Stabilised Nonlinear Inverse Diffusion for Approximating Hyperbolic PDEs

Stabilised backward diffusion processes have shown their use for a number of image enhancement tasks. The goal of this paper is to show that they are also highly useful for designing shock capturing numerical schemes for hyperbolic conservation laws. We propose and investigate a novel flux corrected transport (FCT) type algorithm. It is composed of an advection step capturing the flow dynamics, and a stabilised nonlinear backward diffusion step in order to improve the resolution properties of the scheme. In contrast to classical FCT procedures, we base our method on an analysis of the discrete viscosity form. This analysis shows that nonlinear backward diffusion is necessary. We employ a slope limiting type approach where the antidiffusive flux determined by the viscosity form is controlled by a limiter that prohibits oscillations. Numerical experiments confirm the high accuracy and shock capturing properties of the resulting scheme. This shows the fruitful interaction of PDE-based image processing ideas and numerical analysis.