  Solving compressible flows containing discontinuities remains a major
challenge for numerical methods especially on unstructured grids. Thus in this
work, we make contributions to shock capturing schemes on unstructured grids
with aim of resolving discontinuities with low numerical dissipation. Different
from conventional shock capturing schemes which only use polynomials as
interpolation functions on unstructured grids, the proposed scheme employs the
linear polynomial as well as non-polynomial as reconstruction candidates. For
linear polynomial, the second order MUSCL scheme with the MLP
(Multi-dimensional Limiting Process) slope limiter is adopted. The
multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function
with quadratic surface representation and Gaussian quadrature, so-called
THINC/QQ, is used as the non-polynomial reconstruction candidate. With these
reconstruction candidates, a multi-stage boundary variation diminishing (BVD)
algorithm which aims to minimize numerical dissipation is designed on
unstructured grids to select the final reconstruction function. The resulted
shock capturing scheme is named as MUSCL-THINC/QQ-BVD. The performance of the
proposed scheme is demonstrated through solving compressible single-phase and
multi-phase problems where the discontinuity is the typical flow structure. The
numerical results show that the proposed scheme is capable of capturing sharp
discontinuous profiles without numerical oscillations as well as resolving
vortices associated with Kelvin-Helmholtz instabilities along shear layers and
material interfaces. In comparison with schemes only replying on high order
polynomials, the proposed scheme shows significant improvement of resolution
across discontinuities. Thus, this work provides an accurate and robust
shock-capturing scheme to resolve discontinuities in compressible flows.
