High-fidelity numerical methods based on Boundary Variation Diminishing principle

This paper presents a brief review on a novel framework to design high-fidelity numerical schemes for both continuous and discontinuous flow structures in compressible fluid dynamics. This framework is based on the Boundary Variation Diminishing (BVD) principle which requires that the spatial reconstruction minimize the jumps of the reconstructed values at cell boundaries to reduces the dissipation errors in numerical solutions effectively. For the targeted flow structures, one can choose the BVD-admissible functions as the candidates for spatial reconstruction. A BVD algorithm can be then devised according to the BVD principle by properly selecting candidate functions for reconstruction so as to effectively control both numerical oscillation and dissipation. BVD schemes are substantially different from the conventional high-resolution schemes that use the polynomial reconstructions and nonlinear limiting projections to prevent numerical oscillation. We also present some BVD schemes of practical significance. Numerical verifications show that these schemes share the following desirable properties: 1) effectively suppressing spurious numerical oscillation in the presence of strong shock or discontinuity; 2) substantially reducing numerical dissipation errors; 3) retrieving the underlying high-order linear schemes for smooth solutions over all wave numbers; 4) the capability of resolving both smooth and discontinuous flow structures of wide-range scales with substantially improved solution quality; 5) preventing contact discontinuity and material interface from smearing-out even for long-term computation.