A finite volume multilevel WENO scheme for multidimensional scalar conservation laws

We develop a general framework for solving scalar conservation laws using finite volume weighted essentially non oscillatory (WENO) techniques on general computational meshes in multiple space dimensions. We address two fundamental issues. First, polynomial approximations on general stencils of mesh cells can be of poor quality, even for what appear to be geometrically nice stencils. We present a robust and efficient procedure for producing accurate stencil polynomial approximations. Bad stencils are identified by considering the condition number of the linear system used to define the stencil polynomial. Second, we develop a novel and efficient finite volume, multilevel WENO (ML-WENO) reconstruction that is flexible enough to be applied effectively in a variety of settings and with essentially any reasonable set of stencils. It combines stencil polynomial approximations of various degrees with a nonlinear weighting biasing the reconstruction away from both inaccurate oscillatory polynomials of high degree (i.e., those crossing a shock or steep front) and smooth polynomials of low degree, thereby selecting the smooth polynomial(s) of maximal degree of approximation. We conduct numerical tests showing poor quality mesh stencils, the behavior of the reconstruction for both smooth and discontinuous functions, and applications to scalar conservation laws.