method is presented of formulating the Lagrangian and anti-Lagrangian equations for networks consisting of non-linear RLCM (M = memristor) multiports with hybrid representation.
The formulation is obtained with no restriction on the network topology. An explicit procedure is given to construct the scalar functions needed. This procedure is based on the concept of L and H functions introduced. The Brayton-Moser equations in a generalized form can be directly obtained from the anti-Lagrangian equations. From these equations, new equations of Brayton-Moser type can also be derived.
The invariance property of Lagrangian and anti-Lagrangian equations under a transformation of variables is also discussed.
5 citations
TL;DR: In this article, the authors studied the efficiency of computational methods for the stiff ODEs of chemical kinetics that arise when the partial differential equations of chemically reacting gas flow are treated by a fractional step technique.
3 citations
TL;DR: In this paper, it was shown that the assumption that exact solution values are used to start the step is not required for certain estimates, and that the local error estimates for such methods are obtained as linear combinations of computed values.
Abstract: Most methods used for the numerical solution of ordinary differential equations may be characterized by a pair of matrices $(A,B)$. In this article, local error estimates for such methods are obtained as linear combinations of computed values. (This is often equivalent to constructing pairs of embedded methods.) Although these local error estimates are obtained by assuming that exact solution values are used to start the step, it is shown that this assumption is not required for certain estimates. Some hybrid methods have such (computable) estimates, and also have simple step length change procedures.
01 Jan 1981
TL;DR: In this article, a stiff system of linear evolution equations in Banach space is investigated, and it is shown that the standard singular perturbation algorithm can be considerably simplified and the asymptotic solutions can be obtained in each order independently.
Abstract: A stiff system of linear evolution equations in Banach space is investigated. It is shown that the standard singular perturbation algorithm can be considerably simplified. In this modified algorithm the asymptotic solutions can be obtained in each order independently. The initial conditions are given explicitly so there is no need to solve the so‐called ’’inner’’ equations. Two examples of application are considered.
01 Jan 1981
TL;DR: A new technique is developed for the real-time solution of stiff differential systems that is A-stable and suitable for hardware-in-the-loop application and properties of the techniques are examined.
Abstract: A new technique is developed for the real-time solution of stiff differential systems. The method is A-stable and is suitable for hardware-in-the-loop application. A computational architecture for implementation of the algorithm is presented, and properties of the techniques are examined.