A note on implementation of boundary variation diminishing algorithm to high-order local polynomial-based schemes

TL;DR

This paper explores the application of the boundary variation diminishing (BVD) algorithm to high-order local polynomial schemes like DG and FR, aiming to improve the stability and accuracy of capturing discontinuities without ad hoc parameters.

Contribution

It introduces the BVD reconstruction method within the FR framework, demonstrating its effectiveness in selecting stable approximations without relying on traditional TVB parameters.

Findings

BVD effectively reduces boundary jumps in polynomial reconstructions.

Results are comparable to traditional TVB limiters in test cases.

Applicable to third or lower order polynomials, with potential for higher orders in future work.

Abstract

A novel approach for selecting appropriate reconstructions is implemented to the hyperbolic conservation laws in the high-order local polynomial-based framework, e.g., the discontinuous Galerkin (DG) and flux reconstruction (FR) schemes. The high-order polynomial approximation generally fails to correctly capture a strong discontinuity inside a cell due to the Runge phenomenon, which is replaced by more stable approximation on the basis of a troubled-cell indicator such as that used in the total variation bounded (TVB) limiter. This paper examines the applicability of a new algorithm, so-called boundary variation diminishing (BVD) reconstruction, to the weighted essentially non-oscillatory (WENO) methodology in the FR framework including the nodal type DG method. The BVD reconstruction adaptively chooses a proper approximation for the solution function so as to minimize the jump between values at the left- and right-side of cell boundaries without any ad hoc constant such as the TVB parameter. Several numerical tests are conducted for a linear advection equation as well as the selection of an appropriate reconstruction, where the results of the BVD algorithm are comparable to those using the conventional TVB limiter. Note that the present work is limited to third or lower order polynomials, leaving the implementations for higher-order schemes a future work.