Generation of Finite Difference Formulas on Arbitrarily Spaced Grids

Fornberg, Bengt

doi:10.2307/2008770

Cited by 112 publications

(123 citation statements)

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“…We find that these upwind stencils provide a significant increase in the numerical accuracy of the puncture motion at a given resolution (see Appendix A). The particular stencils which we use can be generated via a recursion algorithm, as outlined in [54].…”

Section: Finite Differencingmentioning

confidence: 99%

High accuracy binary black hole simulations with an extended wave zone

et al. 2011

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We present results from a new code for binary black hole evolutions using the moving-puncture approach, implementing finite differences in generalised coordinates, and allowing the spacetime to be covered with multiple communicating non-singular coordinate patches. Here we consider a regular Cartesian near zone, with adapted spherical grids covering the wave zone. The efficiencies resulting from the use of adapted coordinates allow us to maintain sufficient grid resolution to an artificial outer boundary location which is causally disconnected from the measurement. For the well-studied test-case of the inspiral of an equal-mass non-spinning binary (evolved for more than 8 orbits before merger), we determine the phase and amplitude to numerical accuracies better than 0.010% and 0.090% during inspiral, respectively, and 0.003% and 0.153% during merger. The waveforms, including the resolved higher harmonics, are convergent and can be consistently extrapolated to r → ∞ throughout the simulation, including the merger and ringdown. Ringdown frequencies for these modes (to ( , m) = (6, 6)) match perturbative calculations to within 0.01%, providing a strong confirmation that the remnant settles to a Kerr black hole with irreducible mass Mirr = 0.884355 ± 20 × 10 −6 and spin S f /M 2 f = 0.686923 ± 10 × 10 −6 .

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Section: Finite Differencingmentioning

confidence: 99%

High accuracy binary black hole simulations with an extended wave zone

et al. 2011

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“…Horizontal gradients and Laplacian operators are calculated using high-order finite difference approximations with regular or irregular point distribution. Unless otherwise specified, first and second order derivatives in x are approximated using fourth-order schemes following the method of [28]. In the test cases described here, the lateral boundary conditions are either periodic or vertical reflective walls, where @ @n D 0.…”

Section: General Modeling Strategymentioning

confidence: 99%

Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves

Yates

Benoît

2015

Numerical Methods in Fluids

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International audienceSUMMARY The accuracy and efficiency of two methods of resolving the exact potential flow problem for nonlinear waves are compared using three different 1DH test cases. The two model approaches use high-order finite difference schemes in the horizontal dimension and differ in the resolution of the vertical dimension. The first model uses high-order finite difference schemes also in the vertical, while the second model applies a spectral approach. The convergence, accuracy, and efficiency of the two models are demonstrated as a function of the temporal, horizontal, and vertical resolution for (1) the propagation of regular nonlinear waves in a periodic domain, (2) the motion of nonlinear standing waves in a domain with fully reflective boundaries, and (3) the propagation and shoaling of a train of waves on a slope. The spectral model approach converges more rapidly as a function of the vertical resolution. In addition, with equivalent vertical resolution, the spectral model approach shows enhanced accuracy and efficiency in the parameter range used for practical model applications

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“…It involves discretising the region into a set of grid points and calculating an approximation to the solution at each grid point. The simulation is stepped forward in time using a stencil-like calculation with weights chosen to minimise the local truncation error [11].…”

Section: A Fdtd For Ultrasound Simulationmentioning

confidence: 99%

Acceleration of GPU-Based Ultrasound Simulation via Data Compression

Haigh

McCreath

2014

2014 IEEE International Parallel &Amp; Distributed Processing Symposium Workshops

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Abstract-The realistic simulation of ultrasound wave propagation is computationally intensive. The large size of the grid and low degree of reuse of data means that it places a great demand on memory bandwidth. Graphics Processing Units (GPUs) have attracted attention for performing scientific calculations due to their potential for efficiently performing large numbers of floating point computations. However, many applications may be limited by memory bandwidth, especially for data sets whose size is larger than that of the GPU platform. This problem is only partially mitigated by applying the standard technique of breaking the grid into regions and overlapping the computation of one region with the host-device memory transfer of another.In this paper, we implement a memory-bound GPU-based ultrasound simulation and evaluate the use of a technique for improving performance by compressing the data into a fixedpoint representation that reduces the time required for interhost-device transfers. We demonstrate a speedup of 1.5 times on a simulation where the data is broken into regions that must be copied back and forth between the CPU and GPU. We develop a model that can be used to determine the amount of temporal blocking required to achieve near optimal performance, without extensive experimentation. This technique may also be applied to GPU-based scientific simulations in other domains such as computational fluid dynamics and electromagnetic wave simulation.

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Generation of Finite Difference Formulas on Arbitrarily Spaced Grids

Abstract: Abstract.Simple recursions are derived for calculating the weights in compact finite difference formulas for any order of derivative and to any order of accuracy on onedimensional grids with arbitrary spacing. Tables are included for some special cases (of equispaced grids).

Cited by 112 publications

References 1 publication

High accuracy binary black hole simulations with an extended wave zone

High accuracy binary black hole simulations with an extended wave zone

Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves

Acceleration of GPU-Based Ultrasound Simulation via Data Compression

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