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In this study, a new framework of constructing very high orderdiscontinuity-capturing schemes is proposed for finite volume method. Theseschemes, so-called $\mathrm{P}_{n}\mathrm{T}_{m}-\mathrm{BVD}$ (polynomial of$n$-degree and THINC function of $m$-level reconstruction based on BVDalgorithm), are designed by employing high-order linear-weight polynomials andTHINC (Tangent of Hyperbola for INterface Capturing) functions with adaptivesteepness as the reconstruction candidates. The final reconstruction functionin each cell is determined with a multi-stage BVD (Boundary VariationDiminishing) algorithm so as to effectively control numerical oscillation anddissipation. We devise the new schemes up to eleventh order in an efficient wayby directly increasing the order of the underlying upwind scheme usinglinear-weight polynomials. The analysis of the spectral property and accuracytests show that the new reconstruction strategy well preserves thelow-dissipation property of the underlying upwind schemes with high-orderlinear-weight polynomials for smooth solution over all wave numbers andrealizes $n+1$ order convergence rate. The performance of new schemes isexamined through widely used benchmark tests, which demonstrate that theproposed schemes are capable of simultaneously resolving small-scale flowfeatures with high resolution and capturing discontinuities with lowdissipation. With outperforming results and simplicity in algorithm, the newreconstruction strategy shows great potential as an alternative numericalframework for computing nonlinear hyperbolic conservation laws that havediscontinuous and smooth solutions of different scales.

Constructing high-order discontinuity-capturing schemes with linear-weight polynomials and boundary variation diminishing algorithm