A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law

Castillo, José E.; Grone, Robert

doi:10.1137/s0895479801398025

Cited by 65 publications

(56 citation statements)

References 9 publications

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“…On uniform grids, following the notation of Figure 3.1, the approach of [5,7] can be applied to obtain second-order mimetic discretizations D and G for the divergence and gradient continuum operators, respectively, yielding that the gradient at the boundary points x 0 and x n has the form…”

Section: Advances In Difference Equationsmentioning

confidence: 99%

“…As is worked out by [5], the construction of the discretized mimetic gradient G follows from the discretized mimetic divergence D as a consequence of imposing that both operators must satisfy a discrete version of Green-Gauss theorem. This can be seen by defining an extension of D, denoted by D, which satisfies (…”

Section: Advances In Difference Equationsmentioning

confidence: 99%

“…Particularly, in a recent article [5] a systematic way of constructing high-order mimetic discretizations for gradient, divergence operators with the same order of approximation 2 Advances in Difference Equations at the boundary and inner region was presented. A key point of mimetic discretizations is to build discrete versions of these operators satisfying a discrete analog of the divergence or Green-Gauss theorem which implies that the discrete operators will satisfy a global conservation law.…”

Section: Introductionmentioning

confidence: 99%

“…In this article, we will provide a rigorous proof of quadratic convergence for a particular and unique mimetic finite difference method for the steady-state diffusion equation based on the second-order discrete gradient and divergence operators obtained and studied in [5][6][7]. Although the literature on second-order mimetic methods for the steadystate diffusion is fairly well established, our new method is not standard.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

Guevara-Jordan¹,

Rojas²,

Freites-Villegas³

et al. 2007

Adv. Differ Equ.

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The numerical solution of partial differential equations with finite differences mimetic methods that satisfy properties of the continuum differential operators and mimic discrete versions of appropriate integral identities is more likely to produce better approximations. Recently, one of the authors developed a systematic approach to obtain mimetic finite difference discretizations for divergence and gradient operators, which achieves the same order of accuracy on the boundary and inner grid points. This paper uses the second-order version of those operators to develop a new mimetic finite difference method for the steady-state diffusion equation. A complete theoretical and numerical analysis of this new method is presented, including an original and nonstandard proof of the quadratic convergence rate of this new method. The numerical results agree in all cases with our theoretical analysis, providing strong evidence that the new method is a better choice than the standard finite difference method.

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Section: Advances In Difference Equationsmentioning

confidence: 99%

Section: Advances In Difference Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

Guevara-Jordan¹,

Rojas²,

Freites-Villegas³

et al. 2007

Adv. Differ Equ.

Self Cite

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show abstract

“…(3)- (6), we solve system (2) using an FSG finite-difference method, where all spatial derivatives are substituted by mimetic operators [2][3][4]18]. In addition, we use a leap-frog explicit scheme for the time integration which benefits from the corrections in [11] for reducing storage in this configuration.…”

Section: Viscoelastic Wave Propagationmentioning

confidence: 99%

3D Viscoelastic Anisotropic Seismic Modeling with High-Order Mimetic Finite Differences

Ferrer

Puente

Farrés

et al. 2015

Lecture Notes in Computational Science and Engineering

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We present a scheme to solve three-dimensional viscoelastic anisotropic wave propagation on structured staggered grids. The scheme uses a fully-staggered grid (FSG) or Lebedev grid (Lebedev, J Sov Comput Math Math Phys 4:449-465, 1964; Rubio et al. Comput Geosci 70:181-189, 2014), which allows for arbitrary anisotropy as well as grid deformation. This is useful when attempting to incorporate a bathymetry or topography in the model. The correct representation of surface waves is achieved by means of using high-order mimetic operators (Castillo and Grone, SIAM J Matrix Anal Appl 25:128-142, 2003; Castillo and Miranda, Mimetic discretization methods. CRC Press, Boca Raton, 2013), which allow for an accurate, compact and spatially high-order solution at the physical boundary condition. Furthermore, viscoelastic attenuation is represented with a generalized Maxwell body approximation, which requires of auxiliary variables to model the convolutional behavior of the stresses in lossy media. We present the scheme's accuracy with a series of tests against analytical and numerical solutions. Similarly we show the scheme's performance in high-performance computing platforms. Due to its accuracy and simple pre-and post-processing, the scheme is attractive for carrying out thousands of simulations in quick succession, as is necessary in many geophysical forward and inverse problems both for the industry and academia.

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A performance analysis of a mimetic finite difference scheme for acoustic wave propagation on GPU platforms

Otero

Francés

Rodriguez

et al. 2016

Concurrency and Computation

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, J. A performance analysis of a mimetic finite differencescheme for acoustic wave propagation on GPU platforms. "Concurrency and computation. Practice and experience", 1 Febrer 2017, vol. 29, núm. 4, p. 1-18 SUMMARYRealistic applications of numerical modeling of acoustic wave dynamics usually demand high performance computing because of the large size of study domains and demanding accuracy requirements on simulation results. Forward modeling of seismic motion on a given subsurface geological structure is by itself a good example of such applications, and when used as a component of seismic inversion tools or as a guide for the design of seismic surveys, its computational cost increases enormously. In the case of finite difference methods (or any other volumen-discretization scheme), memory and computing demands rise with grid refinement which may be neccesary to reduce errors on numerical wave patterns and better capture target physical devices. In this work, we present several implementations of a mimetic finite difference method for the simulation of acoustic wave propagation on highly dense staggered grids. These implementations evolve as different optimization strategies are employed starting from appropriate setting of compilation flags, code vectorization by using SSE and AVX instructions, CPU parallelization by exploiting the OpenMP framework, to the final code parallelization on GPU platforms. We present and discuss the increasing processing speed-up of this mimetic scheme achieved by the gradual implementation and testing of all these performance optimizations. In terms of simulation times, the performance of our GPU parallel implementations is consistently better than the best CPU version.Received . . . KEY WORDS: Acoustic waves; mimetic finite differences; SIMD extensions; GPU; CUDA programming * Correspondence to: C/ Jordi Girona 1.3, Edf. 08034, This is the peer reviewed version of the following article: Otero, B., Frances, J., Rodriguez-Cruz, R., Rojas, O., Solano, F., Guevara-Jordan, J. A performance analysis of a mimetic finite difference scheme for acoustic wave propagation on GPU platforms. "Concurrency and computation.

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A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law

Cited by 65 publications

References 9 publications

Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

Convergence of a Mimetic Finite Difference Method for Static Diffusion Equation

3D Viscoelastic Anisotropic Seismic Modeling with High-Order Mimetic Finite Differences

A performance analysis of a mimetic finite difference scheme for acoustic wave propagation on GPU platforms

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