Algebraic Multigrid for Disordered Systems and Lattice Gauge Theories

The construction of multigrid operators for disordered linear lattice operators, in particular the fermion matrix in lattice gauge theories, by means of algebraic multigrid and block LU decomposition is discussed. In this formalism, the effective coarse-grid operator is obtained as the Schur complement of the original matrix. An optimal approximation to it is found by a numerical optimization procedure akin to Monte Carlo renormalization, resulting in a generalized (gauge-path dependent) stencil that is easily evaluated for a given disorder field. Applications to preconditioning and relaxation methods are investigated.Comment: 43 pages, 14 figures, revtex4 styl

Fast and Efficient Numerical Methods for an Extended Black-Scholes Model

An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and accuracy of the numerical scheme considered. In this article we consider a PIDE that arises in option pricing theory (financial problems) as well as in various scientific modeling and deal with two different topics. In the first part of the article, we study several iterative techniques (preconditioned) for the PIDE model. A wavelet basis and a Fourier sine basis have been used to design various preconditioners to improve the convergence criteria of iterative solvers. We implement a multigrid (MG) iterative method. In fact, we approximate the problem using a finite difference scheme, then implement a few preconditioned Krylov subspace methods as well as a MG method to speed up the computation. Then, in the second part in this study, we analyze the stability and the accuracy of two different one step schemes to approximate the model.Comment: 29 pages; 10 figure

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

A Jacobian-free Newton-Krylov method for time-implicit multidimensional hydrodynamics

This work is a continuation of our efforts to develop an efficient implicit solver for multidimensional hydrodynamics for the purpose of studying important physical processes in stellar interiors, such as turbulent convection and overshooting. We present an implicit solver that results from the combination of a Jacobian-Free Newton-Krylov method and a preconditioning technique tailored to the inviscid, compressible equations of stellar hydrodynamics. We assess the accuracy and performance of the solver for both 2D and 3D problems for Mach numbers down to $10^{-6}$. Although our applications concern flows in stellar interiors, the method can be applied to general advection and/or diffusion-dominated flows. The method presented in this paper opens up new avenues in 3D modeling of realistic stellar interiors allowing the study of important problems in stellar structure and evolution.Comment: Accepted for publication in A&

Preconditioning for first-order spectral discretization

Efficient solution of the equations from spectral discretizations is essential if the high-order accuracy of these methods is to be realized. Direct solution of these equations is rarely feasible, thus iterative techniques are required. A preconditioning scheme for first-order Chebyshev collocation operators is proposed herein, in which the central finite difference mesh is finer than the collocation mesh. Details of the proper techniques for transferring information between the meshes are given here, and the scheme is analyzed by examination of the eigenvalue spectra of the preconditioned operators. The effect of artificial viscosity required in the inversion of the finite difference operator is examined. A second preconditioning scheme, involving a high-order upwind finite difference operator of the van Leer type is also analyzed to provide a comparison with the present scheme. Finally, the performance of the present scheme is verified by application to several test problems