Anti-diffusive flux corrections for high order finite difference WENO schemes

In the first part of this thesis work, we generalize a technique of anti-diffusive flux corrections, recently reintroduced by Despres and Lagoutiere for first order schemes, to high order finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes. The objective is to obtain sharp resolution for contact discontinuities in the solutions to conservation law equations, close to the quality of discrete travelling waves which do not smear progressively for longer time, while maintaining high order accuracy in smooth regions and non-oscillatory property for discontinuities. Numerical examples for one and two space dimensional scalar problems and systems demonstrate the good quality of this flux correction. High order accuracy is maintained and contact discontinuities are sharpened significantly compared with the original WENO schemes on the same meshes. In the second part, as an application of the anti-diffusive WENO schemes we developed, we solve the Saint-Venant shallow water equations with the transport of a pollutant by a flow. The position and the concentration of the pollutant is exactly located and correctly determined. In the third part, we extend the idea of flux corrections for conservation law equations to numerical Hamiltonian corrections for Hamilton-Jacobi type equations. The motivation is to obtain sharp resolution for kinks, or discontinuities in the derivatives, in the viscosity solutions to Hamilton-Jacobi equations. The tight link between conservation law equations and Hamilton-Jacobi equations makes this extension possible. Numerical experiments for one and two space dimensional problems show that both expectations, high order accuracy in smooth region and sharp resolution of kinks, are satisfied.