2,032 research outputs found
Application of Richardson extrapolation for multi-dimensional advection equations
A Crank-Nicolson type scheme, which is of order two with respect to all independent variables, is used in the numerical solution of multi-dimensional advection equations. Normally, the order of accuracy of any numerical scheme can be increased by one when the well-known Richardson Extrapolation is used. It is proved that in this particular case the order of accuracy of the combined numerical method, the method consisting of the Crank-Nicolson scheme and the Richardson Extrapolation, is not three but four. (C) 2014 Elsevier Ltd. All rights reserved
Application of Richardson extrapolation with the Crank-Nicolson scheme for multi-dimensional advection
summary:Multi-dimensional advection terms are an important part of many large-scale mathematical models which arise in different fields of science and engineering. After applying some kind of splitting, these terms can be handled separately from the remaining part of the mathematical model under consideration. It is important to treat the multi-dimensional advection in a sufficiently accurate manner. It is shown in this paper that high order of accuracy can be achieved when the well-known Crank-Nicolson numerical scheme is combined with the Richardson extrapolation
Blended numerical schemes for the advection equation and conservation laws
In this paper we propose a method to couple two or more explicit numerical
schemes approximating the same time-dependent PDE, aiming at creating new
schemes which inherit advantages of the original ones. We consider both
advection equations and nonlinear conservation laws. By coupling a macroscopic
(Eulerian) scheme with a microscopic (Lagrangian) scheme, we get a new kind of
multiscale numerical method
Solving advection equations by applying the crank-nicolson scheme combined with the richardson extrapolation
Advection equations appear often in large-scale mathematical models arising in many fields of science and engineering. The Crank-Nicolson scheme can successfully be used in the numerical treatment of such equations. The accuracy of the numerical solution can sometimes be increased substantially by applying the Richardson Extrapolation. Two theorems related to the accuracy of the calculations will be formulated and proved in this paper. The usefulness of the combination consisting of the Crank-Nicolson scheme and the Richardson Extrapolation will be illustrated by numerical examples. Copyright Zahari Zlatev et al
Accurate black hole evolutions by fourth-order numerical relativity
We present techniques for successfully performing numerical relativity
simulations of binary black holes with fourth-order accuracy. Our simulations
are based on a new coding framework which currently supports higher order
finite differencing for the BSSN formulation of Einstein's equations, but which
is designed to be readily applicable to a broad class of formulations. We apply
our techniques to a standard set of numerical relativity test problems,
demonstrating the fourth-order accuracy of the solutions. Finally we apply our
approach to binary black hole head-on collisions, calculating the waveforms of
gravitational radiation generated and demonstrating significant improvements in
waveform accuracy over second-order methods with typically achievable numerical
resolution.Comment: 17 pages, 25 figure
Efficient solutions of two-dimensional incompressible steady viscous flows
A simple, efficient, and robust numerical technique is provided for solving two dimensional incompressible steady viscous flows at moderate to high Reynolds numbers. The proposed approach employs an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function Navier-Stokes equations. Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. For this second problem, mesh refinement and Richardson extrapolation are used to obtain useful benchmark solutions in the full range of Reynolds numbers at which steady laminar flow is established
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
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