In image processing are nonlinear anisotropic diffusion filters used to construct suitable filter algorithms for denoising, edge enhancement, and edge detection. We applied a nonlinear anisotropic diffusion operator in the context of the numerical solution of a scalar hyperbolic conservation law. It turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher-order dissipative terms. In particular we stabilize a central scheme, known for its oscillatory behavior, by the construction of a nonlinear diffusion term. This means constructing a diffusion matrix consisting of eigenvectors parallel and perpendicular to discontinuities and eigenvalues denoting the amount of dissipation depending on the local strength of the gradients. These directions are used to steer the amount of dissipation, which means suppressing diffusion across the shock front and using the perpendicular direction to enable the necessary diffusion to stabilize the underlying second-order scheme. This new approach allows a multidimensional view to the concept of artificial dissipation. We take the concept of entropy production in the vicinity of shock regions as an indicator for the steering of the diffusion matrix. In smooth regions, which obey an entropy equality instead of an entropy inequality no information about the diffusion direction and strength is needed. So we add as a stabilizing term diffusion parallel to the characteristics, which turns out to be the Lax-Wendroff diffusion rate. In the case of unsteady regions we use the entropy production to blend between this diffusion term and an-still second-order-extra diffusion term with diffusion parallel to the gradient of the entropy production. (C) 2002 Elsevier Science (USA).
