Article
Computer Science, Interdisciplinary Applications
Yimin Lin, Jesse Chan, Ignacio Tomas
Summary: In this work, high-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations are studied. A positivity limiting strategy is introduced to ensure the well-posedness of the methods, by blending high-order solutions with a low-order positivity-preserving discretization. The proposed strategy is shown to be accurate and robust through numerical experiments.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Giuseppe Orlando, Paolo Francesco Barbante, Luca Bonaventura
Summary: This paper proposes an efficient, accurate, and robust IMEX solver for the compressible Navier-Stokes equations, which describe non-ideal gases with a general cubic equation of state and Stiffened-Gas EOS. The method utilizes an h-adaptive Discontinuous Galerkin spatial discretization and an Additive Runge Kutta IMEX method for time discretization. It is specifically designed for low Mach number applications and allows for simulation at a reduced computational cost.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Sohail Reddy, Maciej Waruszewski, Felipe A. V. de Braganca Alves, Francis X. Giraldo
Summary: This work presents IMplicit-EXplicit (IMEX) formulations for discontinuous Galerkin (DG) discretizations of the compressible Euler equations governing non-hydrostatic atmospheric flows. Two different IMEX formulations are proposed to address the stiffness problem caused by the governing dynamics and the domain discretization. Efficient Schur complements are derived for both equation sets, and their performance is studied on 2D and 3D test problems, showing their convergence rates and efficiency in mesoscale and global applications.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Mustafa E. Danis, Jue Yan
Summary: This study proposes a new formula for the nonlinear viscous numerical flux and extends it to the compressible Navier-Stokes equations using the direct discontinuous Galerkin method with interface correction (DDGIC). The new method simplifies the implementation and enables accurate calculation of physical quantities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Vit Dolejsi, Magnus Svard
Summary: The aim of this study is to evaluate a recently proposed model for viscous and heat conducting compressible fluids and compare it with the Navier-Stokes model. By accurately simulating a suite of test cases and comparing various measures, it was found that both models exhibit remarkable similarity when alpha equals 1, with differences typically around 1%.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Qiaolin He, Xiaoding Shi
Summary: This paper introduces a fully discrete method combining LDG finite element method and SAV approach for the compressible Navier-Stokes-Allen-Cahn system, allowing separate solution of velocity, density, and mass concentration of fluid mixture, and using SDC method to improve temporal accuracy. Numerical experiments demonstrate the high accuracy in both time and space, discretized energy stability, and efficiency of the proposed method.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Physics, Fluids & Plasmas
L. M. Yang, C. Shu, Z. Chen, Y. Y. Liu, J. Wu, X. Shen
Summary: A high-order gas kinetic flux solver (GKFS) is developed for 2D compressible flows, which evaluates numerical fluxes based on the local asymptotic solution to the Boltzmann equation. It achieves high-order accuracy through a simplified local asymptotic solution and outperforms the second-order counterpart in numerical examples, demonstrating its accuracy and capability.
Article
Computer Science, Interdisciplinary Applications
Chen Liu, Xiangxiong Zhang
Summary: In this paper, a scheme for solving compressible Navier-Stokes equations with desired properties is constructed. The scheme achieves high order spatial accuracy, conservation, and positivity-preserving of density and internal energy. Numerical tests show that higher order polynomial basis produces better numerical solutions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mechanics
Chao Zhang, Qibing Li, Peng Song, Jiequan Li
Summary: A two-stage fourth-order subcell finite volume (SCFV) method combining the gas-kinetic solver (GKS) with subcell techniques was developed to enhance compactness and efficiency, improving accuracy and efficiency in compressible flow simulations.
Article
Computer Science, Interdisciplinary Applications
Michael Baldauf
Summary: A solver based on the Discontinuous Galerkin method is used to solve the Euler equations, with optional diffusion. The horizontally explicit, vertically implicit approach is applied to prevent tiny time-steps, and terrain-following coordinates are used to consider orography in the correct approximation order. The approach demonstrates validity in several test cases relevant for the atmosphere, showing tolerance for steep terrain.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Alberto Bressan, Yi Jiang, Hailiang Liu
Summary: This paper numerically studies a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, revealing the existence of initial value problems with multiple solutions and highlighting fundamental obstacles towards the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Lingquan Li, Jialin Lou, Hiroaki Nishikawa, Hong Luo
Summary: In this study, a new hyperbolic Navier-Stokes system is proposed, introducing gradients as auxiliary variables and developing efficient reconstructed Galerkin methods. By recycling gradient variables, higher order polynomial solutions for primary variables can be obtained without increasing degrees of freedom. Numerical experiments demonstrate that the developed methods can achieve the designed accuracy and provide an attractive alternative for solving the compressible Navier-Stokes equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
R. Al Jahdali, L. Dalcin, R. Boukharfane, I. R. Nolasco, D. E. Keyes, M. Parsani
Summary: This study proposes new optimized explicit Runge-Kutta schemes for the integration of systems of ordinary differential equations arising from high-order entropy stable collocated discontinuous Galerkin methods. By optimizing the stability region of the time integration schemes, the efficiency and robustness of computational fluid dynamics simulations can be improved, leading to significant time and resource savings.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Waldemar Rachowicz, Adam Zdunek, Witold Cecot
Summary: The DPG method is used for simulating three-dimensional compressible viscous flows, with stable schemes constructed for problems with small perturbation parameters. The approach uses weak formulation and specially designed optimal test functions, with a built-in a posteriori error estimation and mesh adaptivity for resolving reliable viscous fluxes in simulations of viscous flows.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mechanics
Ningyu Zhan, Rongqian Chen, Yancheng You
Summary: The meshfree method based on DGKS, named meshfree-DGKS, is proposed for simulating incompressible/compressible flows. Discretization is done using the least squares-based finite difference approach, with the concept of numerical flux introduced to handle compressible problems with discontinuities effectively. The method allows for capturing shock waves easily by reconstructing fluxes at mid-points based on the local solution of the Boltzmann equation.