  This paper serves to treat boundary conditions numerically with high order
accuracy in order to match the two-stage fourth-order finite volume schemes for
hyperbolic problems developed in [{\em J. Li and Z. Du, A two-stage fourth
order time-accurate discretization {L}ax--{W}endroff type flow solvers, {I}.
{H}yperbolic conservation laws, SIAM, J. Sci. Comput., 38 (2016),
pp.~A3046--A3069}]. As such, it is significant when capturing small scale
structures near physical boundaries. Different from previous contributions in
literature, the current approach constructs a fourth order accurate
approximation to boundary conditions by only using the Jacobian of the flux
function (characteristic information) instead of its successive differentiation
leading to tensors of high ranks in the inverse Lax-Wendroff method.
Technically, data in several ghost cells are constructed with interpolation so
that the interior scheme can be implemented over boundary cells, and
theoretical boundary condition has to be modified properly at intermediate
stages so as to make the two-stage scheme over boundary cells fully consistent
with that over interior cells. This highlights the fact that {\em continuous
boundary conditions only match continuous partial differential equations
(PDEs), and they must be approximated in a consistent way (even though it could
be exactly valued) when the PDEs are discretized.} Several numerical examples
are provided to illustrate the performance of the current approach when dealing
with general boundary conditions.
