| Abstract: |
| Fractional fully nonlinear PDEs such as Hamilton-Jacobi-Bellman and Isaacs equations arise naturally in optimal control and differential game theory, with many applications in engineering, science, economics, etc. We study discretizations of such equations by \textit{powers of discrete Laplacians}. Our problems are parabolic and of order $\sigma\in(0,2)$ since they involve fractional Laplace operators $(-\Delta)^{\sigma/2}$, 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 $\sigma$. The accuracy of previous approximations of fractional fully nonlinear equations depend on $\sigma$ and are worse when $\sigma$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-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. |
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