A re-averaged WENO reconstruction and a third order CWENO scheme for hyperbolic conservation laws

A WENO re-averaging (or re-mapping) technique is developed that converts function averages on one grid to another grid to high order. Nonlinear weighting gives the essentially non-oscillatory property to the re-averaged function values. The new reconstruction grid is used to obtain a standard high order WENO reconstruction of the function averages at a select point. By choosing the reconstruction grid to include the point of interest, a high order function value can be reconstructed using only positive linear weights. The re-averaging technique is applied to define two variants of a classic CWENO3 scheme that combines two linear polynomials to obtain formal third order accuracy. Such a scheme cannot otherwise be defined, due to the nonexistence of linear weights for third order reconstruction at the center of a grid element. The new scheme uses a compact stencil of three solution averages, and only positive linear weights are used. The scheme extends easily to problems in higher space dimensions, essentially as a tensor product of the one-dimensional scheme. The scheme maintains formal third order accuracy in higher dimensions. Numerical results show that this CWENO3 scheme is third order accurate for smooth problems and gives good results for non-smooth problems, including those with shocks.