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. Briggs, J.P.; Pennycook, S.J.; Fergusson, J.R.; Jäykkä, J.; Shellard, E.P.S.: Separable projection integrals for higher-order correlators of the cosmic microwave sky: acceleration by factors exceeding 100 (2016)
  2. Fannon, James; Loiseau, Jean-Christophe; Valluri, Prashant; Bethune, Iain; Náraigh, Lennon Ó.: High-performance computational fluid dynamics: a custom-code approach (2016)
  3. Tamain, P.; Bufferand, H.; Ciraolo, G.; Colin, C.; Galassi, D.; Ghendrih, Ph.; Schwander, F.; Serre, E.: The TOKAM3X code for edge turbulence fluid simulations of tokamak plasmas in versatile magnetic geometries (2016)
  4. Veit, Alexander; Merta, Michal; Zapletal, Jan; Lukáš, Dalibor: Efficient solution of time-domain boundary integral equations arising in sound-hard scattering (2016)
  5. White, Robert E.: Computational mathematics. Models, methods, and analysis with MATLAB and MPI (2016)
  6. Zhang, Wei; Cai, Xing: Solving 3D time-fractional diffusion equations by high-performance parallel computing (2016)
  7. Aldasoro, Unai; Escudero, Laureano F.; Merino, María; Monge, Juan F.; Pérez, Gloria: On parallelization of a stochastic dynamic programming algorithm for solving large-scale mixed $0-1$ problems under uncertainty (2015)
  8. Beskos, Alexandros; Jasra, Ajay; Muzaffer, Ege A.; Stuart, Andrew M.: Sequential Monte Carlo methods for Bayesian elliptic inverse problems (2015)
  9. Casoni, E.; Jérusalem, A.; Samaniego, C.; Eguzkitza, B.; Lafortune, P.; Tjahjanto, D.D.; Sáez, X.; Houzeaux, G.; Vázquez, M.: Alya: computational solid mechanics for supercomputers (2015)
  10. Eckstein, Jonathan; Hart, William E.; Phillips, Cynthia A.: PEBBL: an object-oriented framework for scalable parallel branch and bound (2015)
  11. Galizia, Antonella; D’Agostino, Daniele; Clematis, Andrea: An MPI-CUDA library for image processing on HPC architectures (2015)
  12. Garain, Sudip; Balsara, Dinshaw S.; Reid, John: Comparing coarray Fortran (CAF) with MPI for several structured mesh PDE applications (2015)
  13. Gonnet, Pedro: Efficient and scalable algorithms for smoothed particle hydrodynamics on hybrid shared/distributed-memory architectures (2015)
  14. Gorobets, A.V.: Parallel technology for numerical modeling of fluid dynamics problems by high-accuracy algorithms (2015)
  15. Guo, Z.; Xiong, S.M.: On solving the 3-D phase field equations by employing a parallel-adaptive mesh refinement (para-AMR) algorithm (2015)
  16. Mareček, Jakub; Richtárik, Peter; Takáč, Martin: Distributed block coordinate descent for minimizing partially separable functions (2015)
  17. Mason, Luke R.; Mak-Hau, Vicky H.; Ernst, Andreas T.: A parallel optimisation approach for the realisation problem in intensity modulated radiotherapy treatment planning (2015)
  18. Mortensen, Mikael; Valen-Sendstad, Kristian: Oasis: a high-level/high-performance open source Navier-Stokes solver (2015)
  19. Müthing, Steffen; Ribbrock, Dirk; Göddeke, Dominik: Integrating multi-threading and accelerators into DUNE-ISTL (2015)
  20. Oguic, Romain; Viazzo, Stéphane; Poncet, Sébastien: A parallelized multidomain compact solver for incompressible turbulent flows in cylindrical geometries (2015)

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