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Numerical methods for stochastically driven wave equations

Research Project
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01.10.2009
 - 30.09.2012

The interest in stochastic partial differential equations is growing during the last decades. Unfortunately, there are very few equations for which an analytical solution is known. For those stochastic partial differential equations without a known solution, one hope is that numerical methods will help to better understand them. In this project, we will focus on numerical discretisations of stochastic wave equations. For problems such as wave propagation through the ocean, the properties of the water fluctuate randomly due to the presence of turbulence: a deterministic approach is thus not enough to describe the motion of the wave. This leads us to consider stochastic wave equations. As a model problem, we plan to study numerical discretisations of the one-dimensional stochastic wave equation, where the space-time noise is white in time and spatially correlated. This partial differential equation can be seen as a model of a nonlinear string submerged in a turbulent fluid. For the numerical discretisation of such a problem, we first discretise in space (method of lines) and then in time with a geometric integrator . After a pseudo-spectral semi-discretisation in space of the stochastic wave equation, we obtain a system of second-order stochastic differential equations which are oscillators-like equations. A deep understanding of this system of stochastic equations is primordial for a proper numerical treatment of the problem. To get more insight into the behaviour of such kind of system, we firstly concentrate on the study of efficient numerical methods for stochastic oscillators . We then adapt these numerical schemes to systems of second-order stochastic differential equations and finally we look at the full discretisation of the stochastically driven wave equation . Our new geometric integrators will permit a deeper understanding of the numerical discretisation of stochastic oscillators, which will lead to an optimal numerical treatments of stochastically driven wave equations.

Publications

Cohen, D., Larsson, S. and Sigg, M. (2013) ‘A trigonometric method for the linear stochastic wave equation’, SIAM Journal on Numerical Analysis, 51(1), pp. 204–222. Available at: https://doi.org/10.1137/12087030x.

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Members (3)

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David Cohen

Principal Investigator
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Marcus J. Grote

Co-Investigator
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Magdalena Sigg

Project Member