  We develop a numerical solver for three-dimensional wave propagation in
coupled poroelastic-elastic media, based on a high-order discontinuous Galerkin
(DG) method, with the Biot poroelastic wave equation formulated as a first
order conservative velocity/strain hyperbolic system. To derive an upwind
numerical flux, we find an exact solution to the Riemann problem, including the
poroelastic-elastic interface; we also consider attenuation mechanisms both in
Biot's low- and high-frequency regimes. Using either a low-storage explicit or
implicit-explicit (IMEX) Runge-Kutta scheme, according to the stiffness of the
problem, we study the convergence properties of the proposed DG scheme and
verify its numerical accuracy. In the Biot low frequency case, the wave can be
highly dissipative for small permeabilities; here, numerical errors associated
with the dissipation terms appear to dominate those arising from discretisation
of the main hyperbolic system.
  We then implement the adjoint method for this formulation of Biot's equation.
In contrast with the usual second order formulation of the Biot equation, we
are not dealing with a self-adjoint system but, with an appropriate inner
product, the adjoint may be identified with a non-conservative velocity/stress
formulation of the Biot equation. We derive dual fluxes for the adjoint and
present a simple but illuminating example of the application of the adjoint
method.
