Developing numerical software for both scientific exploration and engineering applications is one of my research interests. Currently, I mainly concern solutions of partial differential equations from a variety of areas such as computational fluid dynamics, electronic structure calculations, and computational micromagnetics. Due to, e.g., nonlinearity, high dimensionality, complexity of the domain and boundary conditions, etc., analytical solutions generally are not available. Hence, a numerical solver becomes vital for PDE-based applications, and the efficiency of the numerical solver is one of the most important features desired.

AFEPack (Adaptive Finite Element Package) is a general-purpose C++ library for numerical solutions of partial differential equations, originally developed by Prof. Ruo Li from Peking University and Prof. Wenbin Liu from BNU-HKBU UIC. With over two decades development, AFEPack has been successfully applied for scientific and engineering computational problems in a variety of areas. To promopt the library, recently collaborators and I are working on the following paper Z. Cai et al, AFEPack: a general-purpose C++ library for numerical solutions of partial differential equations, preprint.

My research on design, analysis, and coding of numerical algorithms is based on AFEPack. Three specific-purpose software have been roughly formed in compressible computational fluid dynamics, electronic structure calculations, and computational micromagnetics, respectively, which are introduced below.

AFVM4CFD is designed for the numerical simulations for compressible fluid dynamics. The package is based on finite volume methods, and aims at solving Euler and Navier-Stokes equations. The development of this package started when I was a PhD student. Collaborators include Tao Tang (UIC), Ruo Li(Peking U.), Yana Di (UIC), Nianyu Yi(Xiangtan U.), Xiaohua Zhang (Yunnan Normal University), Xucheng Meng(Beijing Normal U.), Li Chen , Feng Yang (Ant Limited), Jingfeng Wang (U. of Macau), Nizheng Liao(U. of Macau). The features of the package include

• 20K+ lines code for core algorithms;
• Handling both steady and time-dependent (reactive) Euler equations;
• High order numerical accuracy
• Mesh adaptivity module;
• For applications in design of vehicle shape, detonation simulations, etc.

AFEABIC is developed for the electronic structure calculations, which potentially has wide applications in nano sciences, quantum computational chemistry, etc. The development of this package started when I was a postdoc at Michigan State University. Collaborators include Gang Bao (Zhejiang U.), Di Liu (Michigan State U.), Hehu Xie (CAS), Xin Liu (CAS), Zhenning Cai (NUS), Fei Xu (Beijing U. of Tech.), Bin Gao (CAS), Jie Zhou (Xiangtan U.), Yang Kuang (Guangdong U. of Tech.), Yedan Shen (Guangzhou U.), Hongfei Zhan (Peking U.), Ting Wang (Southwest Petroleum U.), Chengyu Liu (U. of Macau). The features of the package include

• 30K+ lines code for core algorithms;
• For both ground-state and dynamics simulations;
• For both full-potential and pseudo-potential calculations;
• Mesh adaptivity module;
• For applications in geometry optimization of given molecules, and nano- and quantum-optics problems;

AFEMAG is developed for numerical simulations in computational micromagnetics. Collaborators include Jingrun Chen (U. of Science and Technology of China), Lei Yang (Macau U. of Science and Technology), Rui Du (Soochow U.), Jing Wu (Soochow U.), Jiajun Zhan and Zixuan Cui and Jiayun He (Macau U .of Science and Technology). The package is developed for supporting our research on numerical methods for micromagnetics, as well as the research on the mechanics for the formation of Skyrmion. The features of the package include

• 10K+ lines code for core algorithms;
• For ground-state calculations of micromagnetics systems;
• Fast calculation of demagnetization field in general domain with complex geometry

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