Special numerical methods for convection-dominated laminar flows at arbitrary Reynolds number

The problem of transport of a substance (heat, matter or momentum) by convection-diffusion is considered for laminar flows of a fluid at arbitrarily large Reynolds numbers. In some neighbourhoods of the solid surfaces bounding the fluid, boundary layers may appear for large Reynolds number. Classical finite difference methods do not yield approximate solutions with a guaranteed accuracy, even with the computing power of supercomputers, if the Reynolds number can be arbitrarily large. For such problems, a numerical method using standard finite difference operators on a special piecewise-uniform mesh is constructed, and it yields approximate solutions with a guaranteed accuracy. Furthermore, the guaranteed accuracy depends only on the amount of computation performed, and not on the Reynolds number. We give numerical examples which agree with these theoretical results.