Topic
Finite difference method
About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.
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TL;DR: In this article, a finite difference method is developed to analyze the guided-wave properties of a class of two-dimensional photonic crystals (irregular dielectric rods) for both in-plane and out-of-plane propagation.
Abstract: In this paper, a finite difference method is developed to analyze the guided-wave properties of a class of two-dimensional photonic crystals (irregular dielectric rods). An efficient numerical scheme is developed to deal with the deterministic equations resulting from a set of finite difference equations for inhomogeneous periodic structures. Photonic band structures within an irreducible Brillouin zone are investigated for both in-plane and out-of-plane propagation. For out-of-plane propagation, the guided waves are hybrid modes; while for in-plane propagation, the guided waves are either TE or TM modes, and there exist photonic bandgaps within which wave propagation is prohibited. Photonic bandgap maps for squares, veins, and crosses are investigated to determine the effects of the filling factor, the dielectric contrast, and lattice constants, on the band-gap width and location. Possible applications of photonic bandgap materials are discussed.
124 citations
TL;DR: A novel, linear, second order semi-discrete scheme in time to solve the governing system of equations in the hydrodynamic Q -tensor model, developed following the novel ‘ energy quadratization ’ strategy so that it is linear and unconditionally energy stable at the semi- Discrete level.
124 citations
TL;DR: In this article, the authors derived explicit and new implicit staggered-grid finite-difference (FD) formulas for derivatives of first order with any order of accuracy by a plane wave theory and Taylor's series expansion.
Abstract: SUMMARY We derive explicit and new implicit staggered-grid finite-difference (FD) formulas for derivatives of first order with any order of accuracy by a plane wave theory and Taylor’s series expansion. Furthermore, we arrive at a practical algorithm such that the tridiagonal matrix equations are formed by the implicit FD formulas derived from the fractional expansion of derivatives. Our results demonstrate that the accuracy of a (2N + 2)th-order implicit formula is nearly equivalent to or greater than that of a (4N)th-order explicit formula. The new implicit method only involves solving tridiagonal matrix equations. We also demonstrate that a( 2N + 2)th-order implicit formulation requires nearly the same amount of memory and computation as those of a (2N + 4)th-order explicit formulation but attains the accuracy achieved by a (4N)th-order explicit formulation when additional cost of visiting arrays is not considered. Our analysis of efficiency and numerical modelling results for elastic wave propagation demonstrates that a high-order explicit staggered-grid method can be replaced by an implicit staggered-grid method of some order, which will increase the accuracy but not the computational cost.
124 citations
TL;DR: In this paper, a 3D finite-difference solution is implemented for simulating induction log responses in the quasi-static limit that include the wellbore and bedding that exhibits transverse anisotropy.
Abstract: A 3-D finite-difference solution is implemented for simulating induction log responses in the quasi-static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite-difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl-free and divergence-free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.
124 citations
TL;DR: In this paper, a finite-difference solution for the inverse problem of determining a control function p(t) in the parabolic partial differential equation is presented, where t is the number of variables in the control function.
Abstract: A finite-difference solution is demonstrated for an inverse problem of determining a control function p(t) in the parabolic partial differential equation ut=uxx+pu+f(x,t), 0
124 citations