Partial Differential Equations - Modelling and ... - ResearchGate

Higher Order Time Stepping for Second Order
Hyperbolic Problems and Optimal CFL
Conditions
J. Charles Gilbert and Patrick Joly
INRIA Rocquencourt, BP 105, 78153 Le Chesnay, France
Jean-Charles.Gilbert@inria.fr
Patrick.Joly@inria.fr
Summary. We investigate explicit higher order time discretizations of linear second
order hyperbolic problems. We study the even order (2m) schemes obtained by
the modified equation method. We show that the corresponding CFL upper bound
for the time step remains bounded when the order of the scheme increases. We
propose variants of these schemes constructed to optimize the CFL condition. The
corresponding optimization problem is analyzed in detail and the analysis results in
a specific numerical algorithm. The corresponding results are quite promising and
suggest various conjectures.
1 Introduction
We are concerned here with a very classical problem, namely the numerical
approximation of second order hyperbolic problems, more precisely problems
of the form
d 2 u
+ Au =0, (1)
dt2 where A is a linear unbounded positive self-adjoint operator in some Hilbert
space V . This appears to be the generic abstract form for a large class of partial
differential equations in which u denotes a function u(x, t) fromΩ ⊂ R d × R +
in R N and A is a second order differential operator in space, of elliptic nature.
Such models are used for wave propagation in various domains of application,
in particular, in acoustics, electromagnetism, and elasticity [Jol03].
During the past four decades, a considerable literature has been devoted
to the construction of numerical methods for the approximation of (1). The
most recent research deals with the construction of higher order in space
and conservative methods for the space semi-discretization of (1) (see, for
instance, [Coh02] and the references therein). These methods lead us to
consider a family (indexed by h>0, the approximation parameter which