Discontinuity-resolving shock-capturing schemes on unstructured grids

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.