Abstract
In this paper we present an efficient numerical approach based on the renormalization group method for the computation of self-similar dynamics. The latter arise, for instance, as the long-time asymptotic behavior of solutions to nonlinear parabolic partial differential equations. We illustrate the approach with the verification of a conjecture about the long-time behavior of solutions to a certain class of nonlinear diffusion equations with periodic coefficients. This conjecture is based on a mixed argument involving ideas from homogenization theory and the renormalization group method. Our numerical approach provides a detailed picture of the asymptotics, including the determination of the effective or renormalized diffusion coefficient.
Highlights
The time evolution of many nonequilibrium physical systems is well described by nonlinear partial differential equations (PDEs)
Within the PDE framework, this assertion amounts to the observation that the PDEs in question have solutions that behave asymptotically as
This limit, in turn, establishes the self-similarity of the long-time asymptotics of u. It is the numerical algorithm consisting of steps 1–3 above supplemented by the calculation of the prefactors {An} and {Bn} that we use to verify Conjecture 2.1 about the asymptotic behavior of solutions to (2.1)
Summary
The time evolution of many nonequilibrium physical systems is well described by nonlinear partial differential equations (PDEs). We conjecture the long-time asymptotic behavior of a class of PDEs. Second, following the strategy of [4, 13], we present a systematic numerical approach, based on the RG ideas, to study self-similar dynamics.
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