Methods of Systematic Upscaling

Systematic Upscaling (SU)is a new multiscale computational methodology for the accurate derivation of equations (or statistical relations) that govern a given physical system at increasingly larger scales. Starting at a fine (e.g., atomistic) scale where first-principle laws (e.g., differential equations) are known, SU advances, scale after scale, to obtain suitable variables and operational rules for simulating the system at any large scale of interest. SU combines the complementary advantages of two multilevel computational paradigms that have emerged over the last 35 years: multigrid in applied mathematics and renormalization group in theoretical physics. It includes systematic procedures to iterate back and forth between all the scales of the physical problem, with a general criterion for choosing appropriate variables that operate at each level, and general techniques to derive their governing relations. Indefinitely large systems can in this way be simulated, with computation at each level being needed only within limited representative windows. No scale separation is assumed; unlike conventional ad-hoc multiscale modeling, SU is in principle quite generally applicable, free of slowdowns and bears fully-controlled