Prior to abour 1990, the modeling of electromagnetic engineering systems was primarily implemented using solution techniques for the sinusoidal steady-state Maxwell's equations. Before about 1960, the principal approaches in this area involved closed-form and infinite-series analytical solutions, with numerical results from these analyses obtained using mechanical calculators. After 1960, the increasing availability of programmable electronic digital computers permitted such frequency-domain approaches to rise markedly in sophistication. Researchers were able to take advantage of the capabilities afforded by powerful new high-level programming languages such as Fortran, rapid random-access storage of large arrags of numbers, and computational speeds that were orders of magnitude faster than possible with mechanical calculators. In this period, the principal computational approaches for Maxwell's equations included the high-frequency asymptotic methods of Keller (1962) as well as Kouyoumjian and Pathak (1974) and the integral equation techniques of Harrington (1968) .

9 – Computational Electromagnetics: The Finite-Difference Time-Domain Method

We define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane. Our estimates show the error to be comparable to a standard Galerkin finite element method using piecewise linear polynomials.

Some errors estimates for the box method

The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH 1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h 2). Results on the effects of numerical integration are also included.

On the finite volume element method

. Asrecently as ﬁve to ten years ago, mesh generation was fre-quently the most time consuming task in a typical CFD cycle.Adaptive Cartesian mesh generation methods are capable ofproducing millions of cells around complex geometries inminutes and have substantially removed this bottleneck.Why write yet another Euler solver? With robust mesh gener-ation largely in-hand, solution time resurfaces as the pacingitem in the CFD cycle. The current work attacks this issue bydesigning a scalable, accurate Cartesian solver with robustmultigrid convergence acceleration. Our primary motivationis to gain efﬁciency by capitalizing on the simpliﬁcations andspecialized data structures available on Cartesian grids. Sig-niﬁcant savings in both CPU time and storage may be real-ized by taking advantage of the fact that cell faces arecoordinate aligned. In addition, higher-order methods withgood limiters are generally easier to design and perform morerobustly on uniform Cartesian meshes.Secondly, in any embedded-boundary Cartesian solver, thebody-intersectingcut-cellsdemand special attention. Thesecells can impose a substantial burden on the numerical dis-cretization since the arbitrary nature of geometric intersec-tion implies that a cut-cell may be orders of magnitudesmaller than its neighboring cells. This fact contrasts sharplywith the comparatively smooth meshes that are generallyfound on a good quality structured or unstructured mesh.Substantial research into these cut-cell issues have been stud-ied by references [9],[10],[6],[8], and [12] (among others)and we hope to take advantage of this investment.Thirdly, this work investigates a multigrid strategy that isspecialized for adaptively reﬁned Cartesian meshes. In ourapproach, all grids in the multigrid hierarchy cover the entiredomain and include cells at many reﬁnement levels. Thesmoother therefore iterates over the entire domain when it isinvoked on any grid in the hierarchy. In this respect, theapproach shares more with agglomeration or algebraic multi-grid techniques than with many other Cartesian or AMRmethods which iterate over only cells at the same level ofreﬁnement

A Parallel Multilevel Method for Adaptively Refined Cartesian Grids with Embedded Boundaries

We consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains. We demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids. Extensions to nonselfadjoint and parabolic problems are discussed.

On convergence of block-centered finite differences for elliptic-problems

As Tikhonov and Samarskil showed for k = 2, it is not essential that k th-order compact difference schemes be centered at the arithmetic mean of the stencil's points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the k th difference quotient is set equal to the value of the k th derivative at the middle point of the stencil; the proof is particularly transparent for k = 2. For any k, in fact, there is a ( k/2J -parameter family of symmetric averages of the values of the k th derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerov's tridiagonal scheme (approximating D2y = f with third-order truncation error) yields fourth-order con- vergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three- periodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.

/pdf/supra-convergent-schemes-on-irregular-grids-2o2y0acpr8.pdf

B Swartz

Papers

Supra-convergent schemes on irregular grids