A new high-fidelity Total Variation Diminishing scheme based on the Boundary Variation Diminishing principle for compressible flows

Abstract

High-accuracy and stable simulations of compressible flows have been highly demanded in many engineering situations. However, it remains a challenging task due to the complicated fluid structures in compressible flows, which include both continuous and discontinuous solutions. The total variation diminishing (TVD) scheme is a mainstream numerical method because it has second-order accuracy for the smooth solution and suppresses numerical oscillations near the discontinuous solution. The properties of the TVD scheme depend on the design of the second-order slope-limiter function, Φ(r), for which minmod, superbee, and van Leer types are commonly adopted. However, the limiter function that provides both superior numerical accuracy and stability has not yet been established. In this paper, we provide an adaptive model for determining the suitable limiter function in the second-order TVD range, unlike the conventional methods, which use one type of limiter function for the entire calculation region. To designate the suitable limiter value, we use the boundary variation diminishing (BVD) principle, which suggests the most appropriate reconstruction method within the specific candidates. The numerical results of benchmark tests show that the proposed scheme has fewer numerical dissipation errors and provides a suitable limiter value that ensures both accuracy and stability.