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Computer Science, Interdisciplinary Applications
Ziyao Xu, Chi-Wang Shu
Summary: Recently, there have been studies on the positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations. However, the current implementation and theoretical proofs of positivity-preserving methods for these equations are unnecessarily complicated and constrained. This paper introduces a more appropriate definition of mass conservation and establishes high-order positivity-preserving limited discontinuous Galerkin methods that are simpler to implement and prove, and can preserve the positivity of solutions for linear and nonlinear equations with fixed wind direction.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Erica R. Johnson, James A. Rossmanith, Christine Vaughan
Summary: The HyQMOM variant of QMOM is proven to have moment-invertibility over a convex region in solution space. A high-order discontinuous Galerkin (DG) scheme is developed to solve the resulting fluid system, with novel limiters introduced to guarantee the system's hyperbolicity. The scheme is also extended to include a BGK collision operator, which is shown to be asymptotic-preserving in the high-collision limit.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Ziyao Xu, Chi -Wang Shu
Summary: In this work, we propose third order maximum-principle-satisfying and positivity-preserving schemes for scalar conservation laws and the Euler equations based on the Lax-Wendroff time discretization and discontinuous Galerkin spatial discretization. The accuracy and effectiveness of the maximum-principle-satisfying and positivity-preserving techniques are demonstrated by ample numerical tests.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Gerardo Hernandez-Duenas, Guillermo Ramirez-Santiago
Summary: The work presents a hyperbolic 1D model for blood flow in compliant tilted vessels and discusses numerical schemes inheriting important features of the model. It characterizes a large class of smooth equilibrium solutions and describes entropy-satisfying numerical schemes. The numerical results show robustness, stability, and accuracy, aiming at medical applications.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2021)
Article
Computer Science, Interdisciplinary Applications
Arpit Babbar, Sudarshan Kumar Kenettinkara, Praveen Chandrashekar
Summary: The Lax-Wendroff method is a single step method for solving partial differential equations, which combines a flux reconstruction version with a Jacobian-free procedure. It has the advantages of high CFL numbers and high accuracy.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Wuchen Li, Siting Liu, Stanley Osher
Summary: In this paper, we propose, study, and compute solutions to a class of optimal control problems for hyperbolic systems of conservation laws and their viscous regularization. We take the barotropic compressible Navier-Stokes equations as a canonical example and develop a metric variational problem for it. The numerical examples demonstrate the effectiveness of the proposed algorithm.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
S. Mousavi Yeganeh, J. Farzi
Summary: This paper utilizes MPP and PP parametrized flux limiters to achieve strict maximum principle and positivity-preserving property for solving hyperbolic conservation laws, demonstrating efficiency and effectiveness through high-order MPP RK-SV and PP RK-SV schemes. The proposed schemes maintain the maximum principle without additional time step restrictions and preserve the high-order accuracy for linear advection problems.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Kailiang Wu, Yulong Xing
Summary: This paper introduces novel high-order accurate DG schemes for compressible Euler equations under gravitational fields, which exhibit properties of well-balancedness and positivity preservation. By carefully designing the spatial discretization with suitable source term reformulation and modified HLLC flux, these schemes achieve the simultaneous satisfaction of both properties. Extensive numerical tests demonstrate the schemes' ability to preserve equilibrium states, capture perturbations, handle low density and/or pressure problems, and provide good resolution for smooth and discontinuous solutions.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Gerard Gallice, Agnes Chan, Raphael Loubere, Pierre-Henri Maire
Summary: This paper presents a novel subface flux-based Finite Volume (FV) method for discretizing multi-dimensional hyperbolic systems of conservation laws on general unstructured grids. The method utilizes a simple Eulerian Riemann solver, which is constructed from its Lagrangian counterpart through the Lagrange-to-Euler mapping, to approximate the subface fluxes. The resulting multi-dimensional FV scheme is characterized by an explicit time step condition that ensures positivity preservation and entropy stability.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Chuan Fan, Xiangxiong Zhang, Jianxian Qiu
Summary: In this paper, a high order weighted essentially non-oscillatory (WENO) finite difference discretization method is constructed for solving the compressible Navier-Stokes (NS) equations. The method achieves positivity preservation of density and internal energy through a positivity-preserving flux splitting and a scaling positivity-preserving limiter. The core advantages of the proposed method are robustness and efficiency, making it particularly suitable for solving challenging problems involving low density and low pressure flow regime.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
T. Dzanic, F. D. Witherden
Summary: In this work, a positivity-preserving adaptive filtering approach is proposed for discontinuous spectral element approximations of the ideal magnetohydrodynamics equations. The approach combines entropy filtering and eight-wave method to capture shocks and enforce a divergence-free magnetic field. An operator-splitting approach is introduced to handle non-conservative source terms and ensure the satisfaction of positivity and entropy constraints. A computationally efficient algorithm for solving the optimization process is presented. Numerical experiments demonstrate the efficacy of the proposed scheme in handling various problems.
COMPUTERS & FLUIDS
(2023)
Article
Computer Science, Interdisciplinary Applications
T. Dzanic, F. D. Witherden
Summary: This work presents a positivity-preserving entropy-based adaptive filtering method for shock capturing in discontinuous spectral element methods. The method adapts the filter strength to enforce positivity and a local discrete minimum entropy principle, allowing it to robustly handle strong discontinuities with sub-element resolution. It does not require problem-dependent parameter tuning and can be easily implemented on general unstructured meshes with relatively low computational cost.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Feng Xiao, Wei Shyy, Lin Fu
Summary: In this study, a new low-dissipation TENO scheme is proposed for compressible flow simulations, which enhances the capability of resolving discontinuities and suppresses numerical oscillations through an improved discontinuity-detecting criterion, a local interpolation-like strategy, and a two-steepness approximation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Man Long Wong, Jordan B. Angel, Michael F. Barad, Cetin C. Kiris
Summary: A robust, highly accurate, and efficient diffuse interface method for simulating compressible gas-liquid two-phase flows is presented using the high-order finite difference WCNS scheme. The method ensures positivity and boundedness by constructing limiting procedures based on assumptions about the ratios of specific heats of gas and liquid. Numerical tests demonstrate the method's robustness and high accuracy, even under extreme conditions.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Jeremy Chouchoulis, Jochen Schutz, Jonas Zeifang
Summary: Based on previous research, this paper presents a novel method for solving hyperbolic conservation laws, which achieves higher-order consistency and improves the flexibility and stability of the time integration by adding Runge-Kutta-type stages. Additionally, a Jacobian-free approximation is used to compute the high-order flux derivatives.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Kailiang Wu, Dongbin Xiu
JOURNAL OF COMPUTATIONAL PHYSICS
(2018)
Article
Computer Science, Interdisciplinary Applications
Yeonjong Shin, Kailiang Wu, Dongbin Xiu
JOURNAL OF COMPUTATIONAL PHYSICS
(2018)
Article
Mathematics, Applied
Kailiang Wu, Huazhong Tang
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2018)
Article
Computer Science, Interdisciplinary Applications
Kailiang Wu, Dongbin Xiu
JOURNAL OF COMPUTATIONAL PHYSICS
(2019)
Article
Mathematics, Applied
Kailiang Wu, Chi-Wang Shu
NUMERISCHE MATHEMATIK
(2019)
Article
Computer Science, Interdisciplinary Applications
Tong Qin, Kailiang Wu, Dongbin Xiu
JOURNAL OF COMPUTATIONAL PHYSICS
(2019)
Article
Computer Science, Interdisciplinary Applications
Kailiang Wu, Dongbin Xiu
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Mathematics, Applied
Zhen Chen, Kailiang Wu, Dongbin Xiu
JOURNAL OF SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Jun Hou, Tong Qin, Kailiang Wu, Dongbin Xiu
Summary: A novel correction algorithm is proposed for multi-class classification problems with corrupted training data, which can deliver correct classification results and significantly better recovery results compared to models without the correction algorithm. The theoretical findings in the paper are verified by numerical examples.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Kailiang Wu, Yulong Xing
Summary: This paper introduces novel high-order accurate DG schemes for compressible Euler equations under gravitational fields, which exhibit properties of well-balancedness and positivity preservation. By carefully designing the spatial discretization with suitable source term reformulation and modified HLLC flux, these schemes achieve the simultaneous satisfaction of both properties. Extensive numerical tests demonstrate the schemes' ability to preserve equilibrium states, capture perturbations, handle low density and/or pressure problems, and provide good resolution for smooth and discontinuous solutions.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Kailiang Wu, Tong Qin, Dongbin Xiu
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Kailiang Wu, Chi-Wang Shu
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Kailiang Wu, Dongbin Xiu
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2019)
Article
Mathematics, Applied
Kailiang Wu, Chi-Wang Shu
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2018)