An efficient class of WENO schemes with adaptive order for unstructured meshes

Recent advances in finite-difference WENO schemes for hyperbolic conservation laws have resulted in WENO schemes with adaptive order of accuracy. For instance, a WENO-AO(5,3) scheme can provide up to fifth order of accuracy when the smoothness of the solution in the fifth order stencil warrants it, and yet, it can adaptively drop down to third order of accuracy when the higher order is not warranted by the solution on the mesh. Having an analogous capability for finite-volume WENO schemes for hyperbolic conservation laws, especially on unstructured meshes, can be very valuable. The present paper documents the design of finite volume WENO-AO(4,3) and WENO-AO(5,3) schemes for unstructured meshes. As with WENO-AO for structured meshes, the key advance lies in realizing that there is a favorable basis set, which is very easily constructed, and in which the computation is dramatically simplified. As with finite-difference WENO, we realize that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order central WENO scheme that is, nevertheless, very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes that work well on unstructured meshes. On both the large and small stencils we have been able to make the stencil evaluation step very efficient owing to the choice of a favorable Taylor series basis. By extending the Parallel Axis Theorem, we show that there is a significant simplification in the finite volume reconstruction. Instead of solving a constrained least squares problem, our method only requires the solution of a smaller least squares problem on each stencil. This also simplifies the matrix assembly and solution for each stencil. The evaluation of smoothness indicators is also simplified. Accuracy tests show that the method meets its design accuracy. Several stringent test problems are presented to demonstrate that the method works very robustly and very well. The test problems are chosen to show that our method can be applied to many different meshes that are used to map geometric complexity or solution complexity.