基于边界变差最小的高精度有限差分格式构造

从加权紧致非线性（WCNS）高精度有限差分格式出发，在其重构形式的基础上，根据边界变差最小（BVD）原理，遵循单元边界两侧重构物理量值之差最小的准则，在每个单元内通过两步空间重构，构造了一种新的高精度有限差分格式。一般对WCNS等加权非线性格式的改进都是基于改善色散耗散特性、优化非线性权、提高分辨率等单一途径，将它们作为重构候补函数进行结合，既保持了各自优势所在，又能控制格式整体黏性，所得格式具有丰富的应用场景。通过数值实验，将结果与单一格式进行对比，新格式既能在流场光滑区保证设计的精度，对激波等间断附近的振荡也有很好的抑制作用，提高了对高波数区的分辨率，而且在长时间计算后也有较为精确的结果。面向广泛发展的数值格式，还可以构造出其他新方法，对包含强间断和多尺度的流动问题可以获得更好的结果。

Constructing high-order finite difference scheme based on boundary variation diminishing principle

A novel high-order finite difference scheme is proposed based on the Weighted Compact Nonlinear Scheme (WCNS). A two-stage spatial reconstruction is implemented in each cell following the Boundary Variation Diminishing (BVD) principle, which requires the difference between reconstructed values of physical quantity at cell boundaries to be a minimum value. Generally, improvements of weighted nonlinear schemes are achieved by improving a single property of the scheme, such as dispersion and diffusion properties, nonlinear weights, and resolution. Here, they are treated in combination as candidate functions for reconstruction. The new scheme can not only retain each candidate’s own advantages, but also control viscosity of the whole scheme, and is thus applicable for various scenarios. Numerical experiments, including accuracy tests, shock tube problems and double Mach reflection, are conducted, and the results are compared with those of other single-scheme ones. Results reveal that the new scheme is capable of attaining designed accuracy in the smooth area of flow field, and suppressing spurious oscillations near shocks, so that the resolution of high wave number regions can be increased. In addition, precise wave profile can be acquired even after long-time simulation. With extensively developed numerical schemes, other new approaches could be formulated for to provide better results for strong discontinuities and multi-scale structures of compressible flows.