The nonlinear convection terms in the governing equations of compressible fluid flows are hyperbolic in nature and are nontrivial for modelling and numerical simulation. Many numerical methods have been developed in the last few decades for this purpose and are typically based on Riemann solvers, which are strongly dependent on the underlying eigen-structure of the governing equations. Objective of the present work is to develop simple algorithms which are not dependent on the eigen-structure and yet can tackle easily the hyperbolic parts. Central schemes with smart diffusion mechanisms are apt for this purpose. For fixing the numerical diffusion, the basic ideas of satisfying the Rankine-Hugoniot (RH) conditions along with generalized Riemann invariants are proposed. Two such interesting algorithms are presented, which capture grid-aligned steady contact discontinuities exactly and yet have sufficient numerical diffusion to avoid numerical shock instabilities. Both the algorithms presented are robust in avoiding shock instabilities, apart from being accurate in capturing contact discontinuities, do not need wave speed corrections and are independent of eigen-strutures of the underlying hyperbolic parts of the systems.
A categorical embedding discontinuity-capturing shallow neural network for anisotropic elliptic interface problems
Department of Mathematics, National Central University（数学系，国立中央大学）；Department of Applied Mathematics, National Yang Ming Chiao Tung University（应用数学系，国立阳明交通大学）；Department of Applied Mathematics, National University of Kaohsiung（应用数学系，国立高雄大学）
A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems
National Central University（国立中央大学）；National Center for Theoretical Sciences, National Taiwan University（国立台湾大学国家理论科学研究中心）；National Yang Ming Chiao Tung University（国立阳明交通大学）
High-Resolution Weighted Essentially Non-Oscillatory Compact Least-Squares Schemes with Implicit Time Integration for Compressible Navier-Stokes Equations on Curvilinear Grids
Ningbo University（宁波大学）；Northeast Normal University（东北师范大学）；Tsinghua University（清华大学）
Comparison of adaptive multiresolution and adaptive mesh refinement applied to simulations of the compressible Euler equations
University of Southampton（南安普顿大学）；Instituto Nacional de Pesquisas Espaciais (INPE)（国家空间研究所）；Universidade Estadual de Campinas (Unicamp)（坎皮纳斯州立大学）；Université d’Aix-Marseille（艾克斯-马赛大学）
Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes
Institute of Mathematics, Johannes Gutenberg University（约翰内斯·古腾堡大学数学研究所）；High-Performance Scientific Computing, Centre for Advanced Analytics and Predictive Sciences, University of Augsburg（奥格斯堡大学高级分析与预测科学中心高性能科学计算）；Mathematical Institute, University of Cologne（科隆大学数学研究所）
Stable and accurate interface capturing advection schemes
CMLA, ENS Cachan, CNRS, Université Paris-Saclay（CMLA，巴黎高等师范学校卡尚分校，法国国家科学研究中心，巴黎萨克雷大学）；Maison de la simulation, USR 3441, CEA Saclay（模拟之家，USR 3441，CEA萨克莱）；CEA DAM DIF；CEA Saclay, DEN（CEA萨克莱，DEN）；CGG
A Fourth-order Conservative Adaptive Multiresolution Wavelet Upwind Scheme for Compressible Flows
School of Civil Engineering and Mechanics, Lanzhou University（兰州大学土木工程与力学学院）；Key Laboratory of Mechanics on Disaster and Environment in Western China (Lanzhou University), the Ministry of Education, Lanzhou University（兰州大学西部灾害与环境力学教育部重点实验室）；Department of Mechanical Engineering, Institute of Science Tokyo（东京科学大学机械工程系）
