Summation-by-parts (SBP) finite difference methods have several desirable properties for second-order wave equations. They combine the computational efficiency of narrow-stencil finite difference ...operators with provable stability on curvilinear multiblock grids. While several techniques for boundary and interface conditions exist, weak imposition via simultaneous approximation terms (SATs) is perhaps the most flexible one. Although SBP methods have been applied to elastic wave equations many times, an SBP-SAT method for general anisotropic elastic wave equations has not yet been presented in the literature. We fill this gap by deriving energy-stable self-adjoint SBP-SAT methods for general anisotropic materials on curvilinear multiblock grids. The methods are based on fully compatible SBP operators. Although this paper focuses on classical SBP finite difference operators, the presented boundary and interface treatments are general and apply to a range of methods that satisfy an SBP property. We demonstrate the stability and accuracy properties of a particular set of fully compatible SBP-SAT schemes using the method of manufactured solutions. We also demonstrate the utility of the new method in elastodynamic cloaking and seismic imaging in mountainous regions.
•New boundary and interface treatments for anisotropic elastic wave equations.•Proofs of energy stability and discrete self-adjointness.•Application problems in elastodynamic cloaking and seismic imaging.
We introduce a geometric stencil selection algorithm for Laplacian in 3D that significantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset ...from a set of irregular points surrounding a given point that admits an accurate numerical differentiation formula. The subset serves as an influence set for the numerical approximation of the Laplacian in meshless finite difference methods using either polynomial or kernel-based techniques. Numerical experiments demonstrate a competitive performance of this method in comparison to the finite element method and to other selection methods for solving the Dirichlet problems for the Poisson equation on several STL models. Discretization nodes for these domains are obtained either by 3D triangulations or from Cartesian grids or Halton quasi-random sequences.
Equivalent projectors for virtual element methods Ahmad, B.; Alsaedi, A.; Brezzi, F. ...
Computers & mathematics with applications (1987),
September 2013, 2013-09-00, 20130901, Letnik:
66, Številka:
3
Journal Article
Recenzirano
Odprti dostop
In the original virtual element space with degree of accuracy k, projector operators in the H1-seminorm onto polynomials of degree ≤k can be easily computed. On the other hand, projections in the L2 ...norm are available only on polynomials of degree ≤k−2 (directly from the degrees of freedom). Here, we present a variant of the virtual element method that allows the exact computations of the L2 projections on all polynomials of degree ≤k. The interest of this construction is illustrated with some simple examples, including the construction of three-dimensional virtual elements, the treatment of lower-order terms, the treatment of the right-hand side, and the L2 error estimates.
In this paper we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier–Stokes–Cahn–Hilliard (q-NSCH) system governing ...a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the Cahn–Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier–Stokes equation – for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh–Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10−8 and energy decreases in time.
•Two novel staggered grid finite difference methods are provided.•Discrete mass conservation is naturally satisfied due to the discretization.•Energy stability is achieved at the fully discrete level for both methods.•An efficient nonlinear multigrid solver is designed for solving the two methods.
In this article, domain of the third order generalized difference operator Δi3 in Hahn sequence space h is introduced. Some topological properties and inclusion relations are shown. Additionally, ...Schauder basis of the new Hahn sequence space h(Δi3) is calculated and α−, β− and γ−dual of the space h(Δi3) are computed. In the last section, some new matrix classes (h;bv), (h;bvp), (h;bv∞), (h;bs), (h;cs), (bv;h), (bv0;h), (bs;h), (cs0;h), (cs;h) are characterized. We conclude the paper with characterization of the classes (h(Δi3);μ) and (μ;h(Δi3)) where μ={c,c0,ℓ1,ℓp,ℓ∞,h,bs,cs,cs0,bv,bv0}.
In the present paper, a parameter-uniform numerical method is constructed and analysed for solving one-dimensional singularly perturbed parabolic problems with two small parameters. The solution of ...this class of problems may exhibit exponential (or parabolic) boundary layers at both the left and right part of the lateral surface of the domain. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution, we consider the implicit Euler method for time stepping on a uniform mesh and a special hybrid monotone difference operator for spatial discretization on a specially designed piecewise uniform Shishkin mesh. The resulting scheme is shown to be first-order convergent in temporal direction and almost second-order convergent in spatial direction. We then improve the order of convergence in time by means of the Richardson extrapolation technique used in temporal variable only. The resulting scheme is proved to be uniformly convergent of order two in both the spatial and temporal variables. Numerical experiments support the theoretically proved higher order of convergence and show that the present scheme gives better accuracy and convergence compared of other existing methods in the literature.
By using non-equispaced grid points near boundaries, we derive boundary optimized first derivative finite difference operators, of orders up to twelve. The boundary closures are based on a ...diagonal-norm summation-by-parts (SBP) framework, thereby guaranteeing linear stability on piecewise curvilinear multi-block grids. The new operators lead to significantly more efficient numerical approximations, compared with traditional SBP operators on equidistant grids. We also show that the non-uniform grids make it possible to derive operators with fewer one-sided boundary stencils than their traditional counterparts. Numerical experiments with the 2D compressible Euler equations on a curvilinear multi-block grid demonstrate the accuracy and stability properties of the new operators.
In this paper, we determine the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the operator on the sequence space . These results generalize the spectrum of the ...difference operator , , , , , and some other cases , , over the space .
•Solving the anisotropic acoustic equation without S-wave artifacts.•Correct the dispersion error and phase error of the finite-difference method.•Deal with free surface and absorbing boundary ...condition in a direct way.
Numerical solutions of the acoustic wave equation, especially in anisotropic media, is crucial to seismic modeling, imaging and inversion as it provides efficient, practical, and stable approximate representation of the medium. However, a clean implementation (free of shear wave artifacts and dispersion) of wave propagation, especially in anisotropic media, requires an integral operator, the direct evaluation of which is extremely expensive. Recently, the low-rank method was proposed to provide a good approximation to the integral operator utilizing Fourier transforms. Thus, we propose to split the integral operator into two terms. The first term provides a differential operator that approximates that can be approximated with a standard finite-difference method. We, then, apply the low-rank approximation on the residual term of the finite-difference operator. We implement the two terms in two complementing steps, in which the spectral step corrects for any errors admitted by the finite difference step. Even though we utilize finite-difference approximations, the resulting algorithm admits spectral accuracy. Also, through the finite difference step, the method can deal approximately with the free surface and absorbing boundary conditions in a straight forward manner. Numerical examples show that the method is of high accuracy and efficiency.
Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman (2008) 17) is an ...embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret (2012) 22), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao (2009) 26). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.
•In this paper, a new method for the numerical approximation of PDEs on surfaces is proposed. Our method has the advantage of being comprised of standard computational components, such as the closest point representation of the surface, and RBF finite difference methods.•Our approach uses a narrow computational tube around the surface and avoids the need for a quasi-uniform distribution of surface points. This makes the method a natural candidate for coupling with grid-based methods such as the grid-based particle method for moving interface problems (Leung and Zhao, J. Comput. Phys. 228 (8) (2009) 2993–3024).•The method is also efficient: it exploits repeated patterns in computational geometry, it uses small computational tubes, and it avoids an explicit interpolation step. Further-more, a change in the order of the method is carried out simply by changing the number of points in the finite difference stencil. See our novelty statement for details on how the method compares with the original closest point method (Ruuth and Merriman, J. Comput. Phys. 227 (3) (2008) 1943–1961) and recent RBF methods (e.g., Piret, J. Comput. Phys. 231 (14) (2012) 4662–4675).•We conduct convergence studies in two and three dimensions and apply the method to a variety of problems, including reaction–diffusion systems and image denoising. Second order accurate results are observed in our experiments.
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