In computational fluid dynamics, the Stochastic Eulerian Lagrangian Method (SELM)[1] is an approach to capture essential features of fluid-structure interactions subject to thermal fluctuations while introducing approximations which facilitate analysis and the development of tractable numerical methods.
Approaches also are introduced for the stochastic fields of the SPDEs to obtain numerical methods taking into account the numerical discretization artifacts to maintain statistical principles, such as fluctuation-dissipation balance and other properties in statistical mechanics.
[1] The SELM fluid-structure equations typically used are The pressure p is determined by the incompressibility condition for the fluid The
the following adjoint conditions are imposed Thermal fluctuations are introduced through Gaussian random fields with mean zero and the covariance structure To obtain simplified descriptions and efficient numerical methods, approximations in various limiting physical regimes have been considered to remove dynamics on small time-scales or inertial degrees of freedom.
The SELM approach has been shown to yield stochastic fluid-structure dynamics that are consistent with statistical mechanics.
Different types of coupling operators have also been introduced allowing for descriptions of structures involving generalized coordinates and additional translational or rotational degrees of freedom.
For numerically discretizing the SELM SPDEs, general methods were also introduced for deriving numerical stochastic fields for SPDEs that take discretization artifacts into account to maintain statistical principles, such as fluctuation-dissipation balance and other properties in statistical mechanics.