Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise

Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

“…For example, the Crank-Nicolson implicit scheme gives an error of order n −2 with h comparable with n −1 . For the equation with noise the results of [3] show that (2) gives an approximation with error of order n −1/2 , provided n 2 h ≤ b < 1 2 ; we shall show that this order of approximation is best possible for schemes of this nature. The main reason why higher order methods do not give improvements is the lack of smoothness of the solution of the equation (1).…”

“…A proof of convergence for a simple approximation scheme (see (2) below), and an estimate of the order of convergence, was given by Gyöngy [3]; his results in fact apply to the more general equationu = ∂ 2 u ∂x 2 + f (x, t, u) + σ(x, t, u)Ẇ (x, t). The purpose of the present paper is to investigate to what extent these results are best possible.…”

“…The following convergence result for the scheme (2) is proved in [3]: Theorem 1.1. Let p ≥ 2, and suppose u 0 is Holder continuous with exponent 1 2 .…”

Full discretization of the stochastic Burgers equation with correlated noise

“…Gyöngy and Shardlow in [18,7] apply finite differences in order to approximate the mild solution of parabolic SPDEs driven by space-time white noise. Yoo investigates the mild solution of parabolic SPDEs by finite differences in [19].…”

Method of lines for parabolic stochastic functional partial differential equations

“…Gyöngy [7] studies the strong convergence in the uniform norm over the space variable for a finite-difference scheme with a regular mesh on [0, 1] for the parabolic…”