SOME STOCHASTIC PARTICLE METHODS FOR NONLINEAR PARABOLIC PDES

We introduce, on some examples, the main tools to analyze the convergence rate of sto- chastic particle methods for the numerical approximation of some nonlinear parabolic PDEs. We treat the case of Lipschitz coecients as well as the case of viscous scalar conservation laws. We also discuss some particular aspects like the case of a bounded spatial domain or the use of a Romberg extrapolation as a speed up procedure.

[1]  Alain-Sol Sznitman,et al.  A propagation of chaos result for Burgers' equation , 1986 .

[2]  S. Méléard.  Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models , 1996 .

[3]  Jonathan Goodman,et al.  Convergence of the Random Vortex Method , 1987 .

[4]  B. Jourdain,et al.  PROBABILISTIC INTERPRETATION AND PARTICLE METHOD FOR VORTEX EQUATIONS WITH NEUMANN’S BOUNDARY CONDITION , 2004, Proceedings of the Edinburgh Mathematical Society.

[5]  S. Pope.  Lagrangian PDF Methods for Turbulent Flows , 1994 .

[6]  Hirofumi Osada,et al.  Propagation of chaos for the two dimensional Navier-Stokes equation , 1986 .

[7]  Denis Talay,et al.  Probabilistic numerical methods for partial differential equations: Elements of analysis , 1996 .

[8]  Elbridge Gerry Puckett,et al.  Convergence of a random particle method to solutions of the Kolmogorov equation _{}=ₓₓ+(1-) , 1989 .

[9]  Benjamin Jourdain,et al.  Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures , 2000 .

[10]  Ding-Gwo Long,et al.  Convergence of the random vortex method in two dimensions , 1988 .

[11]  Kohatsu-Higa Arturo,et al.  Weak rate of convergence for an Euler scheme of nonlinear SDE’s , 1997 .

[12]  E. Peirano,et al.  The pdf approach to turbulent polydispersed two-phase flows , 2001 .

[13]  Dominique Lépingle,et al.  Euler scheme for reflected stochastic differential equations , 1995 .

[14]  Mireille Bossy,et al.  A stochastic particle method for the McKean-Vlasov and the Burgers equation , 1997, Math. Comput..

[15]  B. Jourdain.  Probabilistic Characteristics Method for a One-Dimensional Inviscid Scalar Conservation Law , 2002 .

[16]  A. Chorin.  Numerical study of slightly viscous flow , 1973, Journal of Fluid Mechanics.

[17]  B. Pacchiarotti,et al.  Numerical Approximation for Functionals of Reflecting Diffusion Processes , 1998, SIAM J. Appl. Math..

[18]  Karl Oelschläger,et al.  A law of large numbers for moderately interacting diffusion processes , 1985 .

[19]  D. Talay,et al.  Expansion of the global error for numerical schemes solving stochastic differential equations , 1990 .

[20]  F. Antonelli,et al.  Rate of convergence of a particle method to the solution of the Mc Kean-Vlasov's equation , 2002 .

[21]  Branching process associated with 2d-Navier Stokes equation , 2001 .

[22]  G. Cottet.  Boundary conditions and deterministic vortex methods for the Navier-Stokes equations , 1989 .

[23]  Mireille Bossy,et al.  COMPARISON OF A STOCHASTIC PARTICLE METHOD AND A FINITE VOLUME DETERMINISTIC METHOD APPLIED TO BURGERS EQUATION , 1997, Monte Carlo Methods Appl..

[24]  Alexandre J. Chorin.  Vortex Sheet Approximation of Boundary Layers , 1989 .

[25]  M. Pulvirenti,et al.  Propagation of chaos for Burgers' equation , 1983 .

[26]  A. Majda,et al.  Vortex methods. II. Higher order accuracy in two and three dimensions , 1982 .

[27]  P. Raviart.  An analysis of particle methods , 1985 .

[28]  Y. Brenier,et al.  Sticky Particles and Scalar Conservation Laws , 1998 .

[29]  Sylvie Méléard,et al.  Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data , 2001 .

[30]  J. Gärtner.  On the McKean‐Vlasov Limit for Interacting Diffusions , 1988 .

[31]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[32]  S. Pope.  Application of the velocity‐dissipation probability density function model to inhomogeneous turbulent flows , 1991 .

[33]  M. Pulvirenti,et al.  Hydrodynamics in two dimensions and vortex theory , 1982 .

[34]  Mireille Bossy,et al.  Optimal rate of convergence of a stochastic particle method to solutions of 1D viscous scalar conservation laws , 2003, Math. Comput..

[35]  Mireille Bossy,et al.  A symmetrized Euler scheme for an efficient approximation of reflected diffusions , 2004, Journal of Applied Probability.

[36]  Sylvie Méléard,et al.  A propagation of chaos result for a system of particles with moderate interaction , 1987 .

[37]  Andrew J. Majda,et al.  Vortex methods. I. Convergence in three dimensions , 1982 .

[38]  J. Fontbona.  A probabilistic interpretation and stochastic particle approximations of the 3-dimensional Navier-Stokes equations , 2006 .

[39]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[40]  Shigeyoshi Ogawa,et al.  Weak rate of convergence for an Euler scheme of nonlinear SDE’s , 1997, Monte Carlo Methods Appl..

[41]  Christian Léonard,et al.  Une loi des grands nombres pour des systèmes de diffusions avec interaction et à coefficients non bornés , 1986 .

[42]  Hiroshi Tanaka.  Limit Theorems for Certain Diffusion Processes with Interaction , 1984 .

[43]  Mireille Bossy,et al.  Rate of convergeance of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval , 2002 .

[44]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .