  We present a fundamental improvement of a high polynomial degree time domain
cell method recently introduced by the last three authors. The published work
introduced a method featuring block-diagonal system matrices where the block
size and conditioning scaled poorly with respect to polynomial degree. The
issue is herein bypassed by the construction of new basis functions exploiting
quadrature rule based mass lumping techniques for arbitrary polynomial degrees
in two dimensions for the Maxwell equations and the acoustic wave equation in
the first order velocity pressure formulation. We characterize the degrees of
freedom of all new discrete approximation spaces we employ for differential
forms and show that the resulting block diagonal (inverse) mass matrices have
block sizes independent of the polynomial degree. We demonstrate on an
extensive number of examples how the new technique is applicable and efficient
for large scale computations.
