Heterogenous multiscale method for locally self-similar problems

Abstract

We present a multiscale method for a class of problems that are locally self-similar in scales and hence do not have scale separation. Our method is based on the framework of the heterogeneous multiscale method (HMM). At each point where macroscale data is needed, we perform several small scale simulations using the microscale model, then using the results and local self- similarity to predict the needed data at the scale of interest. We illustrate this idea by computing the effective macroscale transport of a percolation network at the percolation threshold. 1. Introduction and HMM Upscaling In the last several years, there has been a tremendous growth of activity on developing multiscale methods in a number of application areas. For some reviews and perspectives, we refer to (4, 5, 8). The primary goal is to develop computational techniques that can extract accurately the macroscale behavior of the system under consideration, at a cost that is substantially less than the cost of solving the microscale problem. Obviously this can only be done if the problem under consideration has some special features that can be taken advantage of. Up to now, with the exception of renormalization group methods (1), all other existing multiscale techniques assume that there is scale separation in the problem, and this property is used in an essential way in order to design efficient multiscale methods (5, 8). While many problems, particularly problems with multi-physics, do have scale separation, there are other important problems with multiscales that do not have this property. The most well-known example is the fully developed turbulent flow whose active scales typically span continuously from the large energy pumping scale to the small energy dissipation scale, the ratio of these two scales was estimated by Kolomogorov to be of O(Re 3/4 )( 2, 6). HereRe is the Reynolds number of the flow which can easily reach 10 9 for atmospheric turbulence. In the absence of scale separation, we must seek other special features of these problems in order to develop efficient computational methods. The special feature that we will focus on in this paper is (statistical) self-similarity in scales, namely that the averaged properties of the system at different scales are related to each other by simple scaling factors. This feature is approximately satisfied by many important problems such as turbulent flows and sub-surface transport in the inertial range of scales. We will call these type D problems (for the definition of type A, type B and type C problems, see (3)). To develop an efficient computational technique, it is most convenient to use the framework of the heterogeneous multiscale method (HMM) (3). HMM has two main components: • Selecting the macroscale solver;