Boundary Variation Diminishing (BVD) reconstruction: a new approach to improve Godunov scheme

TL;DR

This paper introduces the Boundary Variation Diminishing (BVD) reconstruction method, which reduces numerical dissipation in Godunov schemes by allowing discontinuities within cells, leading to improved accuracy for both smooth and discontinuous solutions.

Contribution

It proposes a novel BVD strategy combining high-order polynomial interpolation with jump-like reconstructions, challenging the assumption that discontinuities only occur at cell interfaces.

Findings

Significantly improved solution quality for scalar and Euler laws.

Effective reduction of numerical dissipation in Godunov schemes.

Versatile approach for both continuous and discontinuous solutions.

Abstract

This paper presents a new approach, so-called boundary variation diminishing (BVD), for reconstructions that minimize the discontinuities (jumps) at cell interfaces in Godunov type schemes. It is motivated by the observation that diminishing the jump at the cell boundary might effectively reduce the dissipation in numerical flux. Different from the existing practices which seek high-order polynomials within mesh cells while assuming discontinuities being always at the cell interfaces, we proposed a new strategy that combines a high-order polynomial-based interpolation and a jump-like reconstruction that allows a discontinuity being partly represented within the mesh cell rather than at the interface. It is shown that new schemes of high fidelity for both continuous and discontinuous solutions can be devised by the BVD guideline with properly-chosen candidate reconstruction schemes. Excellent numerical results have been obtained for both scalar and Euler conservation laws with substantially improved solution quality in comparison with the existing methods. This work provides a simple and accurate alternative of great practical significance to the current Godunov paradigm which overly pursues the smoothness within mesh cell under the questionable premiss that discontinuities only appear at cell interfaces.