Third-order Limiting for Hyperbolic Conservation Laws applied to Adaptive Mesh Refinement and Non-Uniform 2D Grids

TL;DR

This paper extends a third-order limiter function for finite volume schemes to non-uniform and adaptive 2D grids, ensuring high accuracy and oscillation control in complex computational fluid dynamics simulations.

Contribution

The work generalizes the third-order limiter to non-uniform and adaptive grids in 1D and 2D, incorporating an order-fix to preserve third-order accuracy.

Findings

Maintains third-order accuracy on smooth profiles.

Avoids oscillations near discontinuities.

Effective on various grid configurations.

Abstract

In this paper we extend the recently developed third-order limiter function $H_{3\text{L}}^{(c)}$ [J. Sci. Comput., (2016), 68(2), pp.~624--652] to make it applicable for more elaborate test cases in the context of finite volume schemes. This work covers the generalization to non-uniform grids in one and two space dimensions, as well as two-dimensional Cartesian grids with adaptive mesh refinement (AMR). The extension to 2D is obtained by the common approach of dimensional splitting. In order to apply this technique without loss of third-order accuracy, the order-fix developed by Buchm\"uller and Helzel [J. Sci. Comput., (2014), 61(2), pp.~343--368] is incorporated into the scheme. Several numerical examples on different grid configurations show that the limiter function $H_{3\text{L}}^{(c)}$ maintains the optimal third-order accuracy on smooth profiles and avoids oscillations in case of discontinuous solutions.