Iterative schemes for high order compact discretizations to the exterior Helmholtz equation

Erlangga, Yogi A.; Turkel, Eli

doi:10.1051/m2an/2011063

Cited by 19 publications

(22 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, the coordinates associated with the curve that we have introduced in Appendix A in order to build the equation-based extension will need to be replaced with surface-oriented coordinates, see [21]. The only type of solvers for the AP that seem feasible in 3D are iterative solvers, and we plan to use the Krylov-type algorithm Risolv of [52] with complex-shifted preconditioners [13][14][15][16] applied with the help of multigrid.…”

Section: Discussionmentioning

confidence: 99%

“…This is equivalent to specifying the first component of ξ ξ ξ Γ , ξ 0 = φ, because the first component of the trace Tr u is the Dirichlet data, see formula (13). Therefore, using the partition of c into c 0 and c 1 , see formula (51), we claim that the coefficients c 0 = [c 0−M , .…”

Section: The System Of Equationsmentioning

confidence: 99%

“…where the operator Tr is defined by formula (13). Replacing v by P Ω ξ ξ ξ Γ on both sides of (15) we have: P Ω ξ ξ ξ Γ = P Ω Tr P Ω ξ ξ ξ Γ .…”

Section: Constant Coefficientsmentioning

confidence: 99%

“…1]. We note that for the equation with variable coefficients (9) and a(x, y) = b(x, y) it may be more natural and more beneficial, although not necessary, to redefine the functions ξ ξ ξ Γ and the operator Tr, see formula (13), and instead of the standard normal derivative ∂u ∂n use the co-normal derivative a ∂u ∂x n x + b ∂u ∂y n y . We emphasize that the projection P Γ can be built using different APs.…”

Section: Definitionmentioning

confidence: 99%

See 3 more Smart Citations

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

2012

View full text Add to dashboard Cite

The method of difference potentials was originally proposed by Ryaben'kii and can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve.Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils-one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation.Dedicated to our friend Saul Abarbanel on the occasion of his 80th birthday. J Sci Comput (2012) 53:150-193 151 We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: The System Of Equationsmentioning

confidence: 99%

“…where the operator Tr is defined by formula (13). Replacing v by P Ω ξ ξ ξ Γ on both sides of (15) we have: P Ω ξ ξ ξ Γ = P Ω Tr P Ω ξ ξ ξ Γ .…”

Section: Constant Coefficientsmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

See 2 more Smart Citations

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

2012

View full text Add to dashboard Cite

show abstract

“…For a discussion on higher order finite difference approximations for the Helmholtz equation, we refer to e.g. [33].…”

Section: Problem Formulationmentioning

confidence: 99%

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

Sheikh

Lahaye

Ramos

et al. 2016

Journal of Computational Physics

View full text Add to dashboard Cite

Electromagnetic scattering from a cavity embedded in an impedance ground plane

Sun

et al. 2018

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper is concerned with time‐harmonic electromagnetic scattering from a cavity embedded in an impedance ground plane. The fillings (which may be inhomogeneous) do not protrude the cavity and the space above the ground plane is empty. This problem is obviously different from those considered in previous work where either perfectly conducting boundary conditions were used or the cavity was assumed to be empty. By employing the Green's function method, we reduce the scattering problem to a boundary‐value problem in a bounded domain (the cavity), with impedance boundary conditions on the cavity walls and an impedance‐to‐Dirichlet condition on the cavity aperture. Existence and uniqueness of the solution are proved for the weak formulation of the reduced problem. We also propose a numerical method to calculate the radar cross section (RCS), which is a parameter of physical interest. Numerical experiments show that the proposed model and numerical method are efficient for the calculation of RCS from cavities.

show abstract

Iterative schemes for high order compact discretizations to the exterior Helmholtz equation

Cited by 19 publications

References 19 publications

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

Electromagnetic scattering from a cavity embedded in an impedance ground plane

Contact Info

Product

Resources

About