1,927 research outputs found
A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory
We develop a new dispersion minimizing compact finite difference scheme for
the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly
developed ray theory for difference equations. A discrete Helmholtz operator
and a discrete operator to be applied to the source and the wavefields are
constructed. Their coefficients are piecewise polynomial functions of ,
chosen such that phase and amplitude errors are minimal. The phase errors of
the scheme are very small, approximately as small as those of the 2-D
quasi-stabilized FEM method and substantially smaller than those of
alternatives in 3-D, assuming the same number of gridpoints per wavelength is
used. In numerical experiments, accurate solutions are obtained in constant and
smoothly varying media using meshes with only five to six points per wavelength
and wave propagation over hundreds of wavelengths. When used as a coarse level
discretization in a multigrid method the scheme can even be used with downto
three points per wavelength. Tests on 3-D examples with up to degrees of
freedom show that with a recently developed hybrid solver, the use of coarser
meshes can lead to corresponding savings in computation time, resulting in good
simulation times compared to the literature.Comment: 33 pages, 12 figures, 6 table
A fourthâorder compact scheme for the Helmholtz equation: Alphaâinterpolation of FEM and FDM stencils
We propose a fourthâorder compact scheme on structured meshes for the Helmholtz equation given by R(φ):=f(x)+Δφ+ξ2φ=0. The scheme consists of taking the alphaâinterpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alphaâinterpolation method (J. Comput. Appl. Math. 1982; 8(1):15–19) and in 2D making the choice α=0.5 we recover the generalized fourthâorder compact Padé approximation (J. Comput. Phys. 1995; 119:252–270; Comput. Meth. Appl. Mech. Engrg 1998; 163:343–358) (therein using the parameter γ=2). We follow (SIAM Rev. 2000; 42(3):451–484; Comput. Meth. Appl. Mech. Engrg 1995; 128:325–359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasiâstabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128:325–359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((ξâ)4), where ξ, â represent the wavenumber and the mesh size, respectively. An expression for the parameter α is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L2 norm, the H1 semiânorm and the l∞ Euclidean norm are done and the pollution effect is found to be small. Copyright © 2010 John Wiley & Sons, Ltd
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
High order methods for acoustic scattering: Coupling Farfield Expansions ABC with Deferred-Correction methods
Arbitrary high order numerical methods for time-harmonic acoustic scattering
problems originally defined on unbounded domains are constructed. This is done
by coupling recently developed high order local absorbing boundary conditions
(ABCs) with finite difference methods for the Helmholtz equation. These ABCs
are based on exact representations of the outgoing waves by means of farfield
expansions. The finite difference methods, which are constructed from a
deferred-correction (DC) technique, approximate the Helmholtz equation and the
ABCs, with the appropriate number of terms, to any desired order. As a result,
high order numerical methods with an overall order of convergence equal to the
order of the DC schemes are obtained. A detailed construction of these DC
finite difference schemes is presented. Additionally, a rigorous proof of the
consistency of the DC schemes with the Helmholtz equation and the ABCs in polar
coordinates is also given. The results of several numerical experiments
corroborate the high order convergence of the novel method.Comment: 36 pages, 20 figure
A PetrovâGalerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation
A new Petrov–Galerkin (PG) method involving two parameters, namely α1 and α2, is presented, which yields the following schemes on rectangular meshes: (i) a compact stencil obtained by the linear interpolation of the Galerkin FEM and the classical central finite difference method (FDM), should the parameters be equal, that is, α1 = α2 = α; and (ii) the nonstandard compact stencil presented in (Int. J. Numer. Meth. Engng 2011; 86:18–46) for the Helmholtz equation if the parameters are distinct, that is, α1 ≠ α2. The nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms (specified by α1) and the mass terms (specified by α2) that appear in the stencils obtained by the standard Galerkin FEM and the classical central FDM, respectively. On square meshes, these two schemes were shown to provide solutions to the Helmholtz equation that have a dispersion accuracy of fourth and sixth order, respectively (Int. J. Numer. Meth. Engng 2011; 86:18–46). The objective of this paper is to study the performance of this PG method for the Helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions
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