  In this paper, an efficient high-order gas-kinetic scheme (EHGKS) is proposed
to solve the Euler equations for compressible flows. We re-investigate the
underlying mechanism of the high-order gas-kinetic scheme (HGKS) and find a new
strategy to improve its efficiency. The main idea of the new scheme contains
two parts. Firstly, inspired by the state-of-art simplifications on the
third-order HGKS, we extend the HGKS to the case of arbitrary high-order
accuracy and eliminate its unnecessary high-order dissipation terms. Secondly,
instead of computing the derivatives of particle distribution function and
their complex moments, we introduce a Lax-Wendroff procedure to compute the
high-order derivatives of macroscopic quantities directly. The new scheme takes
advantage of both HGKS and the Lax-Wendroff procedure, so that it can be easily
extended to the case of arbitrary high-order accuracy with practical
significance. Typical numerical tests are carried out by EHGKS, with the third,
fifth and seventh-order accuracy. The presence of good resolution on the
discontinuities and flow details, together with the optimal CFL numbers,
validates the high accuracy and strong robustness of EHGKS. To compare the
efficiency, we present the results computed by the EHGKS, the original HGKS and
Runge-Kutta-WENO-GKS. This further demonstrates the advantages of EHGKS.
