In this paper, a low dissipation fifth-order finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) scheme is presented for solving Euler/Navier-Stokes equations in multi-dimensions on structured meshes. In the reconstruction procedures of this finite difference WENO, a series of nested unequal-sized central stencils are used to design spatial reconstruction polynomials based on the local orthogonal Legendre basis functions. So this new WENO scheme could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property by gradually degrading from the fifth-order to third-order, and ultimately to the first-order accuracy near strong discontinuities. Compared with the classical fifth-order finite difference WENO scheme [18], the results show that this new nested multi-resolution WENO scheme could obtain smaller $L^1$ and $L^{\infty }$ errors for solving smooth problems based on the same mesh level. Moreover, the presented WENO scheme also has smaller numerical dissipation without introducing any dissipation preserving technique, and can capture more subtle flow structures for solving viscous flow problems. The proposed multi-resolution WENO scheme can be easily extended to finite volume framework owing to its linear weights can be set as any positive numbers with only one requirement that their summation equals to one. Therefore, the new WENO scheme is more suitable for solving viscous problems containing both discontinuities and complex smooth structures, and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark inviscid extreme and viscous examples are illustrated to verify the good performance of this nested multi-resolution WENO scheme.

本文提出了一种低耗散的五阶有限差分嵌套多分辨率加权本质上非振荡（WENO）方案，用于求解结构化网格上多维的Euler / Navier-Stokes方程。在此有限差分WENO的重建过程中，使用一系列嵌套的不等大小的中心模具，根据局部正交Legendre基函数设计空间重建多项式。因此，这种新的WENO方案可以在平滑区域中保持原始的精度等级，并通过从五阶到三阶逐渐降级，并最终降到强不连续点附近的一阶精度，基本上保持非振荡性。与经典的五阶有限差分WENO方案相比[18]，${\mathrm{大号}}^{\mathrm{1个}}$ 和 ${\mathrm{大号}}^{\infty }$基于相同网格级别解决平滑问题的错误。而且，所提出的WENO方案在不引入任何耗散保持技术的情况下也具有较小的数值耗散，并且可以捕获更多微妙的流动结构以解决粘性流动问题。提出的多分辨率WENO方案可以轻松地扩展到有限体积框架，这是因为它的线性权重可以设置为任何正数，而只需要一个要求它们的总和等于1。因此，新的WENO方案更适合解决包含不连续性和复杂光滑结构的粘性问题，并且可以轻松地实现任意多维精度的高阶精度。