Semi-Lagrangian discontinuous Galerkin schemes for some first- and second-order partial differential equations

Bokanowski, Olivier; Simarmata, Giorevinus

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1699-1730.

Explicit, unconditionally stable, high-order schemes for the approximation of some first- and second-order linear, time-dependent partial differential equations (PDEs) are proposed. The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin (DG) elements. It follows the ideas of the recent works of Crouseilles et al. [N. Crouseilles, M. Mehrenberger and F. Vecil, In CEMRACS’10 research achievements: numerical modeling of fusion. ESAIM Proc. 32 (2011) 211–230], Rossmanith and Seal [J.A. Rossmanith and D.C. Seal, J. Comput. Phys. 230 (2011) 6203–6232], for first-order equations, based on exact integration, quadrature rules, and splitting techniques for the treatment of two-dimensional PDEs. For second-order PDEs the idea of the scheme is a blending between weak Taylor approximations and projection on a DG basis. New and sharp error estimates are obtained for the fully discrete schemes and for variable coefficients. In particular we obtain high-order schemes, unconditionally stable and convergent, in the case of linear first-order PDEs, or linear second-order PDEs with constant coefficients. In the case of non-constant coefficients, we construct, in some particular cases, “almost” unconditionally stable second-order schemes and give precise convergence results. The schemes are tested on several academic examples.

Classification : 65M12, 65M15, 65M25, 65M60
Mots clés : Semi-Lagrangian scheme, weak Taylor scheme, discontinuous Galerkin elements, method of characteristics, high-order methods, advection diffusion equations

Y. Achdou and M. Falcone, A semi-Lagrangian scheme for mean curvature motion with nonlinear Neumann conditions. Interfaces Free Bound. 14 (2012) 455–485. | DOI | Zbl

O. Bokanowski and F. Bonnans, Semi-lagrangian schemes for second order equations. In preparation (2016).

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. | Zbl

F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes. RAIRO Modél. Math. Anal. Numér. 29 (1995) 97–122. | DOI | Numdam | Zbl

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. | DOI | Zbl

P.G. Ciarlet, Finite Element Method for Elliptic Problems. NorthHolland, Amsterdam (1978). | Zbl

B. Cockburn, Discontinuous Galerkin methods. ZAMM Z. Angew. Math. Mech. 83 (2003) 731–754. | DOI | Zbl

B. Cockburn and C.-W. Shu, Runge-kutta discontinuous galerkin methods for convection-dominated problems. J. Comput. Phys. 223 (2007) 398–415.

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. | DOI | Zbl

K. Debrabant, Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise. BIT 50 (2010) 541–558. | DOI | Zbl

K. Debrabant and E.R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comp. 82 (2013) 1433–1462. | DOI | Zbl

F. Faà di Bruno, Note sur une nouvelle formule de calcul différentiel. Quarterly J. Pure Appl. Math. 1 (1857) 359–360. See also http://en.wikipedia.org/wiki/Faa˙di˙Bruno’s˙formula.

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. | DOI | MR | Zbl

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).

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. | DOI | Zbl

R. Ferretti, A technique for high-order treatment of diffusion terms in semi-lagrangian schemes. Commun. Comput. Phys. 8 (2010) 445–470. | DOI | Zbl

R. Ferretti, On the relationship between semi-Lagrangian and Lagrange-Galerkin schemes. Numer. Math. 124 (2013) 31–56. | DOI | Zbl

E. Forest, Canonical integrators as tracking codes (1987) SSC-138.

E. Forest and R. Ruth, Fourth-order symplectic integration. Physica D: Nonlinear Phenomena 43 (1990) 105–117. | DOI | Zbl

P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations. Vol. 23 of Stoch. Model. Appl. Probab. Springer-Verlag, Berlin (1992). | Zbl

H. Kushner, Probability methods for approximations in stochastic control and for elliptic equations. Vol. 129 of Math. Sci. Eng. Academic Press, New York (1977). | Zbl

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). | Zbl

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. | Zbl

J.-L. Menaldi, Some estimates for finite difference approximations. SIAM J. Control Optim. 27 (1989) 579–607. | DOI | Zbl

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.]. | DOI | Zbl

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. | DOI | Zbl

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. | DOI | Numdam | Zbl

É. Pardoux and D. Talay, Discretization and simulation of stochastic differential equations. Acta Appl. Math. 3 (1985) 23–47. | DOI | Zbl

D.A.D. Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Vol. 69 of Math. Appl. Springer-Verlag, Berlin (2012). | Zbl

E. Platen, Zur zeitdiskreten approximation von itoprozessen. Diss. B. IMath, Akad. der Wiss. Der DDR, Berlin (1984).

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. | DOI | Zbl

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. | DOI | Zbl

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). | Zbl

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. | DOI | Zbl

D. Ruth, A canonical integration technique. Technical report (1983).

C. Steiner, M. Mehrenberger and D. Bouche, A semi-Lagrangian discontinuous Galerkin approach. Technical Report (2013) . | HAL

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. | Zbl

P. Wilmott, S. Howison and J. Dewynne, The mathematics of financial derivatives. A student introduction. Cambridge University Press, Cambridge (1995). | Zbl

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. | DOI | Zbl

H. Yoshida, Construction of higher order symplectic integrators. Phys. Lett. A 150 (1990) 262–268. | DOI

H. Yoshida, Recent progress in the theory and application of symplectic integrators. Celest. Mech. Dyn. Astro. 56 (1993) 27–43. | DOI | Zbl