An Efficient Filtered Scheme for Some First Order Time-Dependent Hamilton-Jacobi Equations

We introduce a new class of “filtered” schemes for some first order non-linear Hamilton-Jacobi equations. The work follows recent ideas of Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423-444, 2013) and Oberman and Salvador (J. Comput. Phys., Vol 284, pp. 367-388, 2015) for steady equations. Here we mainly study the time-dependent setting and focus on fully explicit schemes. Furthermore, specific corrections to the filtering idea are also needed in order to obtain high-order efficiency. The proposed schemes are not monotone but still satisfy some -monotone property. A general convergence result together with a precise error estimate of order √ ∆x are given (∆x is the mesh size). The framework allows to construct finite difference discretizations that are easy to implement and high–order in the domain where the solution is smooth. A novel error estimate is also given in the case of the approximation of steady equations. Numerical tests including evolutive conve...

• Type: Journal
• Year: 2016
• Where: SIAMSC
• Authors: Olivier Bokanowski, Maurizio Falcone, Smita Sahu
• Keywords: Computer Graphics, SIAMSC 2016