Numerical Renormalization Group Calculations For Similarity Solutions and Travelling Waves

Abstract

We present a numerical implementation of the renormalization group (RG) for partial differential equations, constructing similarity solutions and travelling waves. We show that for a large class of well-localized initial conditions, successive iterations of an appropriately defined discrete RG transformation in space and time will drive the system towards a fixed point. This corresponds to a scale-invariant solution, such as a similarity or travelling-wave solution, which governs the long-time asymptotic behavior. We demonstrate that the numerical RG method is computationally very efficient.

14 pages, Postscript file of paper and 3 figures distributed as self-unpacking uuencoded compressed tar file.Also available by anonymous ftp to gijoe.mrl.uiuc.edu (128.174.119.153), file /pub/numrg.uu

References in corpus (3)

• Structural Stability and Renormalization Group for Propagating Fronts
• Selection, Stability and Renormalization
• Renormalization Group Theory And Variational Calculations For Propagating Fronts

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