A non-deteriorating algorithm for computational electromagnetism based on quasi-lacunae of Maxwell’s equations

Petropavlovsky, S. V.; Tsynkov, Semyon

doi:10.1016/j.jcp.2011.09.019

Cited by 9 publications

(6 citation statements)

References 44 publications

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“…Provided that the density ξ ξ ξ Γ is obtained as described in Sect. 4.3, inequality (61) also implies that the discrete solution u given by (36) will converge as h → 0 to the continuous solution u of the corresponding problem (24) with the same rate and in the sense of the same norm as in (61).…”

Section: Accuracymentioning

confidence: 90%

“…Two types of methods that have been introduced and analyzed previously can prove useful in this context: the methods for unsteady control of sound [59], and lacunae-based methods that are used for setting the unsteady ABCs [44,56,58], as well as for achieving the improved performance over long times [35][36][37].…”

Section: Discussionmentioning

confidence: 99%

“…This will later be used to have the solution itself approximated on the grid N + , see (31), and reconstructed by means of the discrete Green's formula (36).…”

Section: Representation Of the Solution At The Boundarymentioning

confidence: 99%

“…As indicated in the end of Sect. 3.2, the source term f (x) for the boundary value problem (24) may not even be defined anywhere beyond the domain Ω, but the discrete right-hand side f needs to be known at the nodes K + \ K Ω in order to apply the operator B (h) in formulae (35) and (36). To obtain f at the nodes K + \ K Ω , which are located close to Γ yet outside Ω, we will also employ the Taylor formula [cf.…”

Section: Equation-based Extensionmentioning

confidence: 99%

“…Then, we can reconstruct ξ ξ ξ Γ using formula (40b), obtain ξ γ with the help of (87), and finally apply the discrete Green's formula (36), which yields the difference solution u on N + . This completes the solution algorithm.…”

Section: The System Of Equationsmentioning

confidence: 99%

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

2012

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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.

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Section: Accuracymentioning

confidence: 90%

Section: Discussionmentioning

confidence: 99%

“…This will later be used to have the solution itself approximated on the grid N + , see (31), and reconstructed by means of the discrete Green's formula (36).…”

Section: Representation Of the Solution At The Boundarymentioning

confidence: 99%

Section: Equation-based Extensionmentioning

confidence: 99%

Section: The System Of Equationsmentioning

confidence: 99%

See 3 more Smart Citations