Satisfying the Fluctuation Dissipation Balance With Adaptive Mesh Refinement

In computational thermodynamics, an additional term is added to the fluid equations to account for microscopic fluctuations acting as a reverse action to the energy dissipation. Choosing a fluctuation term proportional to the square root of the viscous operator ensures that the fluctuations will neither overwhelm nor be overwhelmed by the dissipation in the system. However, when adaptive mesh refinement is introduced to the computational domain, the natural viscous operator is no long positive definite and satisfying the fluctuation dissipation balance becomes nontrivial. In this poster we 1) outline a second order finite difference scheme for a MAC grid that yields a symmetric laplace operator L and 2) prove this operator is also positive definite by deriving the corresponding divergence operator D such that L = DD^T and D satisfies the fluctuation dissipation balance.