爆炸冲击波强间断问题的高阶伪弧长算法研究

High order pseudo arc-length method for strong discontinuity of detonation wave

为了提高对冲击波强间断处的分辨率,通过引入弧长参数,使网格自适应地朝着间断处移动,并结合高精度WENO数值格式,进而达到了对大梯度物理量的高分辨率捕捉。针对网格移动造成的非均匀和非正交现象,通过坐标变换,使得计算过程在均匀正交的计算空间中进行。通过和有限体积下的数值结果对比,结合数值误差分析,可以看到高阶伪弧长数值算法不仅保证了高精度而且对间断的捕捉更加明显,在间断附近解的整体光滑性较好,网格的自适应移动使得解的奇异性得到了削弱,因此可以削弱高阶格式容易引起数值振荡这个缺点。最后采用高阶伪弧长算法计算了化学反应流问题,结果表明高阶伪弧长算法有着较快的收敛率,对于解决爆炸与冲击强间断问题有着较为明显的优势。

In this paper,in order to improve the resolution of capturing discontinuities,we introduce the pseudo arc-length parameter to make the mesh move to the discontinuities adaptively.By combining the high-precision WENO scheme with the pseudo arc-length algorithm,the advantages of both schemes can be shown,on the one hand,the solution has a higher convergence rate,on the other hand,it has a higher resolution for the region solution with larger physical variation.Because the traditional high-order scheme is based on Cartesian grid,and the grid in the pseudo arc-length numerical calculation is deformed.In view of the non-uniform grid and non-orthogonal deformation grid caused by the grid moving,the original deformed physical space is mapped to the uniform orthogonal arc-length calculation space by introducing coordinate transformation,and then the classical higher order scheme is used to solve the governing equations in the computational coordinate system.Through the comparison of some numerical examples and the analysis of numerical errors,it can be found that the pseudo arc-length algorithm is better than the finite volume method with fixed mesh.The high-order pseudo arc-length algorithm has a very high resolution to capture discontinuities,and the density of the grid near the discontinuities is very high.The adaptive grid movement weakens the singularity of the governing equation near the discontinuity,so the whole solution is smooth and the numerical oscillation is not obvious.This shows that the pseudo arc-length algorithm can overcome the shortcomings of high-order schemes which easily cause numerical oscillations.Finally,the chemical reaction flow problem is calculated.The results show that the high-order pseudo arc-length numerical algorithm with less mesh number has faster convergence rate and higher discontinuous resolution.Therefore,the high-order pseudo arc-length algorithm has obvious advantages in dealing with the strong discontinuity problem of explosion and shock.