Rate-of-convergence bounds of a finite-difference approximation of the Dirichlet problem for the Helmholtz equation with a coefficient from the “negative” space

1994 ◽  
Vol 71 (4) ◽  
pp. 2531-2538
Author(s):  
S. A. Voitsekhovskii ◽  
V. L. Makarov ◽  
Yu. I. Rybak
Keyword(s):  
Dirichlet Problem ◽  
Finite Difference ◽  
Helmholtz Equation ◽  
Rate Of Convergence ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Negative Space

scholarly journals Modified Explicit Decoupled Group Scheme οn Helmholtz Equation

2020 ◽  
Vol Volume 14 ◽  
Keyword(s):  
Computational Complexity ◽  
Finite Difference ◽  
Iterative Method ◽  
Helmholtz Equation ◽  
Group Scheme ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Group Method ◽  
Two Dimensional ◽  
New Formulation

In this paper, the formulation of a new group iterative method called the Modified Explicit Decoupled Group method in solving the two dimensional Helmholtz equation is described. The method is derived using a combination of the five-point finite difference approximation on the rotated grid stencil together with the five-point centred difference approximation on the standard grid stencils. Numerical experimentations of this new formulation shows significant improvement in computational complexity and execution timings over the original Explicit Decoupled Group method [2].

scholarly journals Adaptive GMRES(m) for the Electromagnetic Scattering Problem

2020 ◽  
Vol 21 (1) ◽  
pp. 191
Author(s):  
Gustavo Espínola ◽  
Juan C. Cabral ◽  
Christian Schaerer
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Control Problem ◽  
Electromagnetic Scattering ◽  
Scattering Problem ◽  
Finite Difference Approximation ◽  
Computational Time ◽  
Difference Approximation ◽  
Standard Methods ◽  
Restarted Gmres

In this article, an adaptive version of the restarted GMRES (GMRES(m)) is introduced for the resolution of the finite difference approximation of the Helmholtz equation. It has been observed that the choice of the restart parameter m strongly affects the convergence of standard GMRES(m). To overcome this problem, the GMRES(m) is formulated as a control problem in order to adaptively combine two strategies: a) the appropriate variation of the restarted parameter m, if a stagnation in the convergence is detected; and b) the augmentation of the search subspace using vectors obtained at previous cycles. The proposal is compared with similar iterative methods of the literature based on standard GMRES(m) with fixed parameters. Numerical results for selected matrices suggest that the switching adaptive proposal method could overcome the stagnation observed in standard methods, and even improve the performance in terms of computational time and memory requirements.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

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