Shock capturing by anisotropic diffusion oscillation reduction

Abstract

This paper introduces an anisotropic diffusion oscillation reduction (ADOR) scheme for shock wave computations. The connection is made between digital image processing, in particular, image edge detection, and numerical shock capturing. Indeed, numerical shock capturing can be formulated on the lines of iterative digital edge detection. Various anisotropic diffusion and super diffusion operators originated from image edge detection are proposed for the treatment of hyperbolic conservation laws and near-hyperbolic hydrodynamic equations of change. The similarity between anisotropic diffusion and artificial viscosity is discussed. Physical origins and mathematical properties of the artificial viscosity are analyzed from the point of view of kinetic theory. A form of pressure tensor is derived from the first principles of the quantum mechanics. Quantum kinetic theory is utilized to arrive at macroscopic transport equations from the microscopic theory. Macroscopic symmetry is used to simplify pressure tensor expressions. The latter provides a basis for the design of artificial viscosity. The ADOR approach is validated by using (inviscid) Burgers' equation, the gas tube problems, the incompressible Navier-Stokes equation and the Euler equation. Both standard central difference schemes and a discrete singular convolution algorithm are utilized to illustrate the approach. Results are compared with those of third-order upwind scheme and essentially non-oscillatory (ENO) scheme. © 2002 Published by Elsevier Science B.V.