Shock capturing by anisotropic diffusion oscillation reduction

TL;DR

The paper addresses robust shock capturing for hyperbolic and near-hyperbolic PDEs by recasting numerical stabilization as anisotropic diffusion guided by edge-detection principles from image processing. It builds a kinetic-theory foundation for artificial viscosity and derives a generalized pressure-tensor framework, then introduces ADOR, combining diffusion operators with edge-enhancing functionals and diffusion-superseded terms, discretized via discrete singular convolution. The method is validated on Burgers' equation in 1D/2D and on incompressible Euler/Navier–Stokes problems, achieving accurate, stable results on relatively coarse grids and leveraging the DSC-ADI scheme for spatial integration. Collectively, ADOR offers a flexible, physically motivated, and computationally efficient approach to capture shocks without excessive smearing or spurious oscillations, with potential extensions to more complex and compressible flows.

Abstract

This paper introduces the method of anisotropic diffusion oscillation reduction (ADOR) 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 is analyzed from the kinetic theory point of view. 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 in one and two spatial dimensions, the incompressible Navier-Stokes equation and the Euler equation. A discrete singular convolution (DSC) algorithm is utilized for the spatial discretization.