Upwind filtering of scalar conservation laws

Abstract

We study a class of multi-dimensional non-local conservation laws of the form $\partial_t u = \operatorname{div}^{\Phi} \mathbf{F}(u)$, where the standard local divergence $\operatorname{div}$ of the flux vector $\mathbf{F}(u)$ is replaced by an average upwind divergence operator $\operatorname{div}^{\Phi}$ acting on the flux along a continuum of directions given by a reference measure and a filter $\Phi$. The non-local operator $\operatorname{div}^{\Phi}$ applies to a general non-monotone flux $\mathbf{F}$, and is constructed by decomposing the flux into monotone components according to wave speeds determined by $\mathbf{F}'$. Each monotone component is then consistently subjected to a non-local 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 $\partial_t u = \operatorname{div} \mathbf{F}(u)$ (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 non-local approach capable of handling non-monotone fluxes.