Alternating direction implicit methods for parabolic equations with a mixed derivative

Alternating direction implicit (ADI) schemes for two-dimensional parabolic equations with a mixed derivative are constructed by using the class of all A sub 0-stable linear two-step methods in conjunction with the method of approximation factorization. The mixed derivative is treated with an explicit two-step method which is compatible with an implicit A sub 0-stable method. The parameter space for which the resulting ADI schemes are second order accurate and unconditionally stable is determined. Some numerical examples are given

Numerical calculations of two dimensional, unsteady transonic flows with circulation

The feasibility of obtaining two-dimensional, unsteady transonic aerodynamic data by numerically integrating the Euler equations is investigated. An explicit, third-order-accurate, noncentered, finite-difference scheme is used to compute unsteady flows about airfoils. Solutions for lifting and nonlifting airfoils are presented and compared with subsonic linear theory. The applicability and efficiency of the numerical indicial function method are outlined. Numerically computed subsonic and transonic oscillatory aerodynamic coefficients are presented and compared with those obtained from subsonic linear theory and transonic wind-tunnel data

An extension of A-stability to alternating direction implicit methods

An alternating direction implicit (ADI) scheme was constructed by the method of approximate factorization. An A-stable linear multistep method (LMM) was used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation. Sufficient conditions for the A-stability of the LMM were determined by applying the theory of positive real functions to reduce the stability analysis of the partial differential equations to a simple algebraic test. A linear test equation for partial differential equations is defined and then used to analyze the stability of approximate factorization schemes. An ADI method for the three-dimensional heat equation is also presented

Metal tube used as solar engine

Ends of metal tube are fastened to axles which are supported on bearings so tube can rotate about its long axis while subjected to invariant bending moment that stresses it along longitudinal axis of rotation. Heat absorbed leads to expansion of metal, which unbalances internal forces and generates rotational moment in tube

Solid medium thermal engine

A device is described which uses a single phase metallic working substance to convert thermal energy directly into mechanical energy. The device consists of a cylindrical metal tube which is free to rotate about its axis while being subjected to continuous bending moment stresses along the longitudinal axis of rotation. The stressing causes portions of the tube to be under compression while other parts are under tension which in turn causes the tube to rotate and provide mechanical energy

Boundary Approximations for Implicit Schemes for .One-Dimensional Inviscid Equations of Gasdynamics

The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary-value problems of the hyperbolic type. For the numerical experiments the one-dimensional inviscid gasdynamic equations in conservation law form are selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit boundary schemes is emphasized. Some numerical examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit vs implicit boundary schemes, and unconditionally stable schemes

Stability Analysis of Numerical Boundary Conditions and Implicit Difference Approximations for Hyperbolic Equations

Implicit noniterative finite-difference schemes have recently been developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time-step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture has been that unconditional stability requires implicit numerical boundary conditions. An apparent counterexample was the space-time extrapolation considered by Gustafsson, Kreiss, and Sundstrom. In this paper we examine space (implicit) and space-time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals. The results indicate that for unconditional stability with a finite number of spatial mesh intervals the numerical boundary conditions must be implicit

Composite absorbing potentials

The multiple scattering interferences due to the addition of several contiguous potential units are used to construct composite absorbing potentials that absorb at an arbitrary set of incident momenta or for a broad momentum interval.Comment: 9 pages, Revtex, 2 postscript figures. Accepted in Phys. Rev. Let