  A novel 5th-order shock capturing scheme is presented in this paper. The
scheme, so-called P4-THINC-BVD (4th degree polynomial and THINC reconstruction
based on BVD algorithm), is formulated as a two-stage cascade BVD (Boundary
Variation Diminishing) algorithm following the BVD principle that minimizes the
jumps of reconstructed values at cell boundaries. In the P4-THINC-BVD scheme,
polynomial of degree four and THINC (Tangent of Hyperbola for INterface
Capturing) functions with adaptive steepness are used as the candidate
reconstruction functions. The final reconstruction function is selected from
the candidate functions by a two-stage cascade BVD algorithm so as to
effectively control numerical oscillation and dissipation. Spectral analysis
and numerical verifications show that the P4-THINC-BVD scheme possesses the
following desirable properties: 1) it effectively suppresses spurious numerical
oscillation in the presence of strong shock or discontinuity; 2) it
substantially reduces numerical dissipation errors; 3) it automatically
retrieves the underlying linear 5th-order upwind scheme for smooth solution
over all wave numbers; 4) it is able to resolve both smooth and discontinuous
flow structures of all scales with substantially improved solution quality in
comparison to other existing methods; and 5) it faithfully maintains the
free-mode solutions in long term computation. P4-THINC-BVD, as well as the
underlying idea presented in this paper, provides an innovative and practical
approach to design high-fidelity numerical schemes for compressible flows
involving strong discontinuities and flow structures of wide range scales.
