Time asynchronous relative dimension in space method for multi-scale problems in fluid dynamics

A novel computational method is presented for solving fluid dynamics equations in the multi-scale framework when the system size is an important parameter of the governing equations. The method (TARDIS) is based on a concurrent transformation of the governing equations in space and time and solving the transformed equations on a uniform Cartesian grid with the corresponding causality conditions at the grid interfaces. For implementation in the framework of TARDIS, the second-order CABARET scheme of Karabasov and Goloviznin [1] is selected for it provides a good combination of numerical accuracy, computational efficiency and simplicity of realisation. Numerical examples are first provided for several isothermal gas dynamics test problems and then for modelling of molecular fluctuations inside a microscopic flow channel and ultrasound wave propagation through a nano-scale region of molecular fluctuations.