Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations
We study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in th...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Springer
2021
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| _version_ | 1826306600757886976 |
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| author | Bokanowski, O Picarelli, A Reisinger, C |
| author_facet | Bokanowski, O Picarelli, A Reisinger, C |
| author_sort | Bokanowski, O |
| collection | OXFORD |
| description | We study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the L2 norm for linear and semi-linear equations, and in the H1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in L2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms. |
| first_indexed | 2024-03-07T06:50:24Z |
| format | Journal article |
| id | oxford-uuid:fc608d0f-1589-4231-9345-4e6af962805e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:50:24Z |
| publishDate | 2021 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:fc608d0f-1589-4231-9345-4e6af962805e2022-03-27T13:20:10ZStability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fc608d0f-1589-4231-9345-4e6af962805eEnglishSymplectic ElementsSpringer2021Bokanowski, OPicarelli, AReisinger, CWe study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the L2 norm for linear and semi-linear equations, and in the H1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in L2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms. |
| spellingShingle | Bokanowski, O Picarelli, A Reisinger, C Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations |
| title | Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations |
| title_full | Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations |
| title_fullStr | Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations |
| title_full_unstemmed | Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations |
| title_short | Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations |
| title_sort | stability and convergence of second order backward differentiation schemes for parabolic hamilton jacobi bellman equations |
| work_keys_str_mv | AT bokanowskio stabilityandconvergenceofsecondorderbackwarddifferentiationschemesforparabolichamiltonjacobibellmanequations AT picarellia stabilityandconvergenceofsecondorderbackwarddifferentiationschemesforparabolichamiltonjacobibellmanequations AT reisingerc stabilityandconvergenceofsecondorderbackwarddifferentiationschemesforparabolichamiltonjacobibellmanequations |