  The higher-order gas-kinetic scheme for solving the Navier-Stokes equations
has been studied in recent years. In addition to the use of higher-order
reconstruction techniques, many terms are used in the Taylor expansion of the
gas distribution functions. Therefore, a large number of coefficients need to
be determined in the calculation of the time evolution of the gas distribution
function at cell interfaces. As a consequence, the higher-order flux function
takes much more computational time than that of a second-order gas-kinetic
scheme. This paper aims to simplify the evolution model by two steps. Firstly,
the coefficients related to the higher-order spatial and temporal derivatives
of a distribution function are redefined to reduce the computational cost.
Secondly, based on the physical analysis, some terms can be removed without
loss of accuracy. Through the simplifications, the computational efficiency of
the higher-order scheme is increased significantly. In addition, a
self-adaptive numerical viscosity is designed to minimize the necessary
numerical dissipation. Several numerical examples are tested to demonstrate the
accuracy and robustness of the current scheme.
