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Math."],"published-print":{"date-parts":[[2021,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton\u2013Jacobi\u2013Bellman (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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> norm for linear and semi-linear equations, and in the <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their H\u00f6lder 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.\n<\/jats:p>","DOI":"10.1007\/s00211-021-01202-x","type":"journal-article","created":{"date-parts":[[2021,5,20]],"date-time":"2021-05-20T12:06:12Z","timestamp":1621512372000},"page":"187-222","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Stability and convergence of second order backward differentiation schemes for parabolic Hamilton\u2013Jacobi\u2013Bellman equations"],"prefix":"10.1007","volume":"148","author":[{"given":"Olivier","family":"Bokanowski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Athena","family":"Picarelli","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christoph","family":"Reisinger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,5,20]]},"reference":[{"issue":"260","key":"1202_CR1","doi-asserted-by":"publisher","first-page":"1861","DOI":"10.1090\/S0025-5718-07-02000-5","volume":"74","author":"G Barles","year":"2007","unstructured":"Barles, G., Jakobsen, E.R.: Error bounds for monotone approximation schemes for parabolic Hamilton\u2013Jacobi\u2013Bellman equations. 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