A 'divide-and-conquer' spatial and temporal multiscale method for transient convection-diffusion-reaction equations

A multiscale method for the numerical solution of transient convection-diffusion-reaction equations is proposed in the present paper. Two main goals have led to the development of the present method: a desired independence of any heuristic parameter such as the stabilization parameter in stabilized methods and a desire for a consistent multiscale approach in space and time. The method is constituted by solution approaches on a coarse- and a fine-scale level and by inter-scale operators for data transfer between those two levels. A particular feature of the method is that no large matrix system has to be solved. The results from three numerical test cases show that for both problematic flow regimes, that is, the regime of dominant convection and the regime of dominant convection and absorption, the present method provides completely stable solutions, which are not achieved by standard stabilized methods, particularly for the later regime. A still to be noted current shortcoming of the proposed method reveals itself in a too smooth resolution of regions with a sharp gradient in the solution field.