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L-stability

About: L-stability is a research topic. Over the lifetime, 792 publications have been published within this topic receiving 22796 citations.


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Book
19 Mar 1985
TL;DR: In this paper, limit process expansions applied to Ordinary Differential Equations (ODE) are applied to partial differential equations (PDE) in the context of Fluid Mechanics.
Abstract: 1 Introduction.- 2 Limit Process Expansions Applied to Ordinary Differential Equations.- 3 Multiple-Variable Expansion Procedures.- 4 Applications to Partial Differential Equations.- 5 Examples from Fluid Mechanics.- Author Index.

2,395 citations

Book
21 Jul 2003
TL;DR: This paper presents a meta-modelling procedure called “Stabilized Explicit Runge-Kutta Methods”, which automates the very labor-intensive and therefore time-heavy and therefore expensive process of integrating discrete-time components into a coherent system.
Abstract: I Basic Concepts and Discretizations.- II Time Integration Methods.- III Advection-Diffusion Discretizations.- IV Splitting Methods.- V Stabilized Explicit Runge-Kutta Methods.

1,391 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present an extension of Hooke's Law for determining the stability under stress of thin shells of isotropic elastic material, which they use to determine the equilibrium of an elementary volume of the substance by considering the forces acting upon it.
Abstract: The object of the present paper is to derive equations that are adequate to decide questions of the stability under stress of thin shells of isotropic elastic material. Equations for the same purpose have been given by R. V. Southwell, who used a method that is closely followed in a part of this paper. Such equations must contain terms that may be, and are, neglected in applications of the theory of elasticity to problems in which the stability of configurations is not considered. The truth of Kirchhoff's uniqueness theorem, which has reference to the ordinary equations of elasticity, in which powers of the displacement co-ordinates above the first are neglected, is sufficient proof of this statement. In practice it is generally sufficient to retain only the first and second order terms, and no terms of higher order are considered here. To obtain such equations an extended form of Hooke's Law is necessary; the extension made by Southwell is used in this paper. There are then two methods available for the derivation of the equations. Either we may obtain the three conditions for the equilibrium of an elementary volume of the substance by considering the forces acting upon it, or we may calculate the energy of strain correct to the third order of displacement co-ordinates, and deduce the equations by variation of this function. The first method has been used in one place here, as it would appear to be the simpler in the particular case of a plane plate, in which only one of the equations, and that the simplest, is required. However, the stability equations for a cylindrical shell are also obtained, and then all three equations are necessary. The derivation by the first method of each one of these is a laborious matter, while using the second method there is only one calculation, that of the strain energy function, to be made. Consequently, for this purpose, as in general, the second method seems to be preferable.

1,003 citations

Journal ArticleDOI
TL;DR: Test results indicate that many problems can be solved more efficiently using this scheme than with a single class of methods, and that the overhead of choosing the most efficient methods is relatively small.
Abstract: This paper describes a scheme for automatically determining whether a problem can be solved more efficiently using a class of methods suited for nonstiff problems or a class of methods designed for stiff problems. The technique uses information that is available at the end of each step in the integration for making the decision between the two types of methods. If a problem changes character in the interval of integration, the solver automatically switches to the class of methods which is likely to be most efficient for that part of the problem. Test results, using a modified version of the LSODE package, indicate that many problems can be solved more efficiently using this scheme than with a single class of methods, and that the overhead of choosing the most efficient methods is relatively small.

889 citations

Book
01 Jun 1970
TL;DR: Part I. Fundamental Methods: The calculation of functions, Roots of transcendental equations, and the care and treatment of singularities.
Abstract: Part I. Fundamental Methods: 1. The calculation of functions 2. Roots of transcendental equations 3. Interpolation - and all that 4. Quadrature 5. Ordinary differential equations - initial conditions 6. Ordinary differential equations - boundary conditions 7. Strategy versus tactics - roots of polynomials 8. Eigenvalues I 9. Fourier series Part II. Double Trouble: 10. Evaluation of integrals 11. Power series, continued fractions, and rational approximations 12. Economization of approximations 13. Eigenvalues II - rotational methods 14. Roots of equations - again 15. The care and treatment of singularities 16. Instability in extrapolation 17. Minimum methods 18. Laplace's equation - an overview 19. Network problems.

669 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20236
202211
20212
20203
20192
20184