Recasting an operator splitting solver into a standard finite volume flux-based algorithm. The case of a Lagrange-Projection-type method for gas dynamics
Rémi Bourgeois, Pascal Tremblin, Samuel Kokh, Thomas Padioleau
In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes.
Lagrange-flux schemes: reformulating second-order accurate Lagrange-remap schemes for better node-based HPC performance
Center for Mathematics and their applications CMLA, École Normale Supérieure de Cachan, Université Paris-Saclay, CNRS（数学及其应用中心 CMLA，巴黎高等师范学校卡尚分校，巴黎萨克雷大学，法国国家科学研究中心）；Maison de la Simulation, URS 3441, CEA Saclay（模拟之家，URS 3441，CEA 萨克莱）；CEA DAM DIF；CEA Saclay, DEN（CEA 萨克莱，DEN）；CGG
A-posteriori-steered $p$-robust multigrid and domain decomposition methods with optimal step-sizes for mixed finite element discretizations of elliptic problems
TU Wien（维也纳工业大学）；Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Inria（法国国家数字科学与多领域研究所）；Ecole nationale des ponts et chaussées（国立路桥学校）；University of Pittsburgh（匹兹堡大学）
University of California, Berkeley（加州大学伯克利分校）；Berkeley Center for Quantum Information and Computation, University of California, Berkeley（加州大学伯克利分校伯克利量子信息与计算中心）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
本文提出容错时代的 Qu-FEM 量子有限元算法，通过相互作用单元和局部到全局指示矩阵保留 FEM 几何灵活性，并给出线路、门复杂度、变系数积分与边界条件处理框架。
