Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation

Abstract

Abstract. In this article, we prove the convergence of a semi-discrete scheme applied to the stochastic Korteweg-de Vries equation driven by an additive and localized noise. It is the Crank-Nicholson scheme for the deterministic part and is implicit. This scheme was used in previous numerical experiments on the influence of a noise on soliton propagation [8,9]. Its main advantage is that it is conservative in the sense that in the absence of noise, the L 2 norm is conserved. The proof of convergence uses a compactness argument in the framework of L 2 weighted spaces and relies mainly on the path-wise uniqueness in such spaces for the continuous equation. The main difficulty relies in obtaining a priori estimates on the discrete solution. Indeed, contrary to the continuous case, Ito formula is not available for the discrete equation.

Grade-two fluids on non-smooth domain driven by multiplicative noise: Existence, uniqueness and regularity

“…Note that the solution pu, zq of (1.5) belongs to Cpr0, Ts; HˆW´1 , 4 3 pOqq X L 8 p0, T; VˆL 2 pOqq with probability 1. We also note that while the papers [48], [49], [50], [51] and [52] motivated us to use time discretization, our problem does not fit their framework.…”

Dispersive Schemes for the Critical Korteweg–de Vries Equation

“…, this fact will be used during the rest of the article without further notice. 3) and the inhomogeneous counterpart stands for any (g 3j (t))…”

“…Though travelling waves are problably the most well known solutions of gKdV equations and are very smooth, the approximation of very rough solutions can be important in some contexts. For example stochastic versions of the KdV equation appear in modellisation of plasma fluids, it has been studied theoretically [16] and numerically [2,3] by Debussche and Printems. In this framework the initial data only belong to L 2 . As was pointed out by Ignat and Zuazua [7] in their work on the nonlinear Schrödinger equation, numerical schemes often fail to reproduce dispersive properties of solutions.…”

“…The technics rely on basic harmonic and functional analysis, and we tried to keep a reasonnable balance between self-containedness and heaviness of the calculus : when proofs are slight modifications of already existing arguments we often point to a reference instead of repeating them, and if several quantities are bounded by the same method the precise argument is detailed only once. Our main result is that there exists N ∈ N and an interpolation operator Π : l 2 (N hZ) → l 2 (hZ) actually constructed for N = 6 such that the semi-discrete finite difference problem ∂ t u n + ∂ 3 h u n + (∂ h ΠEu) 5 n /5 = 0, (n, t) ∈ Z × R, u| t=0 = Πu 0,h , is globally well-posed for u 0,h ∈ l 2 (N hZ) sufficiently small, and the solution satisfies dispersive estimates analogous to (D1, D2) uniformly in h. The notations E, Π, ∂ h and ∂ 3 h are defined in the first and second sections. The convergence of the discrete solution is proved by standard weak convergence/compactness arguments.…”

Convergent finite element based discretization of a stochastic two‐phase flow model

“…Developing efficient numerical methods to simulate SPDEs is very important but also very challenging, see for instance [1,4,5,9,10,13,[15][16][17] and the references therein. In [10], the authors proved the convergence in probability of an explicit and an implicit numerical scheme for the numerical approximation of the unique strong solution for the 2D stochastic Navier-Stokes equations.…”