High order schemes based on upwind schemes with modified coefficients

The upwind schemes play an important role in CFD. In this paper, some high order schemes are constructed without expanding the stencil and by modifying coefficients (MC) of the upwind schemes. According to our theoretical analysis, we show that MC approach preserves the desirable properties which the underlying schemes possess. We apply these new schemes to the linear scalar equation, Burgers equation and the hyperbolic system of conservation laws for simulating Rayleigh–Taylor instability, and show that MC approach increases the accurate order and improves efficiency in comparison with the underlying scheme.