Compact Central WENO Schemes for Multidimensional Conservation Laws

We present a new third-order central scheme for approximating solutions of systems of conservation laws in one and two space dimensions. In the spirit of Godunov-type schemes, our method is based on reconstructing a piecewise-polynomial interpolant from cell-averages which is then advanced exactly in time. In the reconstruction step, we introduce a new third-order, compact, central weighted essentially nonoscillatory (CWENO) reconstruction, which is written as a convex combination of interpolants based on different stencils. The heart of the matter is that one of these interpolants is taken as a suitable quadratic polynomial, and the weights of the convex combination are set as to obtain third- order accuracy in smooth regions. The embedded mechanism in the WENO-like schemes guarantees that in regions with discontinuities or large gradients, there is an automatic switch to a one-sided second-order reconstruction, which prevents the creation of spurious oscillations. In the one-dimensional case, our new third-order reconstruction is based on an extremely compact three-point stencil. Analogous compactness is retained in more space dimensions. The accuracy, robustness, and high-resolution properties of our scheme are demonstrated in a variety of one- and two-dimensional problems.