Scaling laws always reveal a very important property of the phenomena under consideration: their self-similarity, i.e., the property of reproducing themselves on different time and space scales. This property gives important evidence of a type of stabilization, the so called intermediate asymptotic regime, which describes the behavior of general solutions in the range where these solutions no longer depend on the details of the initial and/or boundary conditions. This is the regime where the essential physics of the phenomena is revealed.; This dissertation involves a preliminary look at a powerful way to numerically obtain such self-similar behavior by introducing a physically based twist on the direct numerical integration of the associated PDE. The impetus for this twist stems from the philosophy of the RG, or Renormalization Group, for which the numerical implementation has been dubbed nRG. The power of this method lies in its ability to capture the full asymptotic behavior in a wide variety of cases (anomalous exponents, relevant perturbations etc.), with no a priori information other than the PDE and some simple scaling relations. In addition, the clever twist allows nRG to be performed efficiently on even modest computing facilities.
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