Upwind and central WENO schemes

摘要

There are two different ways to construct a WENO (weighted ENO) scheme for numerical solution of hyperbolic conservation laws. The more popular approach is to construct a WENO interpolation directly for computing the interface values of the solution. The resulting hyperbolic solver is upstream central in the smooth regions when the nonlinear weights reduce to the optimal linear weights. We refer it to as upwind WENO scheme. On the other hand, based on the same set of multiple lower-order polynomials, the so-called “Reconstruction via a Primitive Function” approach constructs a WENO interpolation to first approximate the derivative of the primitive function of the solution. The resulting WENO interpolation of the interface values of the solution is symmetric with respect to the interface and leads to a central hyperbolic solver in the smooth regions when the nonlinear weights reduce to the optimal linear weights. We refer it to as central WENO scheme. However, both types of WENO schemes are upwind in the non-smooth regions and therefore stable. The stability of a central WENO scheme implies that upwinding is only needed in the non-smooth regions.