Accelerated numerical schemes for PDEs and SPDEs

Abstract

We give a survey of some results from Gyöngy and Krylov (SIAM Journal on Mathematical Analysis, 37, 1070-1097, 2006, Collect. Math. Vol. Extra, 255-273, 2006, Illinois Journal of Mathematics, 50, 473-514, 2006, Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space, To appear, Accelerated finite difference schemes for linear stochastic partial differential equations in the whole space. To appear in SIAM Journal on Mathematical Analysis) on accelerated numerical schemes for some classes of deterministic and stochastic PDEs. First, we consider monotone finite difference schemes for parabolic (possibly degenerate) PDEs in the spatial variable. We present some theorems from Gyöngy and Krylov ("Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space, To appear in Mathematics of Computation")) on power series expansions of finite difference approximations in terms of the mesh-size of the grid. These theorems imply that one can accelerate the convergence of finite difference approximations (in the spatial variables) to any order by taking suitable mixtures of approximations corresponding to different mesh-sizes. We extend these results to degenerate elliptic equations in spaces with supremum norm. Then, we establish power series expansions for finite difference approximations of linear stochastic PDEs, and hence we get, as before that the rate of convergence of these approximations can be accelerated to any order, provided the data and the coefficients of the equations are sufficiently smooth. Finally, for a large class of equations and various types of time discretizations for them, we present some results from Gyöngy and Krylov (SIAM Journal on Mathematical Analysis, 37, 1070-1097, 2006, Collect. Math. Vol. Extra, 255-273, 2006, Illinois Journal of Mathematics, 50, 473-514, 2006) on power series expansion in the parameters of the approximations and get theorems on their acceleration. © 2011 Springer-Verlag Berlin Heidelberg.