Meshless discretization of the discrete-ordinates transport equation with integration based on Voronoi cells

TL;DR

This paper introduces a meshless discretization method for the time-dependent radiation transport equation using reproducing kernels and Voronoi tessellation, enabling automatic integration resolution and improved numerical stability.

Contribution

It presents a novel meshless discretization approach with Voronoi-based integration and stabilization techniques for the transport equation, enhancing flexibility and accuracy.

Findings

First-order convergence in time

Second-order convergence in space

Effective stabilization with streamline-upwind Petrov-Galerkin

Abstract

The time-dependent radiation transport equation is discretized using the meshless-local Petrov-Galerkin method with reproducing kernels. The integration is performed using a Voronoi tessellation, which creates a partition of unity that only depends on the position and extent of the kernels. The resolution of the integration automatically follows the particles and requires no manual adjustment. The discretization includes streamline-upwind Petrov-Galerkin stabilization to prevent oscillations and improve numerical conditioning. The angular quadrature is selectively refineable to increase angular resolution in chosen directions. The time discretization is done using backward Euler. The transport solve for each direction and the solve for the scattering source are both done using Krylov iterative methods. Results indicate first-order convergence in time and second-order convergence in space for linear reproducing kernels.