Seminar Series - 2000-2001

are examined and compared. These two families of methods fall into the category of method of discretization in time, in which the governing equation is first discretized in time by using finite differences, and then approximated in the space variables. The first approach consists of using the classical family of Newmark methods for temporal discretization in conjuction with finite element methods for spatial discretization. The second approach is based on the multi-stage high-order Taylor-Galerkin schemes, which consists of using a rational function approximation in the time variable followed by the Bubnov-Galerkin approximation in the space variables.
The Navier-Lamé equation of linear elasticity and the equations of linear acoustics are cast into the abstract Cauchy problem setting, and both families of methods are implemented. Several numerical experiments are performed to compare accuracy, stability, temporal convergence, and cost of both methods.

The numerical examples show that theTaylor-Galerkin schemes give a high-order temporal accuracy which is at most of order 2s for s stages, as compared with the Newmark methods which are only of second order. In addition, the Taylor-Galerkin methods are unconditionally stable on arbitrary non-uniform finite element meshes of varying element sizes and spectral orders, while some members of the Newmark family are only conditionally stable. It is finally concluded that the high-order methods are more cost effective and robust for certain classes of problems.


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