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

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 with 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.Using the BVD algorithm, several numerical tests are conducted for selecting an appropriate function between the original high-order polynomial and the WENO reconstruction approximating either smooth or discontinuous profiles. The selected functions based on the BVD algorithm are not always the same as those by the conventional TVB limiter, which indicates that the present BVD algorithm offers a radically new criteria for selecting a reconstruction function without any ad hoc TVB parameter. Subsequently, the computation of a linear advection equation is examined for the BVD and TVB criteria with the WENO methodology in the FR framework. The results of the BVD algorithm are comparable to those using the conventional TVB limiter in terms of oscillation suppression and numerical dissipation, which would be also possible in the system equations such as Euler equations. Furthermore, since the present BVD algorithm does not need any ad hoc constant such as the TVB parameter, it could be more reliable than the conventional TVB limiter that is prevailing in the existing DG and FR approaches for shocks and other discontinuities. Note that the present work is limited to third or lower order polynomials, leaving the implementations for higher-order schemes a future work.