基于BVD原理的高保真空间重构方法

High-fidelity numerical methods based on Boundary Variation Diminishing principle

简要综述了一类基于单元边界变差最小化(Boundary Variation Diminishing,BVD)原理,设计双曲守恒律高保真数值格式的空间重构方法。BVD原理要求尽量减少通过重构得到的网格边界两侧物理量之间的差,从而能够有效地控制黎曼求解器中的数值黏性。BVD方法针对数值解的空间分布特征,选择多个函数作为空间重构的候补函数,并根据BVD判定准则从候补函数中选取最合适的函数进行空间重构。BVD判据不需要根据求解对象进行经验参数(阈值)的调整。选用适当的候补函数和BVD准则,可以完全避免现有算法中为抑制数值振荡而必须采用的非线性限制。BVD格式能在抑制数值振荡的同时,有效地控制数值耗散,可以对光滑解与间断解都获得高保真的计算结果。本文概述了BVD方法的基本思想、设计相关格式的基本思路,以及一些具有很强实用价值的BVD格式。并通过单相和两相可压缩流动的一些典型算例验证BVD格式的特点和优势。

This paper presents a brief review on a novel framework to design high-fidelity numerical schemes for both continuous and discontinuous flow structures in compressible fluid dynamics.This framework is based on the Boundary Variation Diminishing(BVD)principle which requires that the spatial reconstruction minimize the jumps of the reconstructed values at cell boundaries to reduces the dissipation errors in numerical solutions effectively.For the targeted flow structures,one can choose the BVD-admissible functions as the candidates for spatial reconstruction.A BVD algorithm can be then devised according to the BVD principle by properly selecting candidate functions for reconstruction so as to effectively control both numerical oscillation and dissipation.BVD schemes are substantially different from the conventional high-resolution schemes that use the polynomial reconstructions and nonlinear limiting projections to prevent numerical oscillation.We also present some BVD schemes of practical significance.Numerical verifications show that these schemes share the following desirable properties:1)effectively suppressing spurious numerical oscillation in the presence of strong shock or discontinuity;2)substantially reducing numerical dissipation errors;3)retrieving the underlying high-order linear schemes for smooth solutions over all wave numbers;4)the capability of resolving both smooth and discontinuous flow structures of wide-range scales with substantially improved solution quality;5)preventing contact discontinuity and material interface from smearing-out even for long-term computation.