A new finite volume multi-resolution central WENO scheme for Navier–Stokes equations on staggered meshes

Abstract

In this paper, a new fifth-order finite volume multi-resolution central WENO (MRC-WENO) scheme is proposed for solving Navier–Stokes equations on staggered meshes. A series of high-order spatial reconstruction procedures using unequal-sized stencils and a time discretization method with the natural continuous extension (NCE) of the Runge–Kutta method are applied to design this scheme. By gradually deteriorating from the fifth-order to the third-order, and eventually to the first-order in non-smooth areas, this new MRC-WENO method may preserve the original fifth-order accuracy in smooth regions while maintaining sharp shock transitions without solving any local Riemann problems on staggered meshes. Unlike many traditional central WENO schemes which employ many equal-sized stencils on staggered meshes, the new MRC-WENO scheme employs smaller unequal-sized stencils, easily makes the choice of the stencils, and uses arbitrary positive linear weights provided their summation is one. This is the first time that only three unequal-sized hierarchical central spatial stencils are used in multi-dimensional spatial reconstruction procedures on staggered meshes. This MRC-WENO scheme is simple to construct and can be easily implemented to arbitrary high-order accuracy in higher dimensions. And it is suitable for large scale engineering applications. Some benchmark one-dimensional and two-dimensional viscous examples are supplied to demonstrate the good performance of this new finite volume MRC-WENO scheme on staggered meshes.