New Developments for Increased Performance of the SBP-SAT Finite Difference Technique

Nordström, Jan; Eliasson, Peter

doi:10.1007/978-3-319-12886-3_22

Cited by 5 publications

(4 citation statements)

References 7 publications

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“…ESSENSE [1,2] is a computational fluid dynamics (CFD) solver for compressible Navier-Stokes equations using a high order finite difference method (HOFDM). The solver can handle multiple structured blocks with coupling terms between the blocks.…”

Section: Introductionmentioning

confidence: 99%

GPU-acceleration of A High Order Finite Difference Code Using Curvilinear Coordinates

Kupiainen

Gong

Axner

et al. 2020

Proceedings of the 2020 International Conference on Computing, Networks and Internet of Things

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GPU-accelerated computing is becoming a popular technology due to the emergence of techniques such as OpenACC, which makes it easy to port codes in their original form to GPU systems using compiler directives, and thereby speeding up computation times relatively simply. In this study we have developed an OpenACC implementation of the high order finite difference CFD solver ESSENSE for simulating compressible flows. The solver is based on summation-by-part form difference operators, and the boundary and interface conditions are weakly implemented using simultaneous approximation terms. This case study focuses on porting code to GPUs for the most time-consuming parts namely sparse matrix vector multiplications and the evaluations of fluxes. The resulting OpenACC implementation is used to simulate the Taylor-Green vortex which produces a maximum speed-up of 61.3 on a single V100 GPU by compared to serial CPU version.

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Section: Introductionmentioning

confidence: 99%

GPU-acceleration of A High Order Finite Difference Code Using Curvilinear Coordinates

Kupiainen

Gong

Axner

et al. 2020

Proceedings of the 2020 International Conference on Computing, Networks and Internet of Things

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“…The compatibility of discrete integration and differentiation mimics integration-by-parts on a discrete level. Combined with the weak enforcement of boundary conditions via simultaneous approximation terms (SATs) [8], highly efficient and stable semidiscretisations can be obtained, as described also in [12,28,57,76] and references cited therein.…”

Section: Introductionmentioning

confidence: 99%

On conservation and stability properties for summation-by-parts schemes

Nordström

Ruggiu

2017

Journal of Computational Physics

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High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework. However, if generalised SBP operators are used, the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely -interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain. Here, new formulations are proposed, allowing the creation of discretisations using general SBP operators that are both conservative and stable. Thus, several shortcomings that might be attributed to generalised SBP operators are overcome (cf.There are classical finite difference operators that can be interpreted in an analytical setting similar to the one described above [19]. However, in general -to the authors knowledge -it is not known whether finite difference SBP operators correspond to an analytical basis. Nevertheless, the basic requirement of the SBP property (4) can be enhanced by accuracy conditions in order to get a useful definition of (numerical) SBP operators, cf. [11,59].Definition 1. Using a numerical representation u = (u 0 , . . . , u p ) T of a function u in a nodal basis, the operators D , R , M , and B described above form a qth order SBP discretisation, if 1. the derivative matrix D is exact for polynomials of degree q, 2. the mass / norm matrix M is symmetric and positive definite, 3. the SBP property (4) is fulfilled, and

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“…These techniques originate in finite difference (FD) methods and have been used as building blocks of provably stable discretisations, especially for linear (or linearised) problems. Reviews of these schemes as well as historical and recent developments have been published by Fernández, Hicken, and Zingg (2014); Nordström and Eliasson (2015); Svärd and Nordström (2014). Generalised SBP operators have been introduced inter alia by Fernández, Boom, and Zingg (2014); Gassner (2013) and extensions to multiple dimensions not relying on a tensor product structure by Hicken, Fernández, and Zingg (2015); Ranocha (2016).…”

Section: Introductionmentioning

confidence: 99%

Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation

Ranocha,

Glaubitz,

Öffner

et al. 2016

Preprint

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The correction procedure via reconstruction (CPR, also known as flux reconstruction) is a framework of high order semidiscretisations used for the numerical solution of hyperbolic conservation laws. Using a reformulation of these schemes relying on summation-by-parts (SBP) operators and simultaneous approximation terms (SATs), artificial dissipation / spectral viscosity operators are investigated in this first part of a series. Semidiscrete stability results for linear advection and Burgers' equation as model problems are extended to fully discrete stability by an explicit Euler method. As second part of this series, Glaubitz, Ranocha, Öffner, and Sonar (Enhancing stability of correction procedure via reconstruction using summation-byparts operators II: Modal filtering, 2016) investigate connections to modal filters and their application instead of artificial dissipation.

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New Developments for Increased Performance of the SBP-SAT Finite Difference Technique

Cited by 5 publications

References 7 publications

GPU-acceleration of A High Order Finite Difference Code Using Curvilinear Coordinates

GPU-acceleration of A High Order Finite Difference Code Using Curvilinear Coordinates

On conservation and stability properties for summation-by-parts schemes

Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation

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