Upwind Filtering of Scalar Conservation Laws

We study a class of multidimensional nonlocal conservation laws of the form \partial tu= div\Phi F(u), where the standard local divergence div of the flux vector F(u) is replaced by an average upwind divergence operator div\Phi acting on the flux along a continuum of directions given by a reference measure and a filter \Phi . The nonlocal operator div\Phi applies to a general nonmonotone flux F, and is constructed by decomposing the flux into monotone components according to wave speeds determined by F\prime . Each monotone component is then consistently subjected to a nonlocal derivative operator that utilizes an anisotropic kernel supported on the ``correct"" half of the real axis. We establish well-posedness, derive a priori and entropy estimates, and provide an explicit continuous dependence result on the kernel. This stability result is robust with respect to the size of the kernel, allowing us to specify \Phi as a Dirac delta \delta 0 to recover entropy solutions of the local conservation law \partial tu= div F(u) (with an error estimate). Other choices of \Phi (and the reference measure) recover known numerical methods for (local) conservation laws. This work distinguishes itself from many others in the field by developing a consistent nonlocal approach capable of handling nonmonotone fluxes.