Discretization of fractional fully nonlinear equations by powers of discrete Laplacians
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arXiv
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| Format: | Preprint |
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2024
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| author | Chowdhury, Indranil Jakobsen, Espen Robstad Lien, Robin Østern |
| author_facet | Chowdhury, Indranil Jakobsen, Espen Robstad Lien, Robin Østern |
| contents | We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $σ\in(0,2)$ 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, $L^\infty$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09926 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discretization of fractional fully nonlinear equations by powers of discrete Laplacians Chowdhury, Indranil Jakobsen, Espen Robstad Lien, Robin Østern Numerical Analysis Analysis of PDEs 65-02 (Primary) 65M22, 35A99 (Secondary) We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $σ\in(0,2)$ 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, $L^\infty$-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. |
| title | Discretization of fractional fully nonlinear equations by powers of discrete Laplacians |
| topic | Numerical Analysis Analysis of PDEs 65-02 (Primary) 65M22, 35A99 (Secondary) |
| url | https://arxiv.org/abs/2401.09926 |