Dynamic time step and stability criteria comparison for the heat diffusion equation
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Dynamic time step estimates were determined for four single step numerical schemes in one dimension, and three finite element and three finite difference single step schemes in two dimensions. Unlike other stability-based time step criteria, the dynamic time step estimates are functions of grid size, material properties, and boundary conditions. For a certain problem, an initial coarse mesh is required to determine its lowest eigenvalue. The problem needs to be re-meshed and solved with the new mesh to satisfy a desired accuracy. Correlations between the dynamic time step and the stability criteria were conducted. A stability criteria equivalence was defined as the stability criteria that gives a time step estimate equivalent to the dynamic time step estimate. For the range of problems solved and for the central difference time stepping scheme, the stability criteria equivalence ranges from 0.5 to 2.7 for one-dimensional problems and 0.46 to 9.13 for two-dimensional problems, depending on the problem boundary conditions. The study shows that, unlike the dynamic time step estimate, the stability criteria used for selecting a time step is not adequate, since it does not change with the problem boundary conditions. Elsevier, Paris.
