In this article, we propose a novel high-order multi-moment finite volume method (MMFVM) for solving linear and nonlinear hyperbolic systems on unstructured grids. Different from the previous versions of volume-average/point-value multi-moment (VPM) scheme, the present scheme is developed by using a new reconstruction procedure based on the constrained least-square method which can be naturally extended to arbitrary high-order of accuracy in two and three dimensions. The so-called VPM-CLS (VPM based on constrained least-square method) scheme substantially increases the solution accuracy on a compact stencil which largely simplifies the computations. In order to eliminate the numerical oscillations associated with high-order schemes, we propose a new limiting projection approach named AOD (adaptive order detection) to choose the optimal degree of reconstruction polynomial for trouble cells in presence of discontinuous solutions. The resulting reconstruction method, i.e. VPM-AOD that combines VPM-CLS scheme with AOD limiting projection technique is able to attain 5th-order accuracy on the stencil of all immediately adjacent cells and achieve essentially non-oscillatory and less-dissipative numerical results for discontinuous solutions. The numerical methods are extensively verified by various benchmark tests for the Euler equations of compressible gas dynamics. Numerical results demonstrate the high solution accuracy and excellent capabilities of proposed schemes to handle both complex physics and geometries.

在本文中，我们提出了一种新颖的高阶多矩有限体积方法（MMFVM），用于求解非结构化网格上的线性和非线性双曲系统。与先前版本的体积平均/点值多矩（VPM）方案不同，本方案是通过使用基于约束最小二乘法的新重构过程开发的，该过程可以自然扩展到任意高阶在两个和三个维度上的准确性。所谓的VPM-CLS（基于约束最小二乘法的VPM）方案大大提高了紧凑型模板上的求解精度，从而大大简化了计算。为了消除与高阶方案相关的数值振荡，我们提出了一种新的限制投影方法，称为AOD（自适应顺序检测），用于在不连续解决方案存在的情况下为故障单元选择最佳的多项式重构程度。最终的重建方法，即将VPM-CLS方案与AOD限制投影技术相结合的VPM-AOD，能够在所有紧邻细胞的模板上达到5阶精度，并且对于不连续的样品，获得了基本上无振荡且耗散较少的数值结果解决方案。数值方法已通过各种基准测试对可压缩气体动力学的欧拉方程进行了广泛验证。数值结果表明，所提出的方案能够解决复杂的物理和几何问题，具有很高的求解精度和出色的功能。