An analysis of time discretization in the finite element solution of hyperbolic problems

The problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined. A modified equation analysis reveals that the classical, second-order accurate, time-stepping algorithms, i.e., the Lax-Wendroff, leap-frog, and Crank-Nicolson methods, properly combine with piecewise linear finite elements in advection problems only for small values of the time step. On the contrary, as the Courant number increases, the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit CFL property are not achieved. These fundamental numerical properties can, however, be recovered, while still remaining in the standard Galerkin finite element setting, by increasing the order of accuracy of the time discretization. This is accomplished by exploiting the Taylor series expansion in the time increment up to the third order before performing the Galerkin spatial discretization using piecewise linear interpolations. As a result, it appears that the proper finite element equivalents of second-order finite difference schemes are implicit methods of incremental type having third- and fourth-order global accuracy on uniform meshes (Taylor-Galerkin methods). Numerical results for several linear examples are presented to illustrate the properties of the Taylor-Galerkin schemes in one- and two-dimensional calculations.