414 research outputs found
On the convergence of Lawson methods for semilinear stiff problems
Since their introduction in 1967, Lawson methods have achieved constant
interest in the time discretization of evolution equations. The methods were
originally devised for the numerical solution of stiff differential equations.
Meanwhile, they constitute a well-established class of exponential integrators.
The popularity of Lawson methods is in some contrast to the fact that they may
have a bad convergence behaviour, since they do not satisfy any of the stiff
order conditions. The aim of this paper is to explain this discrepancy. It is
shown that non-stiff order conditions together with appropriate regularity
assumptions imply high-order convergence of Lawson methods. Note, however, that
the term regularity here includes the behaviour of the solution at the
boundary. For instance, Lawson methods will behave well in the case of periodic
boundary conditions, but they will show a dramatic order reduction for, e.g.,
Dirichlet boundary conditions. The precise regularity assumptions required for
high-order convergence are worked out in this paper and related to the
corresponding assumptions for splitting schemes. In contrast to previous work,
the analysis is based on expansions of the exact and the numerical solution
along the flow of the homogeneous problem. Numerical examples for the
Schr\"odinger equation are included
A biconjugate gradient type algorithm on massively parallel architectures
The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. Recently, Freund and Nachtigal have proposed a novel BCG type approach, the quasi-minimal residual method (QMR), which overcomes the problems of BCG. Here, an implementation is presented of QMR based on an s-step version of the nonsymmetric look-ahead Lanczos algorithm. The main feature of the s-step Lanczos algorithm is that, in general, all inner products, except for one, can be computed in parallel at the end of each block; this is unlike the other standard Lanczos process where inner products are generated sequentially. The resulting implementation of QMR is particularly attractive on massively parallel SIMD architectures, such as the Connection Machine
Mr. Lincoln\u27s Wars: A Novel in Thirteen Stories
Telling Tales Stories pivot around axis of Lincoln There was a time in American Literary History when most of the fiction about the American Civil War kept re-fighting the war from one or the other, more or less, jaundiced point of view. The result was a dead pile of second and ...
Analytical and Numerical Analysis of Linear and Nonlinear Properties of an rf-SQUID Based Metasurface
We derive a model to describe the interaction of an rf-SQUID (radio frequency
superconducting quantum interference device) based metasurface with free space
electromagnetic waves. The electromagnetic fields are described on the base of
Maxwell's equations. For the rf-SQUID metasurface we rely on an equivalent
circuit model. After a detailed derivation, we show that the problem that is
described by a system of coupled differential equations is wellposed and,
therefore, has a unique solution. In the small amplitude limit, we provide
analytical expressions for reflection, transmission, and absorption depending
on the frequency. To investigate the nonlinear regime, we numerically solve the
system of coupled differential equations using a finite element scheme with
transparent boundary conditions and the Crank-Nicolson method. We also provide
a rigorous error analysis that shows convergence of the scheme at the expected
rates. The simulation results for the adiabatic increase of either the field's
amplitude or its frequency show that the metasurface's response in the
nonlinear interaction regime exhibits bistable behavior both in transmission
and reflection.Comment: published in Physical Review B, Phys. Rev. B 99, 07540
Error analysis of second-order locally implicit and local time-stepping methods for discontinuous Galerkin discretizations of linear wave equations
This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong CFL condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time
Error analysis of a fully discrete discontinuous Galerkin alternating direction implicit discretization of a class of linear wave-type problems
This paper is concerned with the rigorous error analysis of a fully discrete scheme obtained by using a central fluxes discontinuous Galerkin (dG) method in space and the Peaceman–Rachford splitting scheme in time. We apply the scheme to a general class of wave-type problems and show that the resulting approximations as well as discrete derivatives thereof satisfy error bounds of the order of the polynomial degree used in the dG discretization and order two in time. In particular, the class of problems considered includes, e.g., the advection equation, the acoustic wave equation, and the Maxwell equations for which a very efficient implementation is possible via an alternating direction implicit (ADI) splitting
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