Stochastic KdV equation with multiplicative noise: ε-expansion and numerical conservation aspects

We highlight the numerical conservation of relevant features of the one-dimensional Korteweg-de Vries equation perturbed with multiplicative noise. Our interest consists in maintaining the squared energy and the average L 2-norm of the solution along the numerical dynamics, since it is known that both quantities are conserved by the exact solution along its paths. The analysis benefits of the so-called ε-expansion technique, in order to stress the relevance of the time window and the magnitude of the diffusion term in the numerical conservation. Moreover, we suggest a Hermite-type corrective quadrature formula in order for the numerical solution to retain the average L 2-norm.