Quantum mechanics

Discretization

Convection–diffusion equation

Applied mathematics

Basis (linear algebra)

Economics

Mathematical optimization

Solver

Classical mechanics

Statistical physics

Thermodynamics

Physics

Boltzmann equation

Mathematics

Angular resolution (graph drawing)

Mathematical analysis

Angular velocity

Geometry

Combinatorics

Finance

Mechanics

Order (exchange)

Basis function

International Journal for Numerical Methods in Engineering

Summary This article presents a new adaptive reduced order model for resolving the angular direction of the Boltzmann transport equation, based on proper orthogonal decomposition (POD) and the method of snapshots. It builds upon previous methods of applying POD to the angular dimension, with modifications to increase accuracy and solver stability. Previous methods used continuous global functions spanning the whole sphere. The new approach, discontinuous POD (DPOD), partitions the surface of the sphere into angular regions, each with an independent set of POD basis functions. Combined, these can approximate flux distributions which span the sphere using optimized basis functions for each angular region. In addition, a novel implementation of adaptive angular resolution known as adaptive discontinuous POD (ADPOD) is presented, which allows the number of DPOD basis functions to vary by angular octant and spatial element. DPOD and ADPOD are applied to two problems in order to demonstrate their benefits compared with POD. Both are shown to reduce the number of solver iterations required to find a solution and decrease the error in the angular flux.

A discontinuous and adaptive reduced order model for the angular discretization of the Boltzmann transport equation