  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.
