Constructing higher order discontinuity-capturing schemes with upwind-biased interpolations and boundary variation diminishing algorithm

• Published in: Computers & fluids 2020-03, Vol.200, p.104433, Article 104433
• Main Authors: Deng, Xi, Shimizu, Yuya, Xie, Bin, Xiao, Feng
• Format: Article
• Language: English
• Subjects: Algorithms; Boundary variation diminishing; Conservation laws; Discontinuity; Engineering Sciences; Finite volume method; Fluids mechanics; Hyperbolas; Mechanics; Polynomials; Reconstruction; Shock capturing; Slopes; Upwind schemes (mathematics); Very high order schemes
• ISSN: 0045-7930; 1879-0747
• DOI: 10.1016/j.compfluid.2020.104433

•A new general framework for constructing high order schemes.•Low dissipation linear scheme is retrieved over all wave numbers.•Sharp discontinuities are captured with suppressed oscillations.•Higher order is realized in an efficient way. Based on the fifth-order scheme in our previous work (Deng et. al (2019) [28]), a new framework of constructing very high order discontinuity-capturing schemes is proposed for finite volume method. These schemes, so-called PnTm−BVD (polynomial of n-degree and THINC function of m-level reconstruction based on BVD algorithm), are designed by employing high-order upwind-biased interpolations and THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive steepness as the reconstruction candidates. The final reconstruction function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. We devise the new schemes up to eleventh order in an efficient way by directly increasing the order of the underlying upwind scheme using high order polynomials. The analysis of the spectral property and accuracy tests show that the new reconstruction strategy well preserves the low-dissipation property of the underlying upwind schemes with high-order polynomials for smooth solution over all wave numbers and realizes n+1 order convergence rate. The performance of new schemes is examined through widely used benchmark tests, which demonstrate that the proposed schemes are capable of simultaneously resolving small-scale flow features with high resolution and capturing discontinuities with low dissipation. With outperforming results and simplicity in algorithm, the new reconstruction strategy shows great potential as an alternative numerical framework for computing nonlinear hyperbolic conservation laws that have discontinuous and smooth solutions of different scales.