Automatic differentiation through the use of hyper-dual numbers for second derivatives. Automatic differentiation techniques are typically derived based on the chain rule of differentiation. Other methods can be derived based on the inherent mathematical properties of generalized complex numbers that enable first-derivative information to be carried in the non-real part of the number. These methods are capable of producing effectively exact derivative values. However, when second-derivative information is desired, generalized complex numbers are not sufficient. Higher-dimensional extensions of generalized complex numbers, with multiple non-real parts, can produce accurate second-derivative information provided that multiplication is commutative. One particular number system is developed, termed hyper-dual numbers, which produces exact first- and second-derivative information. The accuracy of these calculations is demonstrated on an unstructured, parallel, unsteady Reynolds-averaged Navier-Stokes solver.

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  1. Ouro, Pablo; Fraga, Bruño; Lopez-Novoa, Unai; Stoesser, Thorsten: Scalability of an Eulerian-Lagrangian large-eddy simulation solver with hybrid MPI/OpenMP parallelisation (2019)
  2. Xiao, Lin; Yu, Adams Wei; Lin, Qihang; Chen, Weizhu: DSCOVR: randomized primal-dual block coordinate algorithms for asynchronous distributed optimization (2019)
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  4. Avis, David; Jordan, Charles: \textttmplrs: a scalable parallel vertex/facet enumeration code (2018)
  5. Baty, Hubert; Nishikawa, Hiroaki: A hyperbolic approach for dissipative magnetohydrodynamics (2018)
  6. Biala, T. A.; Khaliq, A. Q. M.: Parallel algorithms for nonlinear time-space fractional parabolic PDEs (2018)
  7. Bonelle, Jérôme; Fournier, Yvan; Moulinec, Charles: New polyhedral discretisation methods applied to the Richards equation: CDO schemes in \textitCode_Saturne (2018)
  8. Cajas, J. C.; Houzeaux, G.; Vázquez, M.; García, M.; Casoni, E.; Calmet, H.; Artigues, A.; Borrell, R.; Lehmkuhl, O.; Pastrana, D.; Yáñez, D. J.; Pons, R.; Martorell, J.: Fluid-structure interaction based on HPC multicode coupling (2018)
  9. Che, Yonggang; Xu, Chuanfu; Wang, Tongtong: Parallelizing and optimizing a detonation combustion simulation application on heterogeneous platform (2018)
  10. Cifani, P.; Kuerten, J. G. M.; Geurts, B. J.: Highly scalable DNS solver for turbulent bubble-laden channel flow (2018)
  11. Creech, Angus C. W.; Jackson, Adrian; Maddison, James R.: Adapting and optimising fluidity for high-fidelity coastal modelling (2018)
  12. Gorobets, A.; Soukov, S.; Bogdanov, P.: Multilevel parallelization for simulating compressible turbulent flows on most kinds of hybrid supercomputers (2018)
  13. Jordan, Charles; Joswig, Michael; Kastner, Lars: Parallel enumeration of triangulations (2018)
  14. Kalantzis, Vassilis; Xi, Yuanzhe; Saad, Yousef: Beyond automated multilevel substructuring: domain decomposition with rational filtering (2018)
  15. Lorenz, Eric; Sivadasan, Vishnu; Bonn, Daniel; Hoekstra, Alfons G.: Combined lattice-Boltzmann and rigid-body method for simulations of shear-thickening dense suspensions of hard particles (2018)
  16. Lunet, Thibaut; Bodart, Julien; Gratton, Serge; Vasseur, Xavier: Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening (2018)
  17. Luo, Xiangyu; Wei, Yingying: Nonparametric Bayesian learning of heterogeneous dynamic transcription factor networks (2018)
  18. Morra, Gabriele: Pythonic geodynamics. Implementations for fast computing (2018)
  19. Neukirchen, Helmut: Elephant against Goliath: performance of big data versus high-performance computing DBSCAN clustering implementations (2018)
  20. Polyakov, S. V.; Podryga, V. O.; Puzyrkov, D. V.: High performance computing in multiscale problems of gas dynamics (2018)

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