Mapped discontinuous Galerkin interpolations and sheared boundary conditions

TL;DR

This paper presents a high-order discontinuous Galerkin method for accurately computing coordinate transformations and sheared boundary conditions in complex, multi-dimensional scalar field simulations, with proven convergence and error analysis.

Contribution

It introduces a novel DG algorithm for boundary conditions involving coordinate transformations, ensuring high-order accuracy and moment preservation in multiple dimensions.

Findings

The algorithm achieves (p+1)-order accuracy in the DG representation.

The method attains (p+2)-order accuracy in cell averages.

It effectively handles complex sheared boundary conditions with quantified errors.

Abstract

Translations or, more generally, coordinate transformations of scalar fields arise in several applications, such as weather, accretion disk and magnetized plasma turbulence modeling. In local studies of accretion disks and magnetized plasmas these coordinate transformations consist of an analytical mapping and enter via sheared-shift boundary conditions. This work introduces a discontinuous Galerkin algorithm to compute these coordinate transformations or boundary conditions based on projections and quadrature-free integrals. The procedure is high-order accurate, preserves certain moments exactly and works in multiple dimensions. Tests of the proposed approach with increasing complexity are presented, beginning with translations of one and two dimensional fields, followed by 3D and 5D simulations with sheared (twist-shift) boundary conditions. The results show that the algorithm is (p+1)-order accurate in the DG representation and (p+2)-order accurate in the cell averages, with p being the order of the polynomial basis functions. Quantification of the algorithm's diffusion and, for shearing boundary conditions, discussion of aliasing errors are provided.