A weighted-integral based scheme

A weighted-integral based scheme (WIBS) and a weighted essentially non-oscillatory (WENO)-WIBS are constructed, where the integral of the unknown function with a set of linearly independent test functions are recorded on every cell. The time evolutions of these recordings are computed with TVD Runge–Kutta method. At the boundary of every two cells, the function values are interpolated from the recordings of the neighboring cells to calculate flux and volumetric integral in the weak form.Our basic idea is to increase the order of interpolation by increasing both the interpolating cells and cell recordings simultaneously. The interpolation on more cells naturally permits the use of WENO idea to capture the discontinuity, while more cell recordings can shrink the size of the interpolating stencil. The compactness of the reconstruction stencil can increase the accuracy and fully retain it at the boundary. The WIBS so constructed may include as special cases a quite general class of the numerical methods in computational fluid dynamics, such as finite-volume method, finite difference method, discontinuous Galerkin scheme, spectral volume method, spectral difference method, finite element method, and PnPm scheme recently designed by Dumbser et al. [Journal of Computational Physics 227 (2008) 8209–8253], etc.In this paper the property of WIBS and WENO-WIBS on one-dimensional hyperbolic conservation-law systems is systematically explored. In addition to the high stability and order of accuracy for smooth region, the WENO-WIBS exhibits high resolution and non-oscillatory property in capturing the discontinuity. The numerical experiments of WIBS and WENO-WIBS on various benchmark problems are favorably compared with the results obtained by other high-order methods.