  We propose a moving mesh adaptive approach for solving time-dependent partial
differential equations. The motion of spatial grid points is governed by a
moving mesh PDE (MMPDE) in which a mesh relaxation time \tau is employed as a
regularization parameter. Previously reported results on MMPDEs have invariably
employed a constant value of the parameter \tau. We extend this standard
approach by incorporating a variable relaxation time that is calculated
adaptively alongside the solution in order to regularize the mesh appropriately
throughout a computation. We focus on singular problems involving self-similar
blow-up to demonstrate the advantages of using a variable relaxation ime over a
fixed one in terms of accuracy, stability and efficiency.
