Benjamin Weber
Division of Scientific Computing, IT Department, Uppsala University
Time-step restrictions of standard or Lagrange finite element methods (LFEM), the Hermite finite element method (HFEM) and discontinuous Galerkin methods using the symmetric interior penalty (DG-SIM) for the wave equation are analysed. We implement a matrix Fourier transform to analyse periodic block tridiagonal matrices and use it to calculate stability limits for a simple explicit time-stepping method when using higher order finite elements on a uniform grid in one dimension. HFEM is shown to have a more relaxed time-step limit than LFEMs of the same order, and DG-SIMs of the same order are shown to share a common stability limit. We evaluate time-step restrictions for the methods considered in 1D and observe agreement with the derived relationships. We test the evaluated time-step limits in 1D with both continuous and discontinuous initial data and validate the previous calculations.