Galerkin–collocation approximation in time for the wave equation and its post-processing

Abstract

We introduce and analyze families of Galerkin--collocation discretization schemes in time for the wave equation. Their conceptual basis is the establishment of a connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex algebraic systems. Firstly, continuously differentiable in time discrete solutions are studied. Optimal order error estimates are proved. Then, the concept of Galerkin--collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct link between the two families by a computationally cheap post-processing is presented. A key ingredient of the proposed methods is the application of quadrature rules involving derivatives. The performance properties of the schemes are illustrated by numerical experiments.

Variational time discretizations of higher order and higher regularity

“…Hence, all methods with even k share their strong A-stability with the dG method while methods with odd k are A-stable as the cGP method, cf. Remark 2.1 and [6].…”

“…Hence, an improvement of this quantity by applying the postprocessing is not possible. Table 3 shows the results for calculations using Q 6 6 -VTD 6 6 as discretization in time. The presented error norms indicate that the experimental order of convergence are in agreement with our theory.…”

“…On the basis of Example 6.3 we want to take another view on the interpolation cascade. To this end, the results for Q 6 0 -VTD 6 0 which is just dG(6) are presented in Table 8. The errors in the L 2 norm and the ∞ norm are compared for the standard method and the method with cascadic interpolation.…”

“…The postprocessing technique applied to a numerical solution increases the global smoothness by one differentiation order and the local polynomial degree by one. This results in improved accuracy in integral-based norms, see [1,3,4,6,8,14,24]. Note that the postprocessing comes with almost no computational costs since just jumps of derivatives of the discrete solution are needed.…”

“…Moreover, temporal derivatives of discrete solutions to affine linear systems with time-independent coefficients form also solutions of variational time discretization schemes. This relation was used to prove optimal error estimates of stabilized finite element methods for linear first-order partial differential equations [14] and for parabolic wave equations [6,8].…”