Adaptive angle and parallel multigrid for deterministic shielding problems

Traditionally, solver technology and the space/angle discretisations are intimately linked; sweep-based (wavefront) methods are typically used with DG FEM in space and Sn in angle to solve the Boltzmann-transport equation. These parallelise well (scaling to >100,000 cores) on structured grids, however achieving good scaling on unstructured grids is still an open problem. This talk will focus on alternate space/angle discretisations and solver technology we have been developing within the Applied Modelling and Computation Group (AMCG) at Imperial College. These approaches enables the use of traditional angular discretisations like Pn, Sn, along with new approaches based on linear and haar wavelets. We can use these angular discretisations to perform regular and goal-based anisotropic adaptivity in angle, focusing resolution in important directions. We have also been developing multigrid solver technology which does not require sweep-based methods, allowing the possibility of excellent scaling on unstructured grids.

[1]  T. A. Manteuffel,et al.  An Angular Multigrid Acceleration Technique for Sn Equations with Highly Forward-Peaked Scattering , 1991 .

[2]  G. Yang,et al.  A Fast Multigrid Algorithm for Isotropic Transport Problems I: Pure Scattering , 1995, SIAM J. Sci. Comput..

[3]  Barry Lee.  Improved Multiple-Coarsening Methods for Sn Discretizations of the Boltzmann Equation , 2010, SIAM J. Sci. Comput..

[4]  Thomas A. Manteuffel,et al.  A Moment-Parity Multigrid Preconditioner for the First-Order System Least-Squares Formulation of the Boltzmann Transport Equation , 2003, SIAM J. Sci. Comput..

[5]  R. P. Smedley-Stevenson,et al.  Goal-based angular adaptivity applied to the spherical harmonics discretisation of the neutral particle transport equation , 2014 .

[6]  Thomas J. R. Hughes,et al.  A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method , 2006 .

[7]  Guido Kanschat,et al.  A Robust Multigrid Preconditioner for SNDG Approximation of Monochromatic, Isotropic Radiation Transport Problems , 2014, SIAM J. Sci. Comput..

[8]  Suely Oliveira,et al.  Preconditioned Krylov subspace methods for transport equations , 1998 .

[9]  Christopher C. Pain,et al.  Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes , 2013, J. Comput. Phys..

[10]  Barry Lee,et al.  Space‐angle‐energy multigrid methods for Sn discretizations of the multi‐energetic Boltzmann equation , 2012, Numer. Linear Algebra Appl..

[11]  Thomas M. Evans,et al.  Multigrid in energy preconditioner for Krylov solvers , 2013, J. Comput. Phys..

[12]  Jim E. Morel,et al.  Angular Multigrid Preconditioner for Krylov-Based Solution Techniques Applied to the Sn Equations with Highly Forward-Peaked Scattering , 2012 .

[13]  G. Yang,et al.  A Fast Multigrid Algorithm for Isotropic Transport Problems. II: With Absorption , 1996, SIAM J. Sci. Comput..

[14]  Christopher C. Pain,et al.  Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation , 2015, J. Comput. Phys..

[15]  Mark A. Goffin,et al.  Solving the Boltzmann transport equation with multigrid and adaptive space/angle discretisations , 2015 .

[16]  B. CHANG,et al.  Spatial Multigrid for Isotropic Neutron Transport , 2007, SIAM J. Sci. Comput..

[17]  HyeongKae Park,et al.  Coupled Space-Angle Adaptivity for Radiation Transport Calculations , 2009 .

[18]  Barry Lee.  A Novel Multigrid Method for Sn Discretizations of the Mono-Energetic Boltzmann Transport Equation in the Optically Thick and Thin Regimes with Anisotropic Scattering, Part I , 2010, SIAM J. Sci. Comput..

[19]  T. Grasser,et al.  Adaptive variable-order spherical harmonics expansion of the Boltzmann Transport Equation , 2011, 2011 International Conference on Simulation of Semiconductor Processes and Devices.