This paper focuses on increasing the accuracy of low-order ͑four-node quadrilateral͒ finite elements for the transient analysis of wave propagation. Modified integration rules, originally proposed for time-harmonic problems, provide the basis for the proposed technique. The modified integration rules shift the integration points to locations away from the conventional Gauss or Gauss-Lobatto integration points with the goal of reducing the discretization errors, specifically the dispersion error. Presented here is an extension of the idea to time-dependent analysis using implicit as well as explicit time-stepping schemes. The locations of the stiffness integration points remain unchanged from those in time-harmonic case. On the other hand, the locations of the integration points for the mass matrix depend on the time-stepping scheme and the step size. Furthermore, the central difference method needs to be modified from its conventional form to facilitate fully explicit computation. The superior performance of the proposed algorithms is illustrated with the help of several numerical examples.
SUMMARYContinued fraction absorbing boundary conditions (CFABCs) are highly effective boundary conditions for modelling wave absorption into unbounded domains. They are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries. In this paper, CFABCs are extended to the more general case of polygonal computational domains. The key to the current development is the surprising link found between the CFABCs and the complex co-ordinate stretching of perfectly matched layers (PMLs). This link facilitates the extension of CFABCs to oblique corners and, thus, to polygonal domains. It is shown that the proposed CFABCs are easy to implement, expected to perform better than PMLs, and are effective for general polygonal computational domains. In addition to the derivation of CFABCs, a novel explicit time-stepping scheme is developed for efficient numerical implementation. Numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain. Although this paper focuses on wave propagation, its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients.
Abstract.A technique derived from two related methods suggested earlier by some of the authors for optimization of finite-difference grids and absorbing boundary conditions is applied to discretization of perfectly matched layer (PML) absorbing boundary conditions for wave equations in Cartesian coordinates. We formulate simple sufficient conditions for optimality and implement them. It is found that the minimal error can be achieved using pure imaginary coordinate stretching. As such, the PML discretization is algebraically equivalent to the rational approximation of the square root on [0, 1] conventionally used for approximate absorbing boundary conditions. We present optimal solutions for two cost functions, with exponential (and exponential of the square root) rates of convergence with respect to the number of the discrete PML layers using a second order finitedifference scheme with optimal grids. Results of numerical calculations are presented.Key words. absorbing boundary conditions, exponential convergence, finite differences, hyperbolic problems, perfectly matched layers, wave propagation, optimal rational approximations AMS subject classifications. 65N06, 73C02PII. S00361429013914511. Introduction. This paper is a sequel to a number of papers on so-called optimal finite-difference grids or finite-difference Gaussian rules [11,12,3,4,18,13], where exponential superconvergence of standard second order finite-difference schemes at a priori given points was obtained due to a special grid optimization procedure. This approach was successfully applied to the approximation of many nontrivial practically important problems, including elliptic PDEs for both bounded and unbounded domains. For the latter, in [18] the optimal finite-difference grid was obtained, which can be considered as the boundary condition requiring minimal arithmetic work for a given spectral interval. For hyperbolic problems, however, optimal grids were introduced only for the approximation in the interior part of the domain. Here we consider exterior hyperbolic problems. For this kind of problem a closely related method of continued fraction boundary conditions was suggested in [16], where absorbing boundary conditions were reduced to a three-term equations resembling finite-difference relations. Combining the approaches of [18] and [16] we obtain frequency independent finite-difference discretization of Berenger's perfectly matched layer (PML) absorbing boundary conditions (ABCs) [7], which produces the minimal possible impedance error for a given number of discrete layers. Similarly to the optimal grid for the Laplace equation with a solution from a Sobolev space considered in [18], the obtained discretizations show the exponential of the square root rates of convergence, though they use only the three-point stencil for second derivatives. Our solution exhibits much smaller reflection coefficients compared to examples of optimized PMLs (for the same
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