11 research outputs found
Exponential integrators for the stochastic Manakov equation
This article presents and analyses an exponential integrator for the
stochastic Manakov equation, a system arising in the study of pulse propagation
in randomly birefringent optical fibers. We first prove that the strong order
of the numerical approximation is if the nonlinear term in the system is
globally Lipschitz-continuous. Then, we use this fact to prove that the
exponential integrator has convergence order in probability and almost
sure order , in the case of the cubic nonlinear coupling which is relevant
in optical fibers. Finally, we present several numerical experiments in order
to support our theoretical findings and to illustrate the efficiency of the
exponential integrator as well as a modified version of it
Drift-preserving numerical integrators for stochastic Hamiltonian systems
The paper deals with numerical discretizations of separable nonlinear
Hamiltonian systems with additive noise. For such problems, the expected value
of the total energy, along the exact solution, drifts linearly with time. We
present and analyze a time integrator having the same property for all times.
Furthermore, strong and weak convergence of the numerical scheme along with
efficient multilevel Monte Carlo estimators are studied. Finally, extensive
numerical experiments illustrate the performance of the proposed numerical
scheme
Exponential Integrators for Stochastic Maxwell's Equations Driven by It\^o Noise
This article presents explicit exponential integrators for stochastic
Maxwell's equations driven by both multiplicative and additive noises. By
utilizing the regularity estimate of the mild solution, we first prove that the
strong order of the numerical approximation is for general
multiplicative noise. Combing a proper decomposition with the stochastic
Fubini's theorem, the strong order of the proposed scheme is shown to be
for additive noise. Moreover, for linear stochastic Maxwell's equation with
additive noise, the proposed time integrator is shown to preserve exactly the
symplectic structure, the evolution of the energy as well as the evolution of
the divergence in the sense of expectation. Several numerical experiments are
presented in order to verify our theoretical findings.Comment: 21 Page
Numerical approximation and simulation of the stochastic wave equation on the sphere
Solutions to the stochastic wave equation on the unit sphere are approximated
by spectral methods. Strong, weak, and almost sure convergence rates for the
proposed numerical schemes are provided and shown to depend only on the
smoothness of the driving noise and the initial conditions. Numerical
experiments confirm the theoretical rates. The developed numerical method is
extended to stochastic wave equations on higher-dimensional spheres and to the
free stochastic Schr\"odinger equation on the unit sphere.Comment: 26 pages, 7 figure
Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
We consider the numerical approximation of the mild solution to a semilinear
stochastic wave equation driven by additive noise. For the spatial
approximation we consider a standard finite element method and for the temporal
approximation, a rational approximation of the exponential function. We first
show strong convergence of this approximation in both positive and negative
order norms. With the help of Malliavin calculus techniques this result is then
used to deduce weak convergence rates for the class of twice continuously
differentiable test functions with polynomially bounded derivatives. Under
appropriate assumptions on the parameters of the equation, the weak rate is
found to be essentially twice the strong rate. This extends earlier work by one
of the authors to the semilinear setting. Numerical simulations illustrate the
theoretical results.Comment: 32 pages, 4 figure