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 τ is employed as a regularization parameter. Previously reported results on MMPDEs have invariably employed a constant value of the parameter τ. 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, parabolic problems involving self-similar blow-up to demonstrate the advantages of using a variable relaxation time over a fixed one in terms of accuracy, stability and efficiency.

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