ALE方法中一种新的二阶保界守恒重映算法

A new second-order bound-preserving conservative remapping algorithm in the ALE method

在用拉格朗日格式求解流体力学问题时,随着时间的推进,计算网格会扭曲变形,影响格式精度,甚至导致计算中断。因此,要在网格变形较大时进行网格重分和物理量重映,以保证网格质量和格式精度。针对间断有限元方法求解流体力学问题的二阶拉格朗日格式,给出了一种守恒重映算法。该重映算法包括两步:第一步是用已有重映方法计算新网格上的单元平均值,并用相应修补算法对单元平均值进行调整,保证单元平均值的保界性;第二步是由已得到的新单元平均值重构出新网格上分片一次多项式,再使用Van Leer限制器对新网格上的梯度进行限制,使之不出现新的极值。最后用数值算例表明了该重映算法的保界性和二阶收敛性。

When the Euler equations are solved using Lagrangian scheme,the fact that computational cells exactly follow fluid particles may result in severe grid deformation,even more cause inaccuracy and even breakdown of the computation in some cases.So it needs to rezone meshes and remap physical quantities when the deformation of computational grid is severe.Based on the second-order Lagrangian schemes that using the discontinuous Galerkin method to solve the Euler equations,a conservative remapping scheme is proposed.This remapping scheme has two steps：the first step is using the existing remapping method to obtain the approximate average values in the new cells,then using the repair algorithm to ensure the average values in the range of local bounds;the second step is using the average values to reconstruct the linear polynomial in the new cell,and using Van Leer limiter to limit the gradient of this linear polynomial to ensure no new extremum.Results of some numerical tests are presented and demonstrate that this remapping scheme is second-order accurate,conservative and bound-preserving.