Half-way seminar / Computational and Applied Mathematics seminar

Abstract: Stochastic partial differential equations are used to describe various physical processes that are perturbed by noise. Some of those occur on curved surfaces, for example spheres. In this talk semilinear stochastic heat equations with additive noise on the unit sphere are considered. Approximations in space and time are presented in order to simulate and analyse solutions. The space approximation is derived using the spectral method, with spherical harmonic functions. In order to obtain time discretization on an equidistant time grid the Euler--Maruyama scheme is applied. For the semilinear stochastic heat equations on the sphere with additive isotropic Wiener noise, strong convergence rates in space and time are derived, taking regularity of the initial condition and the driving noise into account. Furthermore convergence of the expectation and the second moment is analysed for the corresponding linear equation. The theoretical results are confirmed by numerical simulations.