An efficient and accurate numerical scheme  is proposed, analyzed and
implemented  for the Kawahara and modified Kawahara
equations which
 model many physical phenomena such as gravity-capillary waves and
magneto-sound propagation in plasmas. The scheme  consists of
dual-Petrov-Galerkin
method in space and Crank-Nicholson-leap-frog in time such that at
each time step only a sparse banded linear system needs to be solved.
Theoretical analysis and numerical results are presented to show
that the proposed numerical is extremely accurate
and efficient for Kawahara type equations and other fifth-order
nonlinear equations.