2024/01/18 by Chowdhury, Indranil, Jakobsen, Espen Robstad, Lien, Robin Østern
#35A99 (Secondary) #65-02 (Primary) 65M22 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2401.09926
We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order σ∈(0,2) since they involve fractional Laplace operators (-Δ)σ/2. They arise e.g. in control and game theory as dynamic programming equations -- HJB and Isaacs equation -- and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of σ. The accuracy of previous approximations of fractional fully nonlinear equations depend on σ and are worse when σ is close to 2. We show that the schemes are monotone, consistent, L^∞-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.