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A positivity-preserving well-balanced wet-dry front reconstruction for shallow water equations on rectangular grids

APPLIED NUMERICAL MATHEMATICS 2024年 198卷 295-317页

作者： Wang, Xue Chen, Guoxian Wuhan Univ Sch Math & Stat Wuhan 430072 Peoples R China Wuhan Univ Hubei Key Lab Computat Sci Wuhan 430072 Peoples R China

In this paper, a positivity -preserving, well-balanced finite volume scheme on a rectangular mesh is designed based on wet -dry front reconstruction to solve the shallow water equations with nonflat bottom topography. The crucial step is a special piecewise linear representation of the bottom. The flat bottom approximation simplifies the reconstruction of the wet -dry front, which becomes a straight line inside the partially flooded cells when the water is at rest. The continuity of the discrete bottom across the cell interface centres avoids the calculation of the source term across the cell boundaries and simplifies the numerical flux evaluation. The lake -at -rest steady state is preserved after a well-balanced source term discretization. The draining time method is used for the positivity -preserving property in partially flooded cells. Numerical experiments demonstrate the robustness of the scheme.

关键词： Saint-Venant system of shallow water equations Well-balanced scheme rectangular grids Positivity-preserving scheme Wet-dry front reconstruction Draining time method

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THE LOWEST ORDER DIFFERENTIABLE FINITE ELEMENT ON rectangular grids

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2011年第4期49卷 1350-1368页

作者： Hu, Jun Huang, Yunqing Zhang, Shangyou Peking Univ LMAM Beijing 100871 Peoples R China Peking Univ Sch Math Sci Beijing 100871 Peoples R China Xiangtan Univ Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Hunan Peoples R China Univ Delaware Dept Math Sci Newark DE 19716 USA

A macro type of biquadratic C(1) finite elements is constructed on rectangle grids. This is a rectangular version of the C(1) Powell-Sabin element, a C(1)-P(2) element on triangular grids. Here, each rectangle of the base grid is refined into four subrectangles. As in the case of the Powell-Sabin element, we have more constraints than the number of degrees of freedom on each macroelement. However, the extra constraints are consistent. It is shown further that the constructed finite element space is the full C(1)-Q(2) space on the grid. It is also shown that the finite element space is a tensor product space of one-dimensional C(1)-P(2) spaces, where the nodal basis is supported on four intervals. The B-spline function of P(2) is supported on three intervals. The Girault-Scott operator is extended to the element. The application and the convergence of the finite element to the biharmonic equation are presented. Numerical tests are provided.

关键词： differentiable finite element biharmonic equation Powell-Sabin element Bogner-Fox Schmit element Girault Scott operator rectangular grids

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EMBEDDING rectangular grids INTO SQUARE grids

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IEEE TRANSACTIONS ON COMPUTERS 1991年第1期40卷 46-52页

作者： ELLIS, JA Dept. of Comput. Sci. Victoria Univ. BC Canada Abstract Authors References Cited By Keywords Metrics Similar Download Citation Email Print Request Permissions

We show that two-dimensional rectangular grids of large aspect ratio can be embedded into rectangular grids of smaller aspect ratios with small expansion and dilation. In particular, width can be reduced by a factor of up to 2 with optimal expansion, i.e., when the host rectangle is the smallest sufficient to contain the guest, and optimal dilation, i.e., 2. A width reduction factor of 3 can be obtained with optimal expansion and dilation 3. In general, any rectangular grid can be embedded into a square grid that is no more than unity larger on the side than the minimum possible, with dilation no more than 3. These results improve on those previously obtained, in which dilation better than 18 could not be guaranteed. They might be applied to more complex grid embedding problems, such as embedding multidimensional grids into hypercubes.

关键词： GRAPH EMBEDDINGS MINIMUM DILATION MINIMUM EXPANSION rectangular grids

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A Simple Conforming Mixed Finite Element for Linear Elasticity on rectangular grids in Any Space Dimension

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JOURNAL OF SCIENTIFIC COMPUTING 2014年第2期58卷 367-379页

作者： Hu, Jun Man, Hongying Zhang, Shangyou Peking Univ LMAM Beijing 100871 Peoples R China Peking Univ Sch Math Sci Beijing 100871 Peoples R China Beijing Inst Technol Sch Math Beijing 100081 Peoples R China Univ Delaware Dept Math Sci Newark DE 19716 USA

We construct a family of lower-order rectangular conforming mixed finite elements, in any space dimension. In the method, the normal stress is approximated by quadratic polynomials , the shear stress by bilinear polynomials , and the displacement by linear polynomials . The number of total degrees of freedom (dof) per element is 10 plus 4 in 2D, and 21 plus 6 in 3D, while the previous record of least dof for conforming element is 17 plus 4 in 2D, and 72 plus 12 in 3D. The feature of this family of elements is, besides simplicity, that shape function spaces for both stress and displacement are independent of the spatial dimension . As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. The well-posedness condition and the optimal a priori error estimate are proved. Numerical tests show the stability and effectiveness of these new elements.

关键词： Mixed finite element Symmetric finite element Linear elasticity Conforming finite element rectangular grids Inf-sup condition

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Bandwidth edge counts for linear arrangements of rectangular grids

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JOURNAL OF GRAPH THEORY 1997年第4期26卷 195-202页

作者： Fishburn, P Wright, P AT&T Labs–Research Florham Park NJ 07932 Email: Peter Fishburn (fish@***)[*]AT&T Labs–Research Florham Park NJ 07932

Let G(n,m) denote a graph with nm vertices arranged in n rows and m greater than or equal to max{n, 2} columns with an edge {u, v} between vertices u and v if they are adjacent horizontally or vertically. The bandwidth of G(n,m) is known to equal n. We prove that the number of edges in a bandwidth-achieving linear arrangement f from the vertex set. of G(n,m) onto {1, 2,...: nm} for which \f(u) -f(v)\ = n can range from 2(n -1) + n(m -n) up to n(m -1). The lower bound is attained for a down-diagonal lexicographic linear arrangement;the upper bound is at-rained by a column-row lexicographic linear arrangement. (C) 1997 John Wiley & Sons, Inc.

关键词： linear arrangements graph bandwidth rectangular grids

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On nonstaggered rectangular grids using streamfunction and velocity potential or vorticity and divergence

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MONTHLY WEATHER REVIEW 2004年第6期132卷 1518-1521页

作者： Gavrilov, MB Univ St Cyril & Methudius Fac Nat Sci & Math Inst Phys Meteorol Skopje 91000 Macedonia

Dispersion characteristics of the shallow water gravity-inertia waves are discussed for two nonstaggered rectangular grids that use the velocity potential and streamfunction, and the divergence and vorticity, respectively. It is shown that the simplest second-order accuracy schemes on the two grids produce identical frequencies on an infinite rotating plane, and that, therefore, the two grids are equivalent. Stability analysis of the linearized shallow water equations using leapfrog and semi-implicit time-differencing techniques indicates that viable numerical schemes can be designed on these grids.

关键词： .Schemes Streamfunction Second-Order shallow water equations rectangular grids

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Cross-rhombus stencil-based finite-difference methods for 2D acoustic modeling and reverse-time migration on rectangular grids

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JOURNAL OF GEOPHYSICS AND ENGINEERING 2018年第6期15卷 2674-2685页

作者： Wang, Enjiang Ba, Jing Liu, Yang Hohai Univ Sch Earth Sci & Engn Nanjing 211100 Jiangsu Peoples R China China Univ Petr State Key Lab Petr Resources & Prospecting Beijing 102249 Peoples R China

The recently developed cross-rhombus stencil-based time-space domain finite-difference (FD) method for modeling two-dimensional acoustic equations, controls the temporal and spatial dispersions synchronously and outperforms cross-stencil-based FD (CS-FD). However, it is only applicable to modeling on equally spaced grids and extensive application is hindered. In this work, we extend it further and develop novel cross-rhombus stencil-based FD (CRS-FD) for modeling on arbitrarily rectangular grids. Two kinds of rhombus stencils involving grid points both on and off the axis are developed first to achieve fourth order and sixth order FD accuracies, respectively, for solving the second order temporal derivative on rectangular grids. The plane wave theory is then used to derive the time-space-domain dispersion relations, when the temporal and spatial derivatives are solved by the new rhombus-stencil-based temporal high order FD and the cross-stencil-based spatial high order FD, respectively. The extrapolation stencil is a mixture of rhombus and cross stencils, and the involved FD coefficients are determined by applying the Tayler expansion on the time-space domain dispersion relation. Our new CRS-FD is high-order accurate in both time and space, and applicable to modeling on arbitrarily rectangular grids. Dispersion analysis, stability analysis and modeling examples on rectangular grids show that the CRS-FD is more accurate and stable than the CS-FD. Meanwhile, we develop the variable-length schemes for CRS-FD to further increase efficiency and apply them to extrapolate wavefields in reverse time migration. The results validate that the CRS-FD is more efficient than the CS-FD because much larger time steps can be used while reaching a similar accuracy. The variable-length scheme further reduces the computational time in comparison with the fixed-length scheme.

关键词： finite difference modeling temporal high-order time-space-domain rectangular grids variable-length FD reverse time migration

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FRACTAL INTERPOLATION SURFACES ON rectangular grids

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BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY 2015年第3期91卷 435-446页

作者： Ruan, Huo-Jun Xu, Qiang Zhejiang Univ Dept Math Hangzhou 310027 Zhejiang Peoples R China Jiangsu Normal Univ Sch Math & Stat Xuzhou 221116 Peoples R China

In this paper, we present a general framework to construct fractal interpolation surfaces (FISs) on rectangular grids. Then we introduce bilinear FISs, which can be defined without any restriction on interpolation poi... 详细信息

关键词： fractal interpolation surfaces iterated function systems vertical scaling factors rectangular grids

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Optimal quadratic element on rectangular grids for H¹ problems

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BIT NUMERICAL MATHEMATICS 2021年第2期61卷 665-689页

作者： Zeng, Huilan Zhang, Chen-Song Zhang, Shuo Chinese Acad Sci Acad Math & Syst Sci LSEC ICMSEC Beijing 100190 Peoples R China Chinese Acad Sci Acad Math & Syst Sci NCMIS Beijing 100190 Peoples R China

In this paper, a piecewise quadratic finite element method on rectangular grids for H-1 problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is proved to be O(h(2)) in the energy norm on uniform grids over a convex domain. Alower bound of the L-2-norm error is also proved, which makes the capacity of this scheme more clear. For the eigenvalue problem, the computed eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented to verify the theoretical findings.

关键词： Optimal quadratic element rectangular grids Boundary value problem Eigenvalue problem Lower bound

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Axiomatizing rectangular grids with no Extra Non-unary Relations

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FUNDAMENTA INFORMATICAE 2020年第2期176卷 129-138页

作者： Kopczynski, Eryk Univ Warsaw Inst Informat Banacha 2 PL-02097 Warsaw Poland

We construct a first-order formula phi such that all finite models of phi are non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A subset of N is a spectrum of a formula which has only planar models if numbers n is an element of A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t(n) and space s(n), where t(n)s(n) <= n and t(n), s(n) =Omega(log(n)).

关键词： spectrum problem planar graphs rectangular grids descriptive complexity finite model theory

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