The first-order Godunov scheme was invented in 1950s to solve compressible fluid dynamics. Tremendous efforts in the past sixty years have made the Godunov-type finite volume method a mainstream numerical methodology to compute compressible flows that involve both smooth and discontinuous flow structures. Nonetheless, the currently available high-resolution schemes of this sort, using polynomial-based reconstructions and nonlinear limiting projections to compromise between spurious oscillation and numerical dissipation, can hardly provide adequate solution quality for either smooth or discontinuous solution due to excessive numerical dissipation.
In this talk, we will present a novel principle, so-called Boundary Variation Diminishing (BVD) principle, to design numerical schemes which are able to capture both smooth and discontinuous flow structures with superior solution quality. The BVD principle minimizes the jumps of the reconstructed physical variables at cell boundaries, and thus effectively reduces the dissipation errors, which works a general guideline to optimize numerical solutions. More profoundly, the BVD formulation provides an alternative to the conventional limiting-projection approach to eliminate numerical oscillation. With proper BVD-admissible functions and BVD algorithms, we have developed a new class of numerical schemes of great practical significance for compressible flows. The numerical schemes have been extensively verified with various benchmark tests of single and multiphase compressible flows involving strong discontinuities and complex flow structure of broad range scales. Numerical results show that the present method can simulate compressible flows with greatly improved solution quality in comparison with other existing methods.