Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems

Abstract In this article we present a quadrature-free essentially non-oscillatory finite volume scheme of arbitrary high order of accuracy both in space and time for solving nonlinear hyperbolic systems on unstructured meshes in two and three space dimensions. For high order spatial discretization, a WENO reconstruction technique provides the reconstruction polynomials in terms of a hierarchical orthogonal polynomial basis over a reference element. The Cauchy–Kovalewski procedure applied to the reconstructed data yields for each element a space–time Taylor series for the evolution of the state and the physical fluxes. This Taylor series is then inserted into a special numerical flux across the element interfaces and is subsequently integrated analytically in space and time. Thus, the Cauchy–Kovalewski procedure provides a natural, direct and cost-efficient way to obtain a quadrature-free formulation, avoiding the expensive numerical quadrature arising usually for high order finite volume schemes in three space dimensions. We show numerical convergence results up to sixth order of accuracy in space and time for the compressible Euler equations on triangular and tetrahedral meshes in two and three space dimensions. Furthermore, various two- and three-dimensional test problems with smooth and discontinuous solutions are computed to validate the approach and to underline the non-oscillatory shock-capturing properties of the method.