Showing papers on "L-stability published in 1980"

CHEMEQ - A Subroutine for Solving Stiff Ordinary Differential Equations

Abstract: : A FORTRAN subroutine (CHEMEQ) which employs the Selected Asymptotic Integration Method (SAIM) for solving the stiff ordinary differential equations associated with reactive flow problems is described and documented. The SAIM algorithm identifies the stiff equations for treatment with a stiffly stable method. The remaining equations are treated with a standard classical method. The algorithm is a very low-overhead method which is particularly efficient when coupled with fluid dynamic calculations. It also gives significant reductions in computational costs for model development.

Efficient higher order implicit one-step methods for integration of stiff differential equations

Abstract: A new implicit integration method is presented which can efficiently be applied in the solution of (stiff) differential equations. The given formulas are of a modified implicit Runge-Kutta type and areA-stable. They may containA-stable embedded methods for error estimation and step-size control.

Fixed Leading Coefficient Implementation of SD-Formulas for Stiff ODEs

Abstract: A code based on Enrlght's second-derivative formulas is described for the numermal solution of stiff ODEs A predictor-corrector approach Is taken, and a fixed leadmg coefficmnt technique is used to unplement the variable step-size formulas This strategy is efficient for stiff systems, and the underlying algorithm IS stable and convergent. The code has been designed consistently with the well-known Adams routine STEP, and advantage Is taken of the mmflarlty between Adams' formulas and Ennght 's formulas for an effiment lmplementatmn.

Shooting Methods for Bifurcation Problems in Ordinary Differential Equations

Abstract: We consider multiple shooting methods for bifurcation problems involving boundary value problems for ordinary differential equations. The case of bifurcation from a simple eigenvalue is treated as well as the solution of perturbed bifurcation problems. The original problem is discretizised via shooting techniques. This yields a finite-dimensional bifurcation problem which is solved by a special iteration scheme, having its origin in the theory of Lyapunov and Schmidt. A numerical example demonstrates that our algorithm workes well.

Connections between Stochastic and Ordinary Integral Equations

Abstract: Let σ: ℝn → M m,n (the space of matrices with m columns and n rows) and b: ℝn → ℝn be two Lipschitz continuousmaps. Suppose that σ is of class C2. Let B = (Bt)t≥0 (B0 = 0) be a standard ℝm valued Brownian motion defined on a probability space (Ω,ƒ, ƒt, P). Consider the solution Xx of the following equation: $$ X_{t}^{X} = x + s.\int_{0}^{t} {\sigma (X_{S}^{X})d{{B}_{S}}} + \int_{0}^{t} {b(X_{S}^{X})ds} ,\;x \in {{\mathbb{R}}^{n}} $$ (1) .

Detection and integration of oscillatory differential equations with initial stepsize, order and method selection

Abstract: Within any general class of problems there typically exist subclasses possessed of characteristics that can be exploited to create techniques more efficient than general methods applied to these subclasses. Two such subclasses of initial-value problems in ordinary differential equations are stiff and oscillatory problems. Indeed, the subclass of oscillatory problems can be further refined into stiff and nonstiff oscillatory problems. This refinement is discussed in detail. The problem of developing a method of detection for nonstiff and stiff oscillatory behavior in initial-value problems is addressed. For this method of detection a control structure is proposed upon which a production code could be based. An experimental code using this control structure is described, and results of numerical tests are presented. 3 figures.

GEARS: a package for the solution of sparse, stiff ordinary differential equations. [In Fortran]

Abstract: This paper describes GEARS, a package of Fortran subroutines designed to solve stiff systems of ordinary differential equations of the form dy/dt = f(y,t), where the Jacobian matrices J = par. delta f/par. delta y are large and sparse. The integrator is based on the stiffly stable methods due to Gear, and this approach leads to a sparse system of nonlinear equations at each time step. These are solved with a modified Newton iteration, using one of two separate sparse matrix packages to solve the sparse linear equations that arise. This paper describes the package, in some detail, discusses a number of issues that affected the design of the package, and presents a numerical example to illustrate the effectiveness of the package. 1 figure, 1 table.

A method of skipping the transient phase in the solution of separably stiff ordinary initial value problems

Abstract: Stiff systems of ordinary differential equations are characterized by an initial phase in which the solution changes rapidly. Often there is no interest in reproducing this transient phase. A method is proposed for modifying the initial value if the system of differential equations is separably stiff, i.e. is characterized by the occurrence of a few (typically one) large negative real eigenvalues which dominate the others. The modified system does not possess a transient phase, and in the constant coefficient linear case its solution does not differ from that of the original one in the nonstiff components.

An efficient algorithm reduces computing time in solving a system of stiff ordinary differential equations

Abstract: The problem of stiff differential equations arises in many computer-aided design techniques, particularly in the transient analysis of network simulation. Special multistep methods are used to solve the first-order stiff nonlinear differential equations. Instability and a large number of steps are encountered during simulation. Different techniques such as step and order selection schemes, procedures for changing step and order, may reduce the number of steps while preserving stability. An improved algorithm is presented using BDF formulas given by Brayton et at. and leads to reducing computer time by controlling the number of integration steps, but also the number of Newton iterations, the number of Jacobian matrix evaluations and other parameters, without producing additional errors or instability phenomena. Experimental results are shown.