A new non-linear two-time-level Central Leapfrog scheme in staggered conservation–flux variables for fluctuating hydrodynamics equations with GPU implementation

For solving Landau–Lifshitz Navier–Stokes fluctuating hydrodynamics equations, a two-time-level modification of the classical Central Leapfrog scheme with non-linear flux correction is developed. The new algorithm is simple for implementation and demonstrates high accuracy in satisfying the fluctuation–dissipation theorem. The numerical results for equilibrium and non-equilibrium problems in one-dimensional and three-dimensional settings are compared with theoretical predictions, the results from direct molecular dynamics simulations and also with those of several popular computational schemes in the literature. Because of simplicity and locality of the computational stencil, the new algorithm can be efficiently implemented on the GPU, which accelerates the calculation in comparison with a single-CPU-core computation by a factor of 300.