Numerical renormalization-group calculations for similarity solutions and traveling waves

Abstract

We present a numerical implementation of the renormalization group (RG) for partial differential equations, constructing similarity solutions and traveling waves. We show that for a large class of well-localized initial conditions, successive iterations of an approximately 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 traveling-wave solution, which governs the long-time asymptotic behavior. We demonstrate that the numerical RG method is computationally very efficient.