CaltechAUTHORS: Combined
https://feeds.library.caltech.edu/people/Keller-H-B/combined.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 02 Aug 2024 19:04:43 -0700The Flow of a Viscous Compressible Fluid Through a Very Narrow Gap
https://resolver.caltech.edu/CaltechAUTHORS:20120925-154506131
Year: 1967
DOI: 10.1137/0115051
The effect of compressibility on the pressure distribution
in the narrow gap between a rotating cylinder and a plane in a viscous fluid was studied by Taylor and Saffman [1] during an investigation of the centripetal pump effect discovered by Reiner [2].https://resolver.caltech.edu/CaltechAUTHORS:20120925-154506131Some Positone Problems Suggested by Nonlinear Heat Generation
https://resolver.caltech.edu/CaltechAUTHORS:20200114-085055623
Year: 1967
DOI: 10.1512/iumj.1967.16.16087
There is much current interest in boundary value problems containing positive linear differential operators and monotone functions of the dependent variable, see for example, M.A. Krasnosel'ski [1] and H. H. Schaefer [2]. We call such problems "positone" and shall examine here a particular class of them (which have been called non-linear eigenvalue problems in [2]).https://resolver.caltech.edu/CaltechAUTHORS:20200114-085055623Accurate difference methods for linear ordinary differential systems subject to linear constraints
https://resolver.caltech.edu/CaltechAUTHORS:20120921-104823831
Year: 1969
DOI: 10.1137/0706002
We consider the general system of n first order linear
ordinary differential equations y'(t)=A(t)y(t)+g(t), ahttps://resolver.caltech.edu/CaltechAUTHORS:20120921-104823831Newton's method under mild differentiability conditions
https://resolver.caltech.edu/CaltechAUTHORS:20170802-073135318
Year: 1970
DOI: 10.1016/S0022-0000(70)80009-5
We study Newton's method for determining the solution of f(x) = 0 when f(x) is required only to be continuous and piecewise continuously differentiable in some sphere about the initial iterate, x^(0). First an existence, uniqueness and convergence theorem is obtained employing the modulus of continuity of the first derivative, f_x(x). Under the more explicit assumption of H6lder continuity several other such results are obtained, some of which extend results of Kantorovich and Akilov [1] and Ostrowski [5]. Of course, when Newton's method converges, it is now of order (1 + α),
where a is the Hö1der exponent. Other results on Newton's method without second derivatives are given by Goldstein [2], Schroeder [3], Rheinboldt [6], and Antosiewicz [7], to mention a few. It seems clear that the error analysis for Newton's method given by Lancaster [4] can be extended to the present case.https://resolver.caltech.edu/CaltechAUTHORS:20170802-073135318Comments on "Numerical studies of viscous flow around circular cylinders"
https://resolver.caltech.edu/CaltechAUTHORS:KELpof70
Year: 1970
DOI: 10.1063/1.1692952
It is claimed by Hamielec and Raal(1) that their computations improve upon the extrapolation procedure of Keller and Takami(2) which is considered "inadequate" and "could presumably lead to appreciable errors." However, the authors clearly do not understand the procedure of Keller and Takami or else do not understand the nature of Imai's asymptotic solution, or both.https://resolver.caltech.edu/CaltechAUTHORS:KELpof70Nonlinear bifurcation
https://resolver.caltech.edu/CaltechAUTHORS:20170802-080145275
Year: 1970
DOI: 10.1016/0022-0396(70)90090-2
We consider problems of the form (a) Lu + λg(λ, x, u) = 0, x ∈ D; (b) Bu = 0, x ∈ ∂D, for a very wide class of linear elliptic operators L, say of order 2m, and linear boundary operators B, of order m. For convenience only, we take the linear problem to be self-adjoint.https://resolver.caltech.edu/CaltechAUTHORS:20170802-080145275A New Difference Scheme for Parabolic Problems
https://resolver.caltech.edu/CaltechAUTHORS:20170802-111549240
Year: 1971
DOI: 10.1016/B978-0-12-358502-8.50014-1
This chapter discusses a new difference scheme for parabolic mixed initial-boundary value problems in one space dimension. The scheme has a number of very desirable features. It is simple, easy to program, and efficient. It is unconditionally stable and has second order accuracy with nonuniform nets. Richardson or h → 0 extrapolation is valid and yields two orders of accuracy improvement per extrapolation (with nonuniform nets). It is A-stable as well, that is, if the exact solution decays in time, so does the numerical scheme, with approximately the same rate; the data, coefficients, and solution need only be piecewise smooth and all the above remain valid. The method is also applicable to parabolic systems, to nonlinear parabolic equations, and even to some hyperbolic systems with special properties. The chapter presents the method, indicates the error estimates, h → 0 extrapolation and discusses an efficient algorithm for its application to the problem.https://resolver.caltech.edu/CaltechAUTHORS:20170802-111549240Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation
https://resolver.caltech.edu/CaltechAUTHORS:20170801-160912525
Year: 1971
DOI: 10.1016/0021-9991(71)90090-8
We present three accurate and efficient numerical schemes for solving the Falkner-Skan equation with positive or negative wall shear. Newton's method is employed, with the aid of the variational equations, in all the schemes and yields quadratic convergence. First, ordinary shooting is used to solve for the case of positive wall shear. Then a nonlinear eigenvalue technique is introduced to solve the inverse problem in which the wall shear is prescribed and the pressure distribution is to be determined. With this approach the reverse flow solutions (i.e., negative wall shear) are obtained. Finally, a parallel shooting method is employed to reduce the sensitivity of the convergence of the iterations to the initial estimates.https://resolver.caltech.edu/CaltechAUTHORS:20170801-160912525Shooting and embedding for two-point boundary value problems
https://resolver.caltech.edu/CaltechAUTHORS:20170802-081314237
Year: 1971
DOI: 10.1016/0022-247X(71)90042-4
The shooting method is an extremely powerful technique for both the theoretical analysis and approximate numerical solution of general two point boundary value problems. To illustrate we consider a problem of the form: (a) y'(t) = F(t,y), T_0 ⩽ t ⩽ T_1; (b) B(y(T_0), y(T_1)) = 0.https://resolver.caltech.edu/CaltechAUTHORS:20170802-081314237An inverse problem in boundary-layer flows: Numerical determination of pressure gradient for a given wall shear
https://resolver.caltech.edu/CaltechAUTHORS:20170801-160535948
Year: 1972
DOI: 10.1016/0021-9991(72)90096-4
The problem of determining a pressure gradient distribution that will produce a specified shear force on a body surface in boundary-layer flows is considered. This leads to an "overdetermined" boundary value problem for a partial differential equation containing an unknown coefficient. A numerical procedure for determining the coefficient is given along with several worked out examples including both similar and nonsimilar flows. The method essentially treats the unknown coefficient as an eigenvalue which is computed using Newton's method. This in turn employes a very accurate and efficient finite difference scheme for computing standard boundary-layer flows. Richardson extrapolation is applicable but only modest improvement was obtained in the present examples (for reasons that are explained).https://resolver.caltech.edu/CaltechAUTHORS:20170801-160535948Buckling of Complete Spherical Shells under Slightly Nonuniform Loads
https://resolver.caltech.edu/CaltechAUTHORS:20170802-111028492
Year: 1973
DOI: 10.1016/B978-0-12-215150-7.50010-7
This chapter discusses buckling of complete spherical shells under slightly nonuniform load. It presents the study of the axisymmetric deformations of a complete thin spherical shell subject to external loads, of the form p(θ) = p_0 + τ d(θ); θ is the latitude and τ measures the deviation of the load from a uniform pressure, p_0. The techniques for solving this and a broad class of related problems are quite new and particularly relevant in elasticity theory. Estimates of the error in any iterate can be given and the results also show that some specific perturbation schemes actually yield asymptotic results. The problem is formulated in the chapter based on a modification of the equations. The chapter presents the analysis that is applied to this finite case is elementary and easily shows how these methods yield a rigorous treatment using only the first two variations of the energy functional.https://resolver.caltech.edu/CaltechAUTHORS:20170802-111028492Axisymmetric buckling of rigidly clamped hemispherical shells
https://resolver.caltech.edu/CaltechAUTHORS:20170801-155214791
Year: 1973
DOI: 10.1016/0020-7462(73)90012-7
The shell is subjected to a uniform compressive surface load (either a pressure or a centrally directed load). The results of a numerical study are summarized and compared with experiments.https://resolver.caltech.edu/CaltechAUTHORS:20170801-155214791Accurate Difference Methods for Nonlinear Two-Point Boundary Value Problems
https://resolver.caltech.edu/CaltechAUTHORS:20120808-142816266
Year: 1974
DOI: 10.1137/0711028
We show that each isolated solution, y(t), of the general nonlinear two-point boundary value problem (*): y'=f(t,y), a < t < b, g(y(a),y(b))=0 can be approximated by the (box) difference scheme (**):[u_j - u_(j-1)]/h_j = f(t_(j-½),[u_j + u_(j-1)]/2), 1 ≦ j ≦ J, g(U_0,U_J) = O. For h = max_(1 ≦j≦J)h_j sufficiently small, the difference equations (**) are shown to have a unique solution {U_j}^J_0} in some sphere about {y(t_j)}^J_0, and it can be computed by Newton's method which converges quadratically. If y(t) is
sufficiently smooth, then the error has an asymptotic expansion of the form u_j - y(t_j) = Σ^(m)_(v=1) h^(2v) e_v(t_j) + O(h^(2m+2), so that Richardson extrapolation is justified.
The coefficient matrices of the linear systems to be solved in applying Newton's method are of order n(J + l) when y(t) ∈ ℝ^n. For separated endpoint boundary conditions: g_1(y(a)) = 0, g_2(y(b)) = 0 with dim g_1 = p, dim g_2 = q and p + q = n, the coefficient matrices have the special block tridiagonal form A ≡ [B_j, A_j, C_j] in which the n x n matrices B_j(C_j) have their last q (first p) rows null. Block elimination and band elimination without destroying the zero pattern are shown to be valid. The numerical scheme is very efficient, as a worked out example illustrates.https://resolver.caltech.edu/CaltechAUTHORS:20120808-142816266Turbulent boundary layers with assigned wall shear
https://resolver.caltech.edu/CaltechAUTHORS:20170802-110504850
Year: 1975
DOI: 10.1016/0045-7930(75)90007-9
In this paper we consider the problem of computing the external velocity distribution on a two-dimensional body in an incompressible boundary-layer flow for a specified wall shear. This leads to an 'overdetermined' boundary-value problem for a partial differential equation containing the edge velocity as an unknown parameter. A numerical procedure for determining the external velocity is given, together with several examples for laminar and turbulent boundary layers.https://resolver.caltech.edu/CaltechAUTHORS:20170802-110504850Difference Methods for Boundary Value Problems in Ordinary Differential Equations
https://resolver.caltech.edu/CaltechAUTHORS:20120809-094752016
Year: 1975
DOI: 10.1137/0712059
A general theory of difference methods for problems of the form
Ny ≡ y' - f(t,y) = O, a ≦ t ≦ b, g(y(a),y(b))= 0,
is developed. On nonuniform nets, t_0 = a, t_j = t_(j-1) + h_j, 1 ≦ j ≦ J, t_J = b, schemes of the form
N_(h)u_j = G_j(u_0,•••,u_J) = 0, 1 ≦ j ≦ J, g(u_0,u_J) = 0
are considered. For linear problems with unique solutions, it is shown that the difference scheme is stable and consistent for the boundary value problem if and only if, upon replacing the boundary conditions by an initial condition, the resulting scheme is stable and consistent for the initial value problem. For isolated solutions of the nonlinear problem, it is shown that the difference scheme has a unique solution converging to the exact solution if (i) the linearized difference equations are stable and consistent for the linearized initial value problem, (ii) the linearized difference operator is Lipschitz continuous, (iii) the nonlinear difference equations are consistent with the nonlinear differential
equation. Newton's method is shown to be valid, with quadratic convergence, for computing the numerical solution.https://resolver.caltech.edu/CaltechAUTHORS:20120809-094752016Finite difference methods for ordinary boundary value problems
https://resolver.caltech.edu/CaltechAUTHORS:20180829-073948869
Year: 1976
DOI: 10.1007/BFb0120607
Finite difference methods have been shown to be extremely effective in the accurate and efficient solution of very general nonlinear two point boundary value problems. As with all practical numerical methods their development is tied very closely to the theoretical understanding of the procedures in question. Not surprisingly then there has been much current work on the theory of difference methods for two point problems. We shall recapitulate some of this theory here and also discuss some of the practical aspects in developing standard computer codes for such problems.https://resolver.caltech.edu/CaltechAUTHORS:20180829-073948869A Numerical Method for Singular Two Point Boundary Value Problems
https://resolver.caltech.edu/CaltechAUTHORS:20120807-133058915
Year: 1977
DOI: 10.1137/0714054
The numerical solution of boundary value problems for linear systems of first order equations with a regular singular point at one endpoint is considered. The standard procedure of expanding about the singularity to get a nonsingular problem over a reduced interval is justified in some detail. Quite general boundary conditions are included which permit unbounded solutions. Error estimates are given and some numerical calculations are presented to check the theory.https://resolver.caltech.edu/CaltechAUTHORS:20120807-133058915Global Homotopies and Newton Methods
https://resolver.caltech.edu/CaltechAUTHORS:20170802-105839555
Year: 1978
DOI: 10.1016/B978-0-12-208360-0.50009-7
This chapter describes the global homotopies and Newton methods. A key to devising global methods is to give up the monotone convergence and to consider more general homotopies. It turns out that singular matrices on the path cause no difficulties in the proof of Smales result. They cause trouble in attempts to implement this and most other global Newton methods numerically. Small steps must be taken in the neighborhood of vanishing Jacobians. This feature is not always pointed out in descriptions of the implementations but it is easily detected. These difficulties can be eliminated by using a somewhat different homotopy. The chapter discusses a pseudo-arc length continuation procedure in which the parameter is distance along a local tangent ray to the path. Using this parameter, this chapter discusses how to accurately locate the roots and the limit points on the path. These latter points are of great interest in many physical applications.https://resolver.caltech.edu/CaltechAUTHORS:20170802-105839555Numerical Methods in Boundary-Layer Theory
https://resolver.caltech.edu/CaltechAUTHORS:20161019-105331989
Year: 1978
DOI: 10.1146/annurev.fl.10.010178.002221
There is a large variety of numerical methods that are used to solve the many flow problems to which boundary-layer theory is applied. Two particular methods, the Crank-Nicolson scheme and the Box scheme, seem to dominate in most practical applications. Of course these methods are not uniquely defined and there are many variations in their formulation as well as in the procedures used to solve the resulting (nonlinear) algebraic equations. They are both implicit with respect to variation normal to the boundary layer, they can both be second-order accurate in all variables,
and both can be improved to yield even higher-order accuracy. However, I prefer and stress the Box scheme as it is very easy to adapt to new classes of problems. It also allows a more rapid net variation and ease in obtaining higher-order accuracy.
This scheme was devised in Keller (1971), for solving diffusion problems, but it has subsequently been applied to a broad class of problems. A recent survey by Blottner (1975a) has stressed the Crank-Nicolson scheme. I suggest that article as well for a
brief presentation of various other numerical methods that have been used. A sketch of the applications of the Box scheme to a variety of boundary-layer flow problems is given in Keller (1975b).
Many of the problems and techniques presented in this survey have been worked out with T. Cebeci. Details of some of these techniques and even listings of a few of the codes appear in a text by Cebeci & Bradshaw (1977).https://resolver.caltech.edu/CaltechAUTHORS:20161019-105331989An Academic In Industry
https://resolver.caltech.edu/CaltechAUTHORS:20170802-105152260
Year: 1979
DOI: 10.1016/B978-0-12-734250-4.50009-6
[no abstract]https://resolver.caltech.edu/CaltechAUTHORS:20170802-105152260Difference Methods and Deferred Corrections for Ordinary Boundary Value Problems
https://resolver.caltech.edu/CaltechAUTHORS:20120726-084458239
Year: 1979
DOI: 10.1137/0716018
Compact as possible difference schemes for systems of nth order equations are developed. Generalizations of the Mehrstellenverfahren and simple theoretically sound implementations of deferred corrections are given. It is shown that higher order systems are more efficiently solved as given rather than as reduced to larger lower order systems. Tables of coefficients to implement these methods are included and have been derived using symbolic computations.https://resolver.caltech.edu/CaltechAUTHORS:20120726-084458239Solving two-point seismic-ray tracing problems in a heterogeneous medium. Part 1. A general adaptive finite difference method
https://resolver.caltech.edu/CaltechAUTHORS:20140813-084439640
Year: 1980
A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays.
A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology.
In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.https://resolver.caltech.edu/CaltechAUTHORS:20140813-084439640Computations of the axisymmetric flow between rotating cylinders
https://resolver.caltech.edu/CaltechAUTHORS:20170801-153724200
Year: 1980
DOI: 10.1016/0021-9991(80)90037-6
We study Taylor vortex flows by solving the steady axisymmetric Navier-Stokes. equations in the primitive variables (u, v, w, p). Fourier expansions in z, the axial direction, and centered finite differences in r, the radial direction, are used. The resulting discretized equations are solved using the pseudoarclength continuation methods of Keller (in "Applications of Bifurcation Theory" (P. Rabinowitz, Ed.), pp. 359–384, Academic Press, New York, 1977.), which are designed to detect bifurcations. In this way we accurately determine the first branch of Taylor vortex solutions bifurcating from Couette flow for both a wide and a narrow gap. Agreement with experiments is extremely good for the wide gap case and solutions are obtained for a larger range of Reynolds numbers than previously reported.https://resolver.caltech.edu/CaltechAUTHORS:20170801-153724200Multiple limit point bifurcation
https://resolver.caltech.edu/CaltechAUTHORS:20170802-082136797
Year: 1980
DOI: 10.1016/0022-247X(80)90090-6
In this paper we present a new bifurcation or branching phenomenon which
we call multiple limit point bifurcation. It is of course well known that bifurcation
points of some nonlinear functional equation G(u, λ) = 0 are solutions (u_0, λ_0) at which two distinct smooth branches of solutions, say [u(ε), λ(ε)] and [u^(ε), λ^(ε)], intersect nontangentially. The precise nature of limit points is less easy to specify but they are also singular points on a solution branch; that is, points (u_0, λ_0) = (u(0), λ(0)), say, at which the Frechet derivative G_u^0 ≡ G_u(u_0, λ_0) is singular.https://resolver.caltech.edu/CaltechAUTHORS:20170802-082136797Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems
https://resolver.caltech.edu/CaltechAUTHORS:CHAsiamjssc82
Year: 1982
DOI: 10.1137/0903012
We investigate multi-grid methods for solving linear systems arising from arc-length continuation techniques applied to nonlinear elliptic eigenvalue problems. We find that the usual multi-grid methods diverge in the neighborhood of singular points of the solution branches. As a result, the continuation method is unable to continue past a limit point in the Bratu problem. This divergence is analyzed and a modified multi-grid algorithm has been devised based on this analysis. In principle, this new multi-grid algorithm converges for elliptic systems, arbitrarily close to singularity and has been used successfully in conjunction with arc-length continuation procedures on the model problem. In the worst situation, both the storage and the computational work are only about a factor of two more than the unmodified multi-grid methods.https://resolver.caltech.edu/CaltechAUTHORS:CHAsiamjssc82On the Birth of Isolas
https://resolver.caltech.edu/CaltechAUTHORS:20120716-150101865
Year: 1982
DOI: 10.1137/0142068
Isolas are isolated, closed curves of solution branches of nonlinear problems. They have been observed to occur in the buckling of elastic shells, the equilibrium states of chemical reactors and other problems. In this paper we present a theory to describe analytically the structure of a class of isolas. Specifically, we consider isolas that shrink to a point as a parameter τ of the problem, approaches a critical value τ_0. The point is referred to as an isola center. Equations that characterize the isola centers are given. Then solutions are constructed in a neighborhood of the isola centers by perturbation expansions in a small
parameter ε that is proportional to (τ-τo), with a appropriately determined. The theory is applied to a
chemical reactor problem.https://resolver.caltech.edu/CaltechAUTHORS:20120716-150101865Convergence Rates for Newton's Method at Singular Points
https://resolver.caltech.edu/CaltechAUTHORS:20120712-112618470
Year: 1983
DOI: 10.1137/0720020
If Newton's method is employed to find a root of a map from a Banach space into itself and the derivative is singular at that root, the convergence of the Newton iterates to the root is linear rather than quadratic. In this paper we give a detailed analysis of the linear convergence rates for several types of singular problems. For some of these problems we describe modifications of Newton's method which will restore quadratic convergence.https://resolver.caltech.edu/CaltechAUTHORS:20120712-112618470Fast Seismic Ray Tracing
https://resolver.caltech.edu/CaltechAUTHORS:20120712-112742998
Year: 1983
DOI: 10.1137/0143064
New methods for the fast, accurate and efficient calculation of large classes of seismic rays joining two points x+s and x_R in very general two-dimensional configurations are presented. The medium is piecewise homogeneous with arbitrary interfaces separating regions of different elastic properties (i.e., differing wave speeds c_P and c_S). In general there are 2^(N+1) rays joining x_S to x_R while making contact with N interfaces. Our methods find essentially all such rays for a given N by using continuation or homotopy methods on the wave speeds to solve the ray equations determined by Snell's law. In addition travel times, ray amplitudes and caustic locations are obtained. When several receiver positions x^(j)_(R) are to be included, as in a gather, our techniques easily yield all the rays for the entire gather by employing continuation in the receiver location. The applications, mainly to geophysical inverse problems, are reported elsewhere.https://resolver.caltech.edu/CaltechAUTHORS:20120712-112742998The Stability of One-Step Schemes for First-Order Two-Point Boundary Value Problems
https://resolver.caltech.edu/CaltechAUTHORS:20120716-104541004
Year: 1983
DOI: 10.1137/0720083
The stability of a finite difference scheme is related explicitly to the stability of the continuous problem being solved. At times, this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem.https://resolver.caltech.edu/CaltechAUTHORS:20120716-104541004The Bordering Algorithm and Path Following Near Singular Points of Higher Nullity
https://resolver.caltech.edu/CaltechAUTHORS:20120712-141249734
Year: 1983
DOI: 10.1137/0904039
We study the behavior of the bordering algorithm (a form of block elimination) for solving nonsingular linear systems with coefficient matrices in the partitioned form (A & B \\ C^* & D) when N(A)≧1. Systems with this structure naturally occur in path following procedures. We show that under appropriate assumptions, the algorithm, which is based on solving systems with coefficient matrix A, works as A varies along a path and goes through singular points. The required assumptions are justified for a large class of problems coming from discretizations of boundary value problems for differential equations.https://resolver.caltech.edu/CaltechAUTHORS:20120712-141249734Steady State and Periodic Solution Paths: their Bifurcations and Computations
https://resolver.caltech.edu/CaltechAUTHORS:20201020-072658771
Year: 1984
DOI: 10.1007/978-3-0348-6256-1_16
In this work we present a brief account of the theory and numerical methods for the analysis and Solution of nonlinear autonomous differential equations of the form
d/dτ w = f(w,λ,α); f:B₁×R²→B₂.https://resolver.caltech.edu/CaltechAUTHORS:20201020-072658771Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems
https://resolver.caltech.edu/CaltechAUTHORS:20120628-135243820
Year: 1985
DOI: 10.1137/0906055
We discuss in this paper a new combination of methods for solving nonlinear boundary value problems containing a parameter. Methods of the continuation type are combined with least squares formulations, preconditioned conjugate gradient algorithms and finite element approximations.
We can compute branches of solutions with limit points, bifurcation points, etc.
Several numerical tests illustrate the possibilities of the methods discussed in the present paper; these include the Bratu problem in one and two dimensions, one-dimensional bifurcation and perturbed bifurcation problems, the driven cavity problem for the Navier–Stokes equations.https://resolver.caltech.edu/CaltechAUTHORS:20120628-135243820Some bifurcation diagrams for Taylor vortex flows
https://resolver.caltech.edu/CaltechAUTHORS:MEYpof85
Year: 1985
DOI: 10.1063/1.865007
The numerical continuation and bifurcation methods of Keller [H. B. Keller, in Applications of Bifurcation Theory (Academic, New York, 1977), pp. 359–384] are used to study the variation of some branches of axisymmetric Taylor vortex flow as the wavelength in the axial direction changes. Closed "loops" of solutions and secondary bifurcations are determined. Variations with respect to Reynolds number show the same phenomena. The results presented here show that Taylor vortices with periodic boundary conditions exist in a wider range of wavelengths, lambda, than observed in the Burkhalter/Koschmieder experiments [Phys. Fluids 17, 1929 (1974)]. They also show that there is possibly a lambda subinterval within the neutral curve of Couette flow such that there are no Taylor vortex flows with smallest period in this interval.https://resolver.caltech.edu/CaltechAUTHORS:MEYpof85Three-Dimensional Ray Tracing and Geophysical Inversion in Layered Media
https://resolver.caltech.edu/CaltechAUTHORS:FAWsiamjam85
Year: 1985
DOI: 10.1137/0145029
In this paper the problem of finding seismic rays in a three-dimensional layered medium is examined. The "layers" are separated by arbitrary smooth interfaces that can vary in three dimensions. The endpoints of each ray and the sequence of interfaces it encounters are specified. The problem is formulated as a nonlinear system of equations and efficient, accurate methods of solution are discussed. An important application of ray tracing methods, which is discussed, is the nonlinear least squares estimation of medium parameters from observed travel times. In addition the "type" of each ray is also determined by the least squares process—this is in effect a deconvolution procedure similar to that desired in seismic exploration. It enables more of the measured data to be used without filtering out the multiple reflections that are not pure P-waves.https://resolver.caltech.edu/CaltechAUTHORS:FAWsiamjam85Exact Boundary Conditions at an Artificial Boundary for Partial Differential Equations in Cylinders
https://resolver.caltech.edu/CaltechAUTHORS:HAGsiamjma86
Year: 1986
DOI: 10.1137/0517026
The numerical solution of partial differential equations in unbounded domains requires a finite computational domain. Often one obtains a finite domain by introducing an artificial boundary and imposing boundary conditions there. This paper derives exact boundary conditions at an artificial boundary for partial differential equations in cylinders. An abstract theory is developed to analyze the general linear problem. Solvability requirements and estimates of the solution of the resulting finite problem are obtained by use of the notions of exponential and ordinary dichotomies. Useful representations of the boundary conditions are derived using separation of variables for problems with constant tails. The constant tail results are extended to problems whose coefficients obtain limits at infinity by use of an abstract perturbation theory. The perturbation theory approach is also applied to a class of nonlinear problems. General asymptotic formulas for the boundary conditions are derived and displayed in detail.https://resolver.caltech.edu/CaltechAUTHORS:HAGsiamjma86A Direct Method for Computing Higher Order Folds
https://resolver.caltech.edu/CaltechAUTHORS:YANsiamjssc86
Year: 1986
DOI: 10.1137/0907024
We consider the computation of higher order fold or limit points of two parameter-dependent nonlinear problems. A direct method is proposed and an efficient implementation of the direct method is presented. Numerical results for the thermal ignition problem are given.https://resolver.caltech.edu/CaltechAUTHORS:YANsiamjssc86The Numerical Calculation of Traveling Wave Solutions of Nonlinear Parabolic Equations
https://resolver.caltech.edu/CaltechAUTHORS:HAGsiamjssc86
Year: 1986
DOI: 10.1137/0907065
Traveling wave solutions have been studied for a variety of nonlinear parabolic problems. In the initial value approach to such problems the initial data at infinity determines the wave that propagates. The numerical simulation of such problems is thus quite difficult. If the domain is replaced by a finite one, to facilitate numerical computations, then appropriate boundary conditions on the "artificial" boundaries must depend upon the initial data in the discarded region. In this work we derive such boundary conditions, based on the Laplace transform of the linearized problems at ±∞, and illustrate their utility by presenting a numerical solution of Fisher's equation which has been proposed as a model in genetics.https://resolver.caltech.edu/CaltechAUTHORS:HAGsiamjssc86A multigrid continuation method for elliptic problems with folds
https://resolver.caltech.edu/CaltechAUTHORS:20120627-134928372
Year: 1986
DOI: 10.1137/0907074
We introduce a new multigrid continuation method for computing solutions of nonlinear elliptic eigenvalue problems which contain limit points (also called turning points or folds). Our method combines the frozen tau technique of Brandt with pseudo-arc length continuation and correction of the parameter on the coarsest grid. This produces considerable storage savings over direct continuation methods,as well as better initial coarse grid approximations, and avoids complicated algorithms for determining the parameter on finer grids. We provide numerical results for second, fourth and sixth order approximations to the two-parameter, two-dimensional stationary reaction-diffusion problem: Δu+λ exp(u/(1+au)) = 0.
For the higher order interpolations we use bicubic and biquintic splines. The convergence rate is observed to be independent of the occurrence of limit points.https://resolver.caltech.edu/CaltechAUTHORS:20120627-134928372Computations of Taylor Vortex Flows using multigrid continuation methods
https://resolver.caltech.edu/CaltechAUTHORS:20211104-064756963
Year: 1989
DOI: 10.1007/978-3-642-83733-3_9
Numerical solutions of axisymmetric Taylor vortex flows have been calculated using Multigrid Continuation Techniques. Both infinite and finite cylinders are considered, and the results agree well with experiments. New solutions are found in the infinite cylinder case and these, surprisingly, may help in understanding some experimental results obtained in relatively short cylinders. The numerical method proved to be efficient and reliable so that computations with fine grids and long cylinders are easily performed.https://resolver.caltech.edu/CaltechAUTHORS:20211104-064756963A preferred approach to the linearization of turbulent boundary-layer equations
https://resolver.caltech.edu/CaltechAUTHORS:20170802-104850872
Year: 1989
DOI: 10.1016/0045-7930(89)90029-7
An efficient procedure for solving the fully linearized form of the boundary-layer equations is described for turbulent flows. The procedure makes use of the so-called bordering algorithm and is applicable to problems in which the structure of the linearized system of equations deviates from the block triagonal matrix form which may be caused by boundary conditions.https://resolver.caltech.edu/CaltechAUTHORS:20170802-104850872Hypercube implementations of parallel shooting
https://resolver.caltech.edu/CaltechAUTHORS:20170801-153045481
Year: 1989
DOI: 10.1016/0096-3003(89)90140-9
We consider parallel shooting for linear two-point boundary-value problems with separated boundary conditions. Two different strategies are considered for mapping associated tasks onto the nodes of a hypercube. One of these (the domain strategy) corresponds to a decomposition by subintervals of the independent variable, and the other (the column strategy) to decomposition by dependent variable. Under suitable assumptions the two strategies are compared in terms of estimated computational times. Estimated regions of efficiency, in terms of various parameters, are obtained for the two strategies.https://resolver.caltech.edu/CaltechAUTHORS:20170801-153045481Complex Bifurcation from Real Paths
https://resolver.caltech.edu/CaltechAUTHORS:20120424-132606071
Year: 1990
DOI: 10.1137/0150027
A new bifurcation phenomenon, called complex bifurcation, is studied. The basic idea is simply that real solution paths of real analytic problems frequently have complex paths bifurcating from them. It is shown that this phenomenon occurs at fold points, at pitchfork bifurcation points, and at isola centers. It is also shown that perturbed bifurcations can yield two disjoint real solution branches that are connected by complex paths bifurcating from the perturbed solution paths. This may be useful in finding new real solutions.
A discussion of how existing codes for computing real solution paths may be trivially modified to compute complex paths is included, and examples of numerically computed complex solution paths for a nonlinear two point boundary value problem, and a problem from fluid mechanics are given.https://resolver.caltech.edu/CaltechAUTHORS:20120424-132606071Parallel homotopy algorithm for large sparse generalized eigenvalue problems: Application to hydrodynamic stability analysis
https://resolver.caltech.edu/CaltechAUTHORS:20170802-124308426
Year: 1992
DOI: 10.1007/3-540-55895-0_427
A parallel homotopy algorithm is presented for finding a few selected eigenvalues (for example those with the largest real part) of Az = λBz with real, large, sparse, and nonsymmetric square matrix A and real, singular, diagonal matrix B. The essence of the homotopy method is that from the eigenpairs of Dz = λBz, we use Euler-Newton continuation to follow the eigenpairs of A(t)z = λBz with A(t) ≡ (1−t)D + tA. Here D is some initial matrix and "time" t is incremented from 0 to 1. This method is, to a large degree, parallel because each eigenpath can be computed independently of the others. The algorithm has been implemented on the Intel hypcrcubc. Experimental results on a 64-nodc Intel iPSC/860 hypercube are presented. It is shown how the parallel homotopy method may be useful in applications like detecting Hopf bifurcations in hydrodynamic stability analysis.https://resolver.caltech.edu/CaltechAUTHORS:20170802-124308426Stabilization of Unstable Procedures: The Recursive Projection Method
https://resolver.caltech.edu/CaltechAUTHORS:20120307-153620928
Year: 1993
DOI: 10.1137/0730057
Fixed-point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a "black-box" time integration scheme is stabilized, enabling it to compute unstable steady states. The RPM can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes.https://resolver.caltech.edu/CaltechAUTHORS:20120307-153620928A local study of a double critical point in Taylor-Couette flow
https://resolver.caltech.edu/CaltechAUTHORS:20170802-123922327
Year: 1995
DOI: 10.1007/BF01176814
The steady flow, governed by Navier-Stokes equations for an incompressible viscous fluid, between concentric cylinders is considered. The inner cylinder is assumed to be rotating and the outer one at rest. The flow is assumed to be axisymmetric and periodic in the axial direction. At an (m, n) critical point, the eigenfunctions of the operator, linearized around the exact solution, the Couette flow, consist ofm andn axial waves. In a neighbourhood of such a double critical point, using Liapunov-Schmidt method, bifurcation equations are obtained, in ℝ^2. Expressions for the leading coefficients in the truncated system of 2 equations are derived. Using these, the coefficients are computed at a (2, 4) critical point for 2 different radii ratios and the local bifurcation diagrams obtained. Available numerical solutions of the Navier-Stokes system near this double critical point confirm that the reduced bifurcation equations reproduce the qualitative behaviour adequately.https://resolver.caltech.edu/CaltechAUTHORS:20170802-123922327Numerical Studies of the Gauss Lattice Problem
https://resolver.caltech.edu/CaltechAUTHORS:20091022-102132378
Year: 1997
The difference between the number of lattice points N(R) that lie in x^2 + y^2 ≤ R^2 and the area of that circle, d(R) = N(R) - πR^2, can be bounded by |d(R)| ≤ KR^θ.
Gauss showed that this holds for θ = 1, but the least value for which it holds is an open problem in number
theory. We have sought numerical evidence by tabulating N(R) up to R ≈ 55,000. From the convex hull bounding log |d(R)| versus log R we obtain the bound θ ≤ 0.575, which is significantly better than the best analytical result θ ≤ 0.6301 ... due to Huxley. The behavior of d(R) is of interest to those studying quantum chaos.https://resolver.caltech.edu/CaltechAUTHORS:20091022-102132378Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem
https://resolver.caltech.edu/CaltechAUTHORS:LUIsiamjmaa97
Year: 1997
DOI: 10.1137/S0895479894273900
A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A1 is presented. From the eigenpairs of some real matrix A0, the eigenpairs of
A(t) ≡ (1 − t)A0 + tA1
are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix:
• bifurcation,
• ill conditioning due to nonorthogonal eigenvectors,
• jumping of eigenpaths.
These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented.https://resolver.caltech.edu/CaltechAUTHORS:LUIsiamjmaa97Space-time domain decomposition for parabolic problems
https://resolver.caltech.edu/CaltechAUTHORS:20190829-131534035
Year: 2002
DOI: 10.1007/s002110100345
We analyze a space-time domain decomposition iteration, for a model advection diffusion equation in one and two dimensions. The discretization of this iteration is the block red-black variant of the waveform relaxation method, and our analysis provides new convergence results for this scheme. The asymptotic convergence rate is super-linear, and it is governed by the diffusion of the error across the overlap between subdomains. Hence, it depends on both the size of this overlap and the diffusion coefficient in the equation. However it is independent of the number of subdomains, provided the size of the overlap remains fixed. The convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases. The duration of the transient linear regime is proportional to the length of the time window. For advection dominated problems, the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases. Moreover, it is independent of the size of the time window and of the number of subdomains. Numerical calculations illustrate our analysis.https://resolver.caltech.edu/CaltechAUTHORS:20190829-131534035