一类面向高阶精度自适应流动计算的流场插值方法

网格自适应技术和高阶精度数值方法是提升计算流体力学复杂问题适应能力的有效技术途径. 将这两项技术结合需要解决一系列技术难题, 其中之一是高阶精度流场插值. 针对高阶精度自适应流动计算, 提出一类高精度流场插值方法, 实现将前一迭代步网格中流场数值解插值到当前迭代步网格中, 以延续前一迭代步中的计算状态. 为实现流场插值过程中物理量守恒, 该方法先计算新旧网格的重叠区域, 然后将物理量从重叠区域的旧网格中转移到新网格中. 为满足高阶精度要求, 先采用k-exact最小二乘方法对旧网格上的数值解进行重构, 获得描述物理量分布的高阶多项式, 随后采用高阶精度高斯数值积分实现物理量精确地转移到新网格单元上. 最后, 通过一个具有精确解的数值算例和一个高阶精度自适应流动计算算例验证了本文算法的有效性. 第一个算例结果表明当网格规模固定不变时, 插值精度阶数越高, 插值误差越小; 第二个算例显示本文方法可以有效缩短高精度自适应流动计算的迭代收敛时间. 

SOLUTION INTERPOLATION FOR HIGH-ORDER ACCURATE ADAPTIVE FLOW SIMULATION

Mesh adaptation and high order numerical methods are regarded as effective techniques to improve the adaptability of computational fluid dynamics (CFD) to complex problems. The combination of these two techniques requires solving a series of technical challenges, one of which is the flow field interpolation for high order numerical methods among different adaptation steps. A high-order accurate solution interpolation method is proposed for the high-order accurate adaptive flow simulation. In this method, it interpolates the numerical flow solution from the mesh in the previous iteration step into the mesh of the current iteration step, to allow the simulation to be restarted from the previous state. To realize the conservation of physical quantities in the process of flow field interpolation, the method first computes the overlapping regions of the new and old meshes and then transfers the physical quantities from the old mesh to the new mesh in the overlapping regions. To achieve high-order accuracy, the k-exact least-squares method is first used to reconstruct the numerical solution on the old mesh, and as a result, a polynomial with the required order that represents the distribution of the physical quantity is obtained over each element of the background mesh. Then Gaussian numerical integration is used to integrate the physical quantities over each element of the new mesh, which accurately transfers the physical quantities from the background mesh to each element of the new mesh. Finally, the effectiveness of the proposed algorithm is verified by a numerical example with an exact solution and an example of high-order accurate adaptive flow simulation. The results of the first example show that a smaller interpolation error exists when higher-order accurate interpolation is adopted, and the second example shows that the method in this paper can effectively shorten the iterative convergence time of high order accurate flow simulation.