Showing papers on "L-stability published in 1991"
TL;DR: Projected Implicit Runge-Kutta (PIRK) as mentioned in this paper is a new class of numerical methods for the solution of index-2 Hessenberg systems of initial and boundary value differential-algebraic equations.
Abstract: In this paper a new class of numerical methods, Projected Implicit Runge–Kutta methods, is introduced for the solution of index-2 Hessenberg systems of initial and boundary value differential-algebraic equations (DAEs). These types of systems arise in a variety of applications, including the modeling of singular optimal control problems and parameter estimation for differential-algebraic equations such as multibody systems. The new methods appear to be particularly promising for the solution of DAE boundary value problems, where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations. Previously defined symmetric methods have severe limitations when applied to these problems, including instability, oscillation, and loss of accuracy; the new methods overcome these difficulties. For linear problems we define an essential underlying boundary value ODE and prove well-conditioning of the differential (or state-space) solutio...
134 citations
TL;DR: In this article, the authors present a numerical procedure for solving a set of nonlinear coupled-mode equations on a finite interval, which originally arose in the study of the dynamics of gap solitons in nonlinear periodic media.
Abstract: We present a numerical procedure for solving a set of nonlinear coupled-mode equations on a finite interval. These equations originally arose in the study of the dynamics of gap solitons in nonlinear periodic media. Our procedure, which uses an implicit fourth-order Runge–Kutta method, is easy to implement, versatile, and quite well suited for vectorization or parallelization.
134 citations
TL;DR: This paper examines diagonally implicit iteration methods for solving implicit Runge–Kutta methods with high stage order on parallel computers and shows that the reduced accuracy often shown when integrating stiff problems by means of DIRK methods is not shown by the D IRK methods developed in this paper.
Abstract: This paper examines diagonally implicit iteration methods for solving implicit Runge–Kutta methods with high stage order on parallel computers. These iteration methods are such that after a finite number of m iterations, the iterated Runge–Kutta method belongs to the class of diagonally implicit Runge–Kutta methods (DIRK methods) using $mk$ implicit stages where k is the number of stages of the generating implicit Runge–Kutta method (corrector method). However, a large number of the stages of this DIRK method can be computed in parallel, so that the number of stages that have to be computed sequentially is only m. The iteration parameters of the method are tuned in such a way that fast convergence to the stability characteristics of the corrector method is achieved. By means of numerical experiments it is also shown that the solution produced by the resulting iteration method converges rapidly to the corrector solution so that both stability and accuracy characteristics are comparable with those of the corrector. This implies that the reduced accuracy often shown when integrating stiff problems by means of DIRK methods already available in the literature (which is caused by a low stage order) is not shown by the DIRK methods developed in this paper, provided that the corrector method has a sufficiently high stage order.
95 citations
TL;DR: It will be shown that these methods require fewer stages to achieve the same order as one-step Runge–Kutta methods, which means the two-step methods are potentially more efficient than one- step methods.
Abstract: Implicit two-step Runge–Kutta methods are studied. It will be shown that these methods require fewer stages to achieve the same order as one-step Runge–Kutta methods, which means the two-step methods are potentially more efficient than one-step methods. Order conditions are derived and examples of two-step one-stage methods of order 2 and two-step two-stage methods of order 4 are presented. Stability properties of these methods with respect to $y' = ay$ are studied and A-stable two-step methods of order 2 are characterized. Two-step two-stage methods of order 4 which are A-stable are found by an extensive computer search. Semi-implicit two-stage methods of order 4 were also constructed. This is in contrast to the situation encountered in the Runge–Kutta theory where the unique two-stage method of order 4 is not semi-implicit.
56 citations
01 Sep 1991
TL;DR: The parallel solution of initial value problems for ordinary differential equations has become an active area of research with particular emphasis on traditional forward-step methods that offer the potential for effective small-scale parallelism on existing machines.
Abstract: The parallel solution of initial value problems for ordinary differential equations has become an active area of research. Recent developments in this area are surveyed with particular emphasis on traditional forward-step methods that offer the potential for effective small-scale parallelism on existing machines. >
50 citations
TL;DR: A new class of numerical methods for solving equations of motion of constrained mechanical systems is presented, the framework of which is based on manifold theoretic methods.
Abstract: A new class of numerical methods for solving equations of motion of constrained mechanical systems is presented, the framework of which is based on manifold theoretic methods. Rewriting the system of differential-algebraic equations (DAEs) that describe constrained motion is ordinary differentia] equations (ODEs) on a constraint manifold, the theoretical framework for solving equations of motion is constructed, using a local
32 citations
TL;DR: In this paper, the necessary conditions of the compatibility of the d'Alembert-Hamilton system in Minkowsky space R (1,n) were established and the problem of reduction of P(1, n)-invariant wave equations to ordinary differential equations was discussed.
26 citations
TL;DR: In this paper, conditions for the B-convergence of Rosenbrock type methods when applied to stiff semi-linear systems are given, and the convergence results are extended to stiff nonlinear systems in singular perturbation form.
Abstract: In this paper we give conditions for theB-convergence of Rosenbrock type methods when applied to stiff semi-linear systems. The convergence results are extended to stiff nonlinear systems in singular perturbation form. As a special case partitioned methods are considered. A third order method is constructed.
18 citations
Proceedings Article•
14 Jul 1991TL;DR: A language QFL for describing qualitative temporal behaviors is presented, and procedures and an implementation QDIFF that solves equations in this form are demonstrated.
Abstract: Numerical simulation, phase-space analysis, and analytic techmques are three methods used to solve quantitative differential equations. Most work in Qualitative Reasoning has dealt with analogs of the first two techniques, producing capabilities applicable to a wide range of systems. Although potentially of benefit, little has been done to provide closed-form, analytic solution techniques for qualitative differential equations (QDEs).
This paper presents one such technique for the solution of a class of ordinary linear and nonlinear differential equations. The technique is capable of deriving closed-form descriptions of the qualitative temporal behavior represented by such equations. A language QFL for describing qualitative temporal behaviors is presented, and procedures and an implementation QDIFF that solves equations in this form are demonstrated.
11 citations
TL;DR: This paper is concerned with how to diagnose stiffness as the reason for unsatisfactory performance by a code based on explicit Runge–Kutta formulas.
Abstract: Explicit Runge–Kutta methods are a popular way to solve the initial value problem for a system of ordinary differential equations. Although very effective for nonstiff problems, they are impractical for stiff problems. This paper is concerned with how to diagnose stiffness as the reason for unsatisfactory performance by a code based on explicit Runge–Kutta formulas.
9 citations
TL;DR: A new procedure for the integration of both stiff and nonstiff ordinary differential equations is presented, which is applicable to general nonlinear systems, is fully explicit, requires little computer memory, and is very easy to program.
Abstract: A new procedure for the integration of both stiff and nonstiff ordinary differential equations is presented. This new approach, which is applicable to general nonlinear systems, is fully explicit, requires little computer memory, and is very easy to program. It does not require computation of a Jacobian matrix or the solution of algebraic equations. The new algorithm has much better stability properties than the classical explicit methods and is generally much more efficient than these methods for stiff problems. The new approach automatically partitions the dependent variables at every step into stiff and nonstiff groups, and subsequently integrates each accordingly, using a nonstiff method for the nonstiff variables and a new explicit exponential method for the stiff variables. The algorithm blends these approximations appropriately. Because of the automatic partitioning, the new algorithm is entirely suitable for both stiff and nonstiff problems.
01 Jan 1991
TL;DR: With the numerical algorithm suggested in this paper, stiff DAE systems can be solved accurately and stably even with explicit integration methods.
Abstract: This paper extends a numerical algorithm for solving nonlinear index problems of Chung and Westeiberg [Ind. Eng. Chem. Res., 29 (1990), pp. 1234-1239] to include stiff DAE systems. Stiff DAE systems are shown to be near index problems. Near high index problems cannot be solved by existing stiff ODE/DAE solvers, such as LSODI. A stiff problem should be classified by index. Solutions to these problems can be expressed and controlled in terms of polynomials in the small coefficients which are responsible for the near singularity at the solution point With the numerical algorithm suggested in this paper, stiff DAE systems can be solved accurately and stably even with explicit integration methods.
TL;DR: A compact, absolutely stable numerical method to integrate stiff systems of pseudo-linear, ordinary, first-order differential equations, commonly found in the simulation of biological models, and can be used with deferred approximation to the limit h = 0.
TL;DR: In this paper, a model for the non-steady-state description of gas fluid and d beds is derived, based on the bubble dispersion model, which is mathematically and numerically not cbmplex and good results are obtained.
TL;DR: In this article, a theoretical comparison is made of several popular methods for solving coupled second-order differential equations such as arise in atomic and molecular scattering calculations, recast in a very simple predictor-corrector form.
TL;DR: A method to transform programs by solving program equations, which makes it possible to reduce such equations into systems of equations only containing simple constructors of programs.
Abstract: Based on the theory of orthogonal program expansion[8–10], the paper proposes a method to transform programs by solving program equations. By the method, transformation goals are expressed in program equations, and achieved by solving these equations. Although such equations are usually too complicated to be solved directly, the orthogonal expansion of programs makes it possible to reduce such equations into systems of equations only containing simple constructors of programs. Then, the solutions of such equations can be derived by a system of solving and simplifying rules, and algebraic laws of programs. The paper discusses the methods to simplify and solve equations and gives some examples.
TL;DR: In this article, the authors proposed a new method for solving optimization problems, where the minimization of a function f(X), where X is a vector containing N design variables, can be solved by searching for the minimum along a curve X(t).
Abstract: This paper proposes a new method for solving optimization problems. This method is entirely different from the known methods. The minimization of a function f(X), where X is a vector containing N design variables, can be solved by searching for the minimum along a curve X(t). This curve is represented by a parameter t and is solved by ordinary differential equations (ODE) of dX(t)/dt ‐ ‐ ??f(X) with the initial conditions of X(0) = Xo, where X0 is the initial estimate solution.
TL;DR: In this article, an iterative method of solving some stiff linear boundary-value problems for equations of fourth and second order is constructed, and the rate of convergence is shown to be not slower than for a geometric progression whose ratio decreases as the stiffness parameter of the original equation increases.
Abstract: An iterative method of solving some stiff linear boundary-value problems for equations of fourth and second order is constructed. The rate of convergence is shown to be not slower than for a geometric progression whose ratio decreases as the stiffness parameter of the original equation increases. Results of some numerical experiments are presented.