一类高效率高分辨率加映射的WENO格式及其在复杂流动问题数值模拟中的应用

由于映射操作会带来额外的计算时间消耗, 传统加映射的WENO格式存在计算效率低的缺陷. 为了提高传统加映射WENO格式的计算效率, 通过利用标准符号函数的一种近似逼近函数构造出一族近似常值映射函数, 本文提出了一种新的加映射WENO格式, 称为WENO-ACM. 新映射函数满足文献中已有WENO-PM6格式映射函数的全部设计要求, 其中WENO-PM6是一种为了克服经典WENO-M格式潜在的精度丢失缺陷而提出的格式. 新格式保留了WENO-PM6在低耗散和高分辨率方面的优势, 同时, 显著的减少了每个时间步映射过程中的数学运算操作数, 进而在计算效率方面获得了明显的提升. 理论分析表明, 新格式在即使包含临界点的光滑区域也能够获得最佳收敛精度. 对近似色散关系的研究表明, 新格式的频谱特性也得到了显著的提升. 对大量标准测试算例进行了模拟计算, 包括精度测试、激波管问题、激波?熵波相互作用、爆炸波相互碰撞、二维黎曼问题、双马赫反射、前台阶流动、瑞利-泰勒不稳定性和开尔文?亥姆霍兹不稳定性问题等. 与广泛认可的WENO-JS, WENO-M, WENO-PM6格式综合比较发现, 新提出的WENO-ACM格式在高效率、低数值耗散和间断捕捉等方面都有显著的改进. 最重要的是, 与WENO-M和WENO-PM6格式相比, WENO-ACM将相对于WENO-JS格式的额外计算时间消耗分别减少了80%和90%以上. 

A HIGH-EFFICIENCY AND HIGH-RESOLUTION MAPPED WENO SCHEME AND ITS APPLICATIONS IN THE NUMERICAL SIMULATION OF PROBLEMS WITH COMPLEX FLOWS

The traditional mapped weighted essentially non-oscillatory (WENO) schemes commonly suffer from the drawback of low-efficiency, since they usually require the mapping processes resulting in extra computational costs. The goal of the present work is to improve the efficiency of the mapped WENO schemes. By designing a set of approximate constant mapping function which is devised using an approximation of the standard signum function, a novel mapped WENO scheme is proposed. The new mapping function is devised to meet the overall criteria for a proper mapping function required in the design of the WENO-PM6 scheme. The WENO-PM6 scheme was presented to overcome the potential loss of accuracy of the well-validated WENO-M scheme in previously published literature. The new proposed mapped WENO scheme is denoted as WENO-ACM. It maintains almost all advantages of the WENO-PM6 scheme, such as low dissipations and high resolutions. However, it decreases the number of mathematical operations remarkably in every mapping process leading to a significant improvement of efficiency. Theoretical analysis indicates that the new scheme can attain the optimal convergence rate of accuracy regardless of critical points. The investigation of approximate dispersion relation (ADR) shows that the spectral properties of the new scheme are significantly improved. A variety of benchmark-test problems, including accuracy tests, standard shock-tube problem, Mach 3 shock-entropy wave interaction, Woodward-Colella interacting blast waves, 2D Riemann problem, double Mach reflection, forward-facing step problem, Rayleigh-Taylor instability and Kelvin-Helmholtz instability are conducted. Compared to the well-established WENO-JS, WENO-M, WENO-PM6 schemes comprehensively, the present scheme exhibits significantly improved high efficiency, very high resolution and sharp discontinuity capturing. Most importantly, the extra computational cost of WENO-ACM compared to WENO-JS is much lower than those of WENO-M and WENO-PM6. Specifically, WENO-ACM can reduce the extra computational cost compared to WENO-JS more than 80% and 90% against WENO-M and WENO-PM6, respectively.