Abstract of
Antisymmetry, pseudospectral methods, weighted
residual discretizations, and energy conserving partial
differential equations
by Robert McLachlan and Nicolas Robidoux
In view of its influence on the stability of numerical integrators,
the dual composition method for producing energy-preserving
discretizations of conservative PDEs, which generalizes the
summation by parts approach, is introduced. Dual composition
discretizations involve two compatible weighted residual
approximations: one which targets variational derivatives of
(nonlinear) functionals, and one which targets linear differential
operators. In the simplest cases, it coincides with a
pseudospectral, Galerkin, discretization. In general, however, the
resulting differentiation matrices are not the standard
pseudospectral---or the standard Galerkin---ones. Nonetheless, fast
implementations can sometimes be recovered.
The emphasis is on pseudospectral bases; however, spectral and
finite element bases are also discussed. Fourier antialiasing is
shown to make some discretizations conservative; a Chebyshev grid
version is presented. Semi-discretizations of the one-way and the
inhomogeneous wave equations on the interval are discussed. Several
semi-discretizations, some of them Hamiltonian, of the
Korteweg-de Vries equation on the circle are compared. The
shortcomings of quadrature approximations of energy functionals are
discussed.
Complaints to
nrobidou@netscape.net (Nicolas Robidoux)
December 18, 2000.