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 nonlinear Hamilton--Jacobi equations. The work follows recent ideas of Froese and Oberman [SIAM J. Numer. Anal., 51 (2013), pp. 423--444] and Oberman and Salvador [J. Comput. Phys., 284 (2015), pp. 367--388] 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 accuracy. The proposed schemes are not monotone but still satisfy some $\epsilon$-monotone property. A general convergence result together with a precise error estimate of order $\sqrt{\Delta x}$ are given ($\Delta x$ is the mesh size). The framework allows us 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 convex and nonconvex Hamiltonians, and obstacle problems are presented to validate the approach. We show with several examples how the filter technique can be applied to stabilize an otherwise unstable high-order scheme.


Read More: https://epubs.siam.org/doi/10.1137/140998482