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Subroutine PLTMG is a FORTRAN program for solving self-adjoint elliptic boundary value problems in general regions of R-squared. It is based on a piecewise linear triangle finite element method, an adaptive grid refinement procedure, and a multi-level iterative method to solve the resulting sets of linear equations. This paper describes the method and presents some numerical results and comparisons.
This paper treats the stability of the initial boundary value problem for the method of lines applied to hyperbolic and parabolic partial differential equations in one space dimension. The theory treats the case of variable coefficients and allows for very general boundary conditions. Several examples are given which illustrate the theory. The theory is analogous to that developed by Gustafsson, Kreiss, and Sundstrom for finite-difference methods.
The stability of approximating the solution of mixed initial-boundary value problems for hyperbolic systems by semidiscrete Galerkin methods is studied. It is shown that a particular straightforward Galerkin method yields an unstable approximation, and that this numerical instability is caused by an improper treatment of the boundary. Stable schemes are then presented, one of which differs from the unstable scheme only insofar as the treatment of the boundary is concerned. These stable schemes make use of a particular matrix which symmetrizes the differential system. It is therefore shown that the use of this matrix is crucial to the stability of the computations as well as for obtaining a priori bounds on the energy of the continuous system. This symmetrizing matrix is also related to the diagonalizing matrix for the system of hyperbolic equations and to the Liapunov matrix for the system of ordinary differential equations resulting from the application of Galerkin's method.
Method of particular solutions used to solve two point boundary value problem for nonlinear controlled system
Neutron linear transport theory boundary value problems, using Green function approach
Linearized and weakly nonlinear Boltzmann equation boundary value problems for gas between parallel plates, noting solution existence and uniqueness
Scheme for choosing approximating system of difference equations for degenerate linear and nonlinear elliptic boundary value problems
The stability of semi-discrete approximations for hyperbolic initial-boundary-value problems is considered. Asymptotic estimates of the eigenvalues of the finite-domain problem are given. These estimates are used to examine questions relating the normal mode analysis of the finite-domain problem and the normal mode quarter-plane analysis of the Gustafsson, Kreiss, and Sundstroem theory.
Method for solution of general boundary value problem of propagation of long and ultralong radiowaves around earth
We solve in closed form the boundary-value problem for the partial differential equations which describe the (azimuthal) rotation velocity and induced magnetic fields in a cylindrical plasma centrifuge with ring electrodes of different radii and an external, axial magnetic field. The electric field, current density, and velocity distributions are discussed in terms of the Hartmann number H and the magnetic Reynolds number R. For small Hall coefficients, the induced magnetic field does not affect the plasma rotation. As a result of the Lorentz forces, the plasma rotates with speeds as high as 100,000 cm/sec around its axis of symmetry at typical conditions, so that the lighter (heavier) ion and atom components are enriched at (off) the center of the discharge cylinder.
Mixed boundary value problem of elastostatics for isotropic elastic cylinder containing strip crack opened by internal pressure, using auxiliary functions
New convenient stability criteria are provided for a large class of finite difference approximations to initial-boundary value problems associated with the hyperbolic system u sub t = Au sub x + Bu + f in the quarter plane x greater than or equal to 0, t greater than or equal to 0. The criteria are used to easily establish stability for numerous combinations of well known basic schemes and boundary conditions, thus generalizing many special cases studied in the recent literature. A number of examples are examined, including the unitary unconditionally stable Crank-Nicholson scheme and an almost-dissipative unconditionally stable backward Euler scheme.
The conventional method of imposing time dependent boundary conditions for Runge-Kutta (RK) time advancement reduces the formal accuracy of the space-time method to first order locally, and second order globally, independently of the spatial operator. This counter intuitive result is analyzed in this paper. Two methods of eliminating this problem are proposed for the linear constant coefficient case: (1) impose the exact boundary condition only at the end of the complete RK cycle, (2) impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the RK accuracy in all cases, results in a scheme with much reduced CFL condition, rendering the RK scheme less attractive. The second method retains the same allowable time step as the periodic problem. However it is a general remedy only for the linear case. For non-linear hyperbolic equations the second method is effective only for for RK schemes of third order accuracy or less. Numerical studies are presented to verify the efficacy of each approach.
This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.
Definitions in the case of multiple series equations and multiple integral equations are examined. In considering the solution of a given mixed boundary value problem perhaps the simplest technique is the direct application of the method of complex potentials provided the problem admits such potentials and the domain and the boundary conditions are suitable for such an application. The direct application of complex potentials is described with the aid of examples, taking into account a problem in potential theory, the case of periodic cuts, and an elasticity problem for a nonhomogeneous plane. The reduction to singular integral equations is discussed along with the numerical solution of singular integral equations of the first kind, integral equations with generalized Cauchy kernels, and singular integral equations of the second kind.
Technical memorandum discusses stability of numerical solutions involving semidiscrete approximations to hyperbolic partial differential equations in initial-and-boundary-value problems. Topic of practical significance because hyperbolic partial differential equations arise in mathematical modeling of waves and blasts. Solutions often needed over restricted regions of unbounded spaces. Outer boundaries artificial, introduced only to limit domains of numerical solutions. Conditions at such artificial boundaries cause numerical instabilities that degrade computed solutions.
The mathematics involved in numerically solving for the plane boundary value of the Laplace equation by the grid method is developed. The approximate solution of a boundary value problem for the domain of the Laplace equation by the grid method consists of finding u at the grid corner which satisfies the equation at the internal corners (u=Du) and certain boundary value conditions at the boundary corners.
New convenient stability criteria are provided in this paper for a large class of finite difference approximations to initial-boundary value problems associated with the hyperbolic system u sub t = au sub x + Bu + f in the quarter plane x or = 0, t or = 0. Using the new criteria, stability is easily established for numerous combinations of well known basic schemes and boundary conditions, thus generalizing many special cases studied in recent literature.