  In this paper, we design a new kind of high order inverse Lax-Wendroff (ILW)
boundary treatment for solving hyperbolic conservation laws with finite
difference method on a Cartesian mesh. This new ILW method decomposes the
construction of ghost point values near inflow boundary into two steps:
interpolation and extrapolation. At first, we impose values of some artificial
auxiliary points through a polynomial interpolating the interior points near
the boundary. Then, we will construct a Hermite extrapolation based on those
auxiliary point values and the spatial derivatives at boundary obtained via the
ILW procedure. This polynomial will give us the approximation to the ghost
point value. By an appropriate selection of those artificial auxiliary points,
high-order accuracy and stable results can be achieved. Moreover, theoretical
analysis indicates that comparing with the original ILW method, especially for
higher order accuracy, the new proposed one would require fewer terms using the
relatively complicated ILW procedure and thus improve computational efficiency
on the premise of maintaining accuracy and stability. We perform numerical
experiments on several benchmarks, including one- and two-dimensional scalar
equations and systems. The robustness and efficiency of the proposed scheme is
numerically verified.
