高性能数值计算团队的主要研究领域为实际工程问题中微分方程模型的数值模拟及其理论分析。作为国内外研究的热门课题之一，微分方程在实际生活中拥有广泛的应用，例如天气预报、石油开发及城市交通等。然而大多数微分方程的真解是无法直接给出具体表达式的，因此设计准确、高效的数值方法模拟真解是行之有效的解决途径。本团队具体的研究方向主要包括：时间相关问题的时空并行区域分解算法的分析与应用；时间依赖区域上的间断有限元算法的设计与分析；流体力学及电磁学问题中数学模型的分析与模拟。
目前该研究团队在此方面“标志性”成果和创新性贡献简述如下：
（1）在时空并行区域分解算法求解抛物方程的研究中取得了较大突破。将时空并行算法推广到多层上，不仅给出了相应的最优收敛性理论，也成功的将该算法在超级算计算机天河二号上实现，给出了该算法在四维网格上求解三维抛物方程的 8192 个CPU 核的可扩展性结果。
（2）构造了高精度的半隐式任意拉格朗日欧拉局部间断有限元方法求解对流扩散方程，并给出了全离散意义下格式的稳定性和误差估计，建立平行于固定网格的间断有限元方法的理论分析框架。
（3）给出了非定常对流扩散方程的非线性保正格式，并对含守恒型非线性能量时间导数项的扩散问题离散格式发展了新的论证技术，对全隐离散差分格式给出了解的存在性、唯一性、收敛性、稳定性的严格证明。
代表性论文列表
1. Shishun Li, Jing-Yuan Wang and Xiao-Chuan Cai, A-stable high-order block implicit methods for parabolic equations. SIAM Journal on Numerical analysis, 61(2023), 1858–1884.
2. Shishun Li, Xinping Shao and Xiao-Chuan Cai, Multilevel space-time additive Schwarz methods for parabolic equations. SIAM Journal on Scientific Computing. 40(2018), A3012-A3037.
3. Shishun Li and Xiao-Chuan Cai, Convergence analysis of two-level space-time additive Schwarz method for parabolic equations, SIAM Journal on Numerical analysis, 53(2015), 2727-2751.
4. Lingling Zhou and Yinhua Xia, Arbitrary Lagrangian-Eulerian local discontinuous Galerkin method for linear convection-diffusion equations, Journal of Scientific Computing, 90(2022).
5. Lingling Zhou, Yinhua Xia and Chi-Wang Shu, Stability analysis and error estimates of arbitrary Lagrangian-Eulerian discontinuous Galerkin method coupled with Runge-Kutta time-marching for linear conservation laws, ESAIM: Mathematical Modelling and Numerical Analysis, 53(2019), 105-144.
6. Haifeng Niu, Danping Yang and Jianwei Zhou, Numerical analysis of an optimal control problem governed by the stationary Navier-Stokes equations with global velocity-constrained, Communications in Computational Physics, 24(2018), 1477-1502.
7. Lingling Zhou and Ruihan Guo, Optimal error estimates of the local discontinuous Galerkin method and high-order time discretization scheme for the Swift-Hohenberg equation, Journal of Scientific Computing, 93(2022).
8. Yanfang Wang, Chencheng Hu and Dongbin Wang. Ferromagnetism in (Cr, Mn)-co-doped 3C–SiC analyzed using density functional theory. AIP Advances, 13(2023), 065330.
9. Yanmei Zhang, Xia Cui and Guangwei Yuan, Nonlinear iteration accelerationsolution for equilibrium radiation diffusion equation, ESAIM: Mathematical Modelling and Numerical Analysis, 54(2020), 1465-1490.
10. Zhenzhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases, Discrete and Continuous Dynamical Systems - Series B, 28(2023), 359–384.
