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Keywords:

  • moving mesh;
  • partial differential equations;
  • implicit

Velocity-based moving mesh methods update the mesh at each time level by using a velocity equation with a time-stepping scheme. A particular velocity-based moving mesh method, based on conservation, uses explicit time-stepping schemes with small time steps to avoid mesh tangling. However, this can prove to be impractical when long-term behavior of the solution is of interest. Here, we present a semi-implicit time-stepping scheme which manipulates the structure of the velocity equation such that it resembles a variable-coefficient heat equation. This enables the use of maximum/minimum principle which ensures that mesh tangling is avoided. It is also shown that this semi-implicit scheme can be extended to a fully implicit time-stepping scheme. Thus, the time-step restriction imposed by explicit schemes is overcome without sacrificing mesh structure. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 321–338, 2014