Free Access
Volume 50, Number 6, November-December 2016
Page(s) 1699 - 1730
Published online 18 October 2016
  1. Y. Achdou and M. Falcone, A semi-Lagrangian scheme for mean curvature motion with nonlinear Neumann conditions. Interfaces Free Bound. 14 (2012) 455–485. [CrossRef] [MathSciNet] [Google Scholar]
  2. O. Bokanowski and F. Bonnans, Semi-lagrangian schemes for second order equations. In preparation (2016). [Google Scholar]
  3. O. Bokanowski, Y. Cheng and C.-W. Shu, Convergence of some Discontinuous Galerkin schemes for nonlinear Hamilton-Jacobi equations. Math. Comp. 85 (2016) 2131–2159. [CrossRef] [MathSciNet] [Google Scholar]
  4. F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes. RAIRO Modél. Math. Anal. Numér. 29 (1995) 97–122. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  5. E. Carlini, R. Ferretti and G. Russo, A weighted essentialy non oscillatory, large time-step scheme for Hamilton Jacobi equations. SIAM J. Sci. Comp. 27 (2005) 1071–1091. [CrossRef] [Google Scholar]
  6. P.G. Ciarlet, Finite Element Method for Elliptic Problems. NorthHolland, Amsterdam (1978). [Google Scholar]
  7. B. Cockburn, Discontinuous Galerkin methods. ZAMM Z. Angew. Math. Mech. 83 (2003) 731–754. [CrossRef] [MathSciNet] [Google Scholar]
  8. B. Cockburn and C.-W. Shu, Runge-kutta discontinuous galerkin methods for convection-dominated problems. J. Comput. Phys. 223 (2007) 398–415. [CrossRef] [MathSciNet] [Google Scholar]
  9. N. Crouseilles, M. Mehrenberger and F. Vecil, Discontinuous Galerkin semi-Lagrangian method for Vlasov–Poisson. In CEMRACS’10 research achievements: numerical modeling of fusion. ESAIM Proc. 32 (2011) 211–230. [CrossRef] [EDP Sciences] [Google Scholar]
  10. K. Debrabant, Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise. BIT 50 (2010) 541–558. [CrossRef] [MathSciNet] [Google Scholar]
  11. K. Debrabant and E.R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comp. 82 (2013) 1433–1462. [CrossRef] [MathSciNet] [Google Scholar]
  12. F. Faà di Bruno, Note sur une nouvelle formule de calcul différentiel. Quarterly J. Pure Appl. Math. 1 (1857) 359–360. See also˙di˙Bruno’s˙formula. [Google Scholar]
  13. M. Falcone and R. Ferretti, Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J. Numer. Anal. 35 (1998) 909–940. [CrossRef] [MathSciNet] [Google Scholar]
  14. M. Falcone and R. Ferretti, Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations. Society for Industrial and Applied Mathematics SIAM, Philadelphia, PA (2014). [Google Scholar]
  15. R. Ferretti, Convergence of semi-Lagrangian approximations to convex Hamilton-Jacobi equations under (very) large Courant numbers. SIAM J. Numer. Anal. 40 (2002) 2240–2253. [CrossRef] [MathSciNet] [Google Scholar]
  16. R. Ferretti, A technique for high-order treatment of diffusion terms in semi-lagrangian schemes. Commun. Comput. Phys. 8 (2010) 445–470. [Google Scholar]
  17. R. Ferretti, On the relationship between semi-Lagrangian and Lagrange-Galerkin schemes. Numer. Math. 124 (2013) 31–56. [CrossRef] [MathSciNet] [Google Scholar]
  18. E. Forest, Canonical integrators as tracking codes (1987) SSC-138. [Google Scholar]
  19. E. Forest and R. Ruth, Fourth-order symplectic integration. Physica D: Nonlinear Phenomena 43 (1990) 105–117. [CrossRef] [MathSciNet] [Google Scholar]
  20. P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Vol. 23 of Stoch. Model. Appl. Probab. Springer-Verlag, Berlin (1992). [Google Scholar]
  21. H. Kushner, Probability methods for approximations in stochastic control and for elliptic equations. Vol. 129 of Math. Sci. Eng. Academic Press, New York (1977). [Google Scholar]
  22. H. Kushner and P. Dupuis, Numerical methods for stochastic control problems in continuous time. Vol. 24 of Appl. Math., 2nd edn. Springer, New York (2001). [Google Scholar]
  23. P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equation. In Mathematical Aspects of Finite Elements in Partial Diffential Equations (1974) 89–145. [Google Scholar]
  24. J.-L. Menaldi, Some estimates for finite difference approximations. SIAM J. Control Optim. 27 (1989) 579–607. [CrossRef] [MathSciNet] [Google Scholar]
  25. G.N. Milstein, Weak approximation of solutions of systems of stochastic differential equations. Theory Probab. Appl. 30 (1986) 750–766. [Transl. from Teor. Veroyatnost. i Primenen. 30 (1985) 706–721.]. [CrossRef] [Google Scholar]
  26. G.N. Milstein and M.V. Tretyakov, Numerical solution of the Dirichlet problem for nonlinear parabolic equations by a probabilistic approach. IMA J. Numer. Anal. 21 (2001) 887–917. [CrossRef] [MathSciNet] [Google Scholar]
  27. K.W. Morton, A. Priestley and E. Süli, Stability of the Lagrange-Galerkin method with nonexact integration. RAIRO Modél. Math. Anal. Numér. 22 (1988) 625–653. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  28. É. Pardoux and D. Talay, Discretization and simulation of stochastic differential equations. Acta Appl. Math. 3 (1985) 23–47. [CrossRef] [MathSciNet] [Google Scholar]
  29. D.A.D. Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Vol. 69 of Math. Appl. Springer-Verlag, Berlin (2012). [Google Scholar]
  30. E. Platen, Zur zeitdiskreten approximation von itoprozessen. Diss. B. IMath, Akad. der Wiss. Der DDR, Berlin (1984). [Google Scholar]
  31. J.-M. Qiu and C.-W. Shu, Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the Vlasov–Poisson system. J. Comput. Phys. 230 (2011) 8386–8409. [CrossRef] [MathSciNet] [Google Scholar]
  32. M. Restelli, L. Bonaventura and R. Sacco, A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows. J. Comput. Phys. 216 (2006) 195–215. [CrossRef] [MathSciNet] [Google Scholar]
  33. R.D. Richtmyer and K.W. Morton, Difference methods for initial-value problems, 2nd edn. Interscience Tracts in Pure and Applied Mathematics, No. 4. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney (1967). [Google Scholar]
  34. J.A. Rossmanith and D.C. Seal, A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations. J. Comput. Phys. 230 (2011) 6203–6232. [CrossRef] [MathSciNet] [Google Scholar]
  35. D. Ruth, A canonical integration technique. Technical report (1983). [Google Scholar]
  36. C. Steiner, M. Mehrenberger and D. Bouche, A semi-Lagrangian discontinuous Galerkin approach. Technical Report (2013) hal-00852411. [Google Scholar]
  37. D. Talay, Efficient numerical schemes for the approximation of expectations of functionals of the solution of a SDE and applications. In Filtering and control of random processes (Paris, 1983). Vol. 61 of Lect. Notes Control Inform. Sci. Springer, Berlin (1984) 294–313. [Google Scholar]
  38. P. Wilmott, S. Howison and J. Dewynne, The mathematics of financial derivatives. A student introduction. Cambridge University Press, Cambridge (1995). [Google Scholar]
  39. Y. Yang and C.-W. Shu, Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50 (2012) 3110–3133. [CrossRef] [Google Scholar]
  40. H. Yoshida, Construction of higher order symplectic integrators. Phys. Lett. A 150 (1990) 262–268. [Google Scholar]
  41. H. Yoshida, Recent progress in the theory and application of symplectic integrators. Celest. Mech. Dyn. Astro. 56 (1993) 27–43. [Google Scholar]

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