Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems

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

Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in Channelflow and Openpipeflow

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.Comment: in Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, ed. Alexander Gelfgat (Springer, 2018

Numerical approach for high precision 3-D relativistic star models

A multi-domain spectral method for computing very high precision 3-D stellar models is presented. The boundary of each domain is chosen in order to coincide with a physical discontinuity (e.g. the star's surface). In addition, a regularization procedure is introduced to deal with the infinite derivatives on the boundary that may appear in the density field when stiff equations of state are used. Consequently all the physical fields are smooth functions on each domain and the spectral method is absolutely free of any Gibbs phenomenon, which yields to a very high precision. The power of this method is demonstrated by direct comparison with analytical solutions such as MacLaurin spheroids and Roche ellipsoids. The relative numerical error reveals to be of the order of $10^{-10}$. This approach has been developed for the study of relativistic inspiralling binaries. It may be applied to a wider class of astrophysical problems such as the study of relativistic rotating stars too.Comment: Minor changes, Phys. Rev. D in pres

Iterative Methods for Problems in Computational Fluid Dynamics

We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible Navier-Stokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equations at each time step, and discretization in space then produces a series of linear algebraic systems. We give an overview of commonly used time and space discretization techniques, and we discuss a variety of algorithmic strategies for solving the resulting systems of equations. The emphasis is on preconditioning techniques, which can be combined with Krylov subspace iterative methods. In many cases the solution of subsidiary problems such as the discrete convection-diffusion equation and the discrete Stokes equations plays a crucial role. We examine iterative techniques for these problems and show how they can be integrated into effective solution algorithms for the Navier-Stokes equations

Post-Newtonian approximation for isolated systems calculated by matched asymptotic expansions

Two long-standing problems with the post-Newtonian approximation for isolated slowly-moving systems in general relativity are: (i) the appearance at high post-Newtonian orders of divergent Poisson integrals, casting a doubt on the soundness of the post-Newtonian series; (ii) the domain of validity of the approximation which is limited to the near-zone of the source, and prevents one, a priori, from incorporating the condition of no-incoming radiation, to be imposed at past null infinity. In this article, we resolve the problem (i) by iterating the post-Newtonian hierarchy of equations by means of a new (Poisson-type) integral operator that is free of divergencies, and the problem (ii) by matching the post-Newtonian near-zone field to the exterior field of the source, known from previous work as a multipolar-post-Minkowskian expansion satisfying the relevant boundary conditions at infinity. As a result, we obtain an algorithm for iterating the post-Newtonian series up to any order, and we determine the terms, present in the post-Newtonian field, that are associated with the gravitational-radiation reaction onto an isolated slowly-moving matter system.Comment: 61 pages, to appear in Phys. Rev.

Lectures on Computational Numerical Analysis of Partial Differential Equations

From Chapter 1: The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp

Existence and Monotone Iteration of Positive Pseudosymmetric Solutions for a Third-Order Four-Point BVP with -Laplacian

We study the existence and monotone iteration of solutions for a third-order four-point boundary value problem with -Laplacian. An existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. Meanwhile, as an application of our result, an example is given

Existence of Positive Solutions for a Nonlinear Higher-Order Multipoint Boundary Value Problem

We study the existence of positive solutions for a nonlinear higher-order multipoint boundary value problem. By applying a monotone iterative method, some existence results of positive solutions are obtained. The main result is illustrated with an example