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.

翻译：本文针对笛卡尔网格上采用有限差分法求解双曲守恒律问题，设计了一种新型高阶逆Lax-Wendroff（ILW）边界处理方法。该新型ILW方法将 inflow 边界附近虚拟点值的构造分解为插值与外推两个步骤：首先，通过多项式插值边界附近的内点，赋给若干人工辅助点数值；随后，基于这些辅助点值及通过ILW过程获取的边界空间导数，构建埃尔米特外推多项式，从而获得虚拟点值的近似。通过合理选取人工辅助点，可实现高阶精度与稳定结果。理论分析表明，与原始ILW方法相比，尤其在高阶精度情况下，新方法所需使用的复杂ILW过程项数更少，从而在保持精度与稳定性的前提下提升了计算效率。我们针对多个基准算例（包括一维与二维标量方程及方程组）开展数值实验，数值结果验证了所提方案的鲁棒性与高效性。