L-stability

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.

Abstract: I Basic Concepts and Discretizations.- II Time Integration Methods.- III Advection-Diffusion Discretizations.- IV Splitting Methods.- V Stabilized Explicit Runge-Kutta Methods.

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.

Abstract: To be A-stable, and possibly useful for stiff systems, a Runge–Kutta formula must be implicit. There is a significant computational advantage in diagonally implicit formulae, whose coefficient matrix is lower triangular with all diagonal elements equal. We derive new, strongly S-stable diagonally implicit Runge–Kutta formulae of order 2 in 2 stages and of order 3 in 3 stages, and show that it is impossible for a strongly S-stable diagonally implicit method to attain order 4 in 4 stages. Merely A-stable diagonally implicit formulae, of order 3 in 2 stages and of order 4 in 3 stages, were previously known; we prove that no 4-stage method of this type has order 5. We describe a computer program for stiff differential equations which uses these methods, and compare them to each other and to the GEAR package.