Paul Concus

A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations

We consider a generalized conjugate gradient method for solving sparse, symmetric, positive-definite systems of linear equations, principally those arising from the discretization of boundary value problems for elliptic partial differential equations. The method is based on splitting off from the original coefficient matrix a symmetric, positive-definite one that corresponds to a more easily solvable system of equations, and then accelerating the associated iteration using conjugate gradients. Optimality and convergence properties are presented, and the relation to other methods is discussed. Several splittings for which the method seems particularly effective are also discussed, and for some, numerical examples are given.

Block Preconditioning for the Conjugate Gradient Method

Block preconditionings for the conjugate gradient method are investigated for solving positive definite block tridiagonal systems of linear equations arising from discretization of boundary value problems for elliptic partial differential equations. The preconditionings rest on the use of sparse approximate matrix inverses to generate incomplete block Cholesky factorizations. Carrying out of the factorizations can be guaranteed under suitable conditions. Numerical experiments on test problems for two dimensions indicate that a particularly attractive preconditioning, which uses special properties of tridiagonal matrix inverses, can be computationally more efficient for the same computer storage than other preconditionings, including the popular point incomplete Cholesky factorization.

A generalized conjugate gradient method for nonsymmetric systems of linear equations

We consider a generalized conjugate gradient method for solving systems of linear equations having nonsymmetric coefficient matrices with positive-definite symmetric part. The method is based on splitting the matrix into its symmetric and skew-symmetric parts, and then accelerating the associated iteration using conjugate gradients, which simplifies in this case, as only one of the two usual parameters is required. The method is most effective for cases in which the symmetric part of the matrix corresponds to an easily solvable system of equations. Convergence properties are discussed, as well as an application to the numerical solution of elliptic partial differential equations.

NUMERICAL SOLUTION OF A NONLINEAR HYPERBOLIC EQUATION BY THE RANDOM CHOICE METHOD

Abstract The numerical solution of a nonlinear hyperbolic equation not fulfilling the strict non-linearity condition is considered. A solution procedure is developed based on the random choice method, which permits the sharp tracking of discontinuities. As an illustration, an application to the two-phase flow of petroleum in underground reservoirs is presented.

Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method

We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential boundary-value problems. Here, extensions to the nonlinear case are considered. We split the original discretized operator into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and accelerate the associated iteration based on this splitting by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildy nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial.ZusammenfassungWir haben früher eine verallgemeinerte Methode der konjugierten Gradienten studiert, um dünnbesetzte positiv definite Systeme von linearen Gleichungen zu lösen, die von der Diskretisierung von elliptischen partiellen Differential-Randwertproblemen herrühren. Wir betrachten hier die Verallgemeinerung auf den nichtlinearen Fall: Wir spalten den ursprünglichen diskretisierten Operator auf in eine Summe von zwei Operatoren. Einer von diesen Operatoren entspricht einem leicht lösbaren System von Gleichungen, und wir beschleunigen die aus dieser Spaltung hervorgehende Iteration mit (nichtlinearen) konjugierten Gradienten. Das Verhalten der Methode wird illustriert durch Anwendung auf die Minimalflächen-Gleichung, mit Spaltungen entsprechend dem nichtlinearen SSOR-Verfahren, der angenäherten Faktorisierung der Jacobi-Matrix, oder den elliptischen Operatoren, die sich für schnelle direkte Methoden eignen. Die Resultate von numerischen Experimenten für ein nur schwach nichtlineares Beispiel sind ebenfalls angegeben. Für den entsprechenden linearen Fall ist in diesem Fall die Konvergenz des konjugierten Gradienten-Algorithmus in einer endlichen Anzahl von Schritten wesentlich.

STANDING CAPILLARY-GRAVITY WAVES OF FINITE AMPLITUDE

Standing surface waves in an inviscid incompressible fluid of finite depth are considered, taking into account the effect of capillary forces. Perturbation solutions for the surface profile, velocity potential, frequency of oscillation, and pressure are found to third order in the amplitude of the waves. A graph is given showing the regions in which the frequency of oscillation increases with amplitude and those in which it decreases with amplitude. These regions are defined as a function of the depth of the fluid and a parameter called the relative capillarity. A graph is also given showing the surface profile of a wave. (auth)

Numerical solution of the nonlinear magnetostatic-field equation in two dimensions

Abstract The numerical solution of the second-order, elliptic, quasi-linear, partial differential equation arising in a two-dimensional magnetostatic-field problem, where the magnetic permeability varies with the field, is considered. A set of nonlinear difference equations approximating the original differential equation is derived, and in solving a test problem, the method of nonlinear successive overrelaxation compares favorably both to Newtons method and to a commonly used method based on a small-magnetic-field approximation. The first method, as here presented, could also be used to numerically solve similar equations, such as those for Plateaus problem or for irrotational compressible fluid flow.

Instability of certain capillary surfaces

A family of capillary surface configurations, shown in an earlier paper to be unstable for sufficiently small Bond numberB, is here shown to be unstable also forB sufficiently large. Numerical evaluations indicate that it is unstable for allB. Some numerical calculations of the corresponding container shapes are included.

Nonuniqueness and Uniqueness of Capillary Surfaces

It is shown that for any gravity field g and contact angle 7, an axially symmetric container can be found that differs arbitrarily little from a circular cylinder and can be half filled with liquid in a continuum of distinct ways, such that no two of the surface interfaces are mutually congruent and such that all of them are in equilibrium with the same mechanical energy. This answers affirmatively a question raised by Gulliver and Hildebrandt [1], who obtained such a container in the particular case g = 0, γ = π/2.

Standing capillary-gravity waves of finite amplitude: Corrigendum

The uniqueness condition that was utilized by the author (Concus 1962) is considered. The condition, which excludes certain fluid depths, is shown to be physically unacceptable because it is essentially impossible to satisfy in practice. The resulting in validation of the perturbation method is discussed, and a revision is presented, which invokes the presence of viscosity and allows retention of the previously obtained solutions. The revision may also be applied to the work of other authors who utilized the same method to solve other standing-wave problems.