High-order conservative positivity-preserving DG-interpolation for deforming meshes and application to moving mesh DG simulation of radiative transfer

TL;DR

This paper introduces a high-order, conservative, positivity-preserving DG interpolation scheme for deforming meshes, enhancing moving mesh DG simulations of radiative transfer by maintaining accuracy, efficiency, and positivity.

Contribution

It develops a novel DG-based interpolation method that handles arbitrary mesh deformations while preserving positivity and conservation, improving moving mesh DG simulations.

Findings

Maintains the same convergence order as standard DG methods.

More efficient than fixed uniform mesh approaches.

Successfully preserves positivity of radiative intensity.

Abstract

Solution interpolation between deforming meshes is an important component for several applications in scientific computing, including indirect arbitrary-Lagrangian-Eulerian and rezoning moving mesh methods in numerical solution of partial differential equations. In this paper, a high-order, conservative, and positivity-preserving interpolation scheme is developed based on the discontinuous Galerkin solution of a linear time-dependent equation on deforming meshes. The scheme works for bounded but otherwise arbitrary mesh deformation from the old mesh to the new one. The cost and positivity preservation (with a linear scaling limiter) of the DG-interpolation are investigated. Numerical examples are presented to demonstrate the properties of the interpolation scheme. The DG-interpolation is applied to the rezoning moving mesh DG solution of the radiative transfer equation, an integro-differential equation modeling the conservation of photons and involving time, space, and angular variables. Numerical results obtained for examples in one and two spatial dimensions with various settings show that the resulting rezoning moving mesh DG method maintains the same convergence order as the standard DG method, is more efficient than the method with a fixed uniform mesh, and is able to preserve the positivity of the radiative intensity.