Article
Engineering, Mechanical
Xuehua Yang, Haixiang Zhang, Qi Zhang, Guangwei Yuan
Summary: We propose a positivity-preserving finite volume scheme on non-conforming quadrilateral distorted meshes with hanging nodes for subdiffusion equations. The scheme uses the two-point flux technique and the L1 scheme to handle weak singularity and hanging nodes effectively.
NONLINEAR DYNAMICS
(2022)
Article
Environmental Sciences
Elizaveta Klein, Susie M. L. Hardie, Wolfgang Kickmaier, Ian G. McKinley
Summary: A critical aspect of selecting sites for deep geological repositories for high-level radioactive waste is their ability to slow down the migration of radionuclides released from engineered barriers. Models play a crucial role in connecting field observations and laboratory studies on rock/water/radionuclide interactions. Despite remaining uncertainties, utilizing knowledge from anthropogenically contaminated sites can enhance safety cases for geological repositories.
SCIENCE OF THE TOTAL ENVIRONMENT
(2021)
Article
Computer Science, Interdisciplinary Applications
Gang Peng, Zhiming Gao, Wenjing Yan, Xinlong Feng
Summary: This paper introduces a new cell-centered positivity-preserving finite volume scheme for solving 3D anisotropic diffusion problems on distorted meshes. The scheme utilizes primary and auxiliary unknowns, with discretization, interpolation, and acceleration methods to improve computational efficiency and numerical accuracy.
COMPUTER PHYSICS COMMUNICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Yuichi Kuya, Wataru Okumura, Keisuke Sawada
Summary: A kinetic energy and entropy preserving finite-volume scheme on unstructured meshes is proposed for stable compressible flow computations. The scheme maintains the characteristics of kinetic energy and entropy during the discretization process and performs stable computations on various unstructured meshes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Physics, Multidisciplinary
Di Yang, Gang Peng, Zhiming Gao
Summary: This paper proposes a positivity-preserving finite volume scheme for simulating the nonequilibrium radiation diffusion equations on distorted meshes. The scheme uses fixed stencils and is locally conservative. It utilizes an interpolation algorithm to calculate auxiliary unknowns, while the primary unknowns are cell-centered. The scheme is shown to be efficient and strong in preserving positivity through numerical results.
Article
Mathematics, Applied
Fei Zhao, Zhiqiang Sheng, Guangwei Yuan
Summary: In this paper, a nonlinear strong positivity-preserving finite volume scheme on tetrahedral meshes for nonstationary convection-diffusion equations is presented. The scheme discretizes the diffusion and convection terms nonlinearly and introduces a nonlinear correction for the right-hand term. The scheme is cell-centered, locally conservative, and does not assume non-negativity of auxiliary unknowns. The existence of a solution and the strong positivity-preserving property for the nonlinear system are proven, and numerical examples demonstrate the positivity and second-order convergence rate of the solution.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2023)
Article
Mathematics, Applied
Jiexing Zhang, Yi Wang, Guoxi Ni
Summary: In this paper, a positivity-preserving residual distribution based finite volume scheme is proposed for steady diffusion problems on unstructured triangular meshes. The scheme combines the RD method and a positivity-preserving finite volume scheme to guarantee a conservative property. Numerical experiments demonstrate the optimal convergence rates and the positivity-preserving property of the new scheme.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Physics, Mathematical
Shuai Su, Huazhong Tang, Jiming Wu
Summary: This paper introduces an efficient finite volume scheme on general polygonal meshes for calculating the two-dimensional nonequilibrium three-temperature radiation diffusion equations. The scheme does not require nonlinear iteration for linear problems and can ensure the positivity, existence, and uniqueness of the cell-centered solutions obtained on the corrector phase. Numerical experiments demonstrate the accuracy, efficiency, and positivity of the scheme on various distorted meshes.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Ashwani Assam, Ganesh Natarajan
Summary: A new second-order accurate finite volume scheme for diffusion equations with discontinuous coefficients on unstructured meshes is proposed in the paper. This scheme is based on least squares reconstruction and the concept of alpha damping for linear exactness, with two variants introduced for different conditions to improve the accuracy of the solution. Numerical experiments show that these schemes are capable of estimating the solution and fluxes accurately on generic polygonal meshes, and are discrete extremum preserving.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Yaping Chen, Kailiang Wu
Summary: This paper presents a highly robust third-order accurate finite volume weighted essentially non-oscillatory (WENO) method for special relativistic hydrodynamics on unstructured triangular meshes. The proposed method is proved to preserve the physical constraints and novel techniques are introduced to address the strong nonlinearity.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Physics, Mathematical
Weijie Zhang, Yulong Xing, Yinhua Xia, Yan Xu
Summary: This paper proposes a high-order accurate DG method for the compressible Euler equations on unstructured meshes under gravitational fields, which preserves a general hydrostatic equilibrium state and guarantees the positivity of density and pressure. Through a special way to recover the equilibrium state and the design of novel interface variables, the scheme achieves well-balanced and positivity-preserving properties.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Gang Peng
Summary: A new positivity-preserving finite volume scheme is proposed for solving the convection-diffusion equation on distorted meshes in 2D or 3D. The scheme utilizes a nonlinear two-point flux approximation for diffusion flux and a second-order upwind method with a slope limiter for convection flux. The scheme exhibits second-order convergence rate and is effective in solving the convection-diffusion problem based on numerical results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Francesco Magaletti, Mirko Gallo, Sergio P. Perez, Jose A. Carrillo, Serafim Kalliadasis
Summary: A finite-difference hybrid numerical method is proposed for solving the isothermal fluctuating hydrodynamic equations. The main focus is on ensuring the positivity-preserving property of the numerical scheme, which is crucial for its functionality and reliability in simulating fluctuating vapour systems. The proposed scheme's accuracy and robustness are verified against benchmark theoretical predictions, and it is applied to the challenging bubble nucleation process, capturing its salient features.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Fan Zhang, Jian Cheng
Summary: In this paper, a physical-constraint-preserving high-order finite volume WENO scheme is developed for compressible two-medium gas-gas and gas-liquid interfacial flows. The scheme presents a high-order finite volume approach for solving the five-equation transport model and carefully designs the discretization of the non-conservative term. A novel analysis is proposed to derive the sufficient condition for physically admissible solutions, and a bound- and positivity-preserving limiting procedure is developed based on the finite volume framework. Numerical results demonstrate the importance of the proposed strategy in enhancing the robustness for high-order methods under severe flow conditions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
S. Busto, L. Rio-Martin, M. E. Vazquez-Cendon, M. Dumbser
Summary: This paper presents a new hybrid semi-implicit finite volume / finite element scheme for the numerical solution of the compressible Euler and Navier-Stokes equations on unstructured staggered meshes. The chosen semi-implicit discretization leads to computational efficiency for flows at all Mach numbers. The method shows promising capabilities through computational results for a wide range of benchmark problems.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Physics, Multidisciplinary
Lulu Li, Haiyan Su, Xinlong Feng
Summary: In this paper, an adaptive defect-correction method for natural convection equations is proposed, which addresses the convection dominance problem caused by a high Rayleigh number. A new recovery-type posteriori estimator is combined to solve the discontinuity of the gradient of the numerical solution. The results of reliability and efficiency analysis as well as numerical investigations confirm the stability, accuracy, and efficiency of the proposed method.
Article
Mathematics, Applied
Yinnian He, Xiaojing Dong, Xinlong Feng
Summary: This paper presents the Stokes, Newton, and Oseen iterative finite element methods for the 3D steady MHD equations. The methods approximate the solution pair ((u, B), p) of the equations by combining iterative methods with the finite element method. The paper provides uniform stability and convergence results for the iterative finite element solutions under convergence conditions and demonstrates the efficiency of the methods through numerical tests.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Gang Peng, Zhiming Gao, Xinlong Feng
Summary: In this paper, an extremum-preserving finite volume scheme is proposed for the two-dimensional three-temperature radiation diffusion equations. The scheme has a fixed stencil, satisfies the local conservation condition and discrete extremum principle, and exhibits efficiency and accuracy in solving the equations on distorted meshes.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Xinlong Feng, Xiaoli Lu, Yinnian He
Summary: In this paper, a difference finite element (DFE) method is proposed for the 3D steady Stokes equations. The method transforms the 3D problem into a series of 2D problems and ensures the existence, uniqueness, stability, and first-order convergence of the solution through the discrete inf-sup condition.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Ke Zhang, Haiyan Su, Xinlong Feng
Summary: In this paper, a second order decoupled time discretization method for solving the unsteady incompressible magneto-hydrodynamic equations is presented. The method utilizes rotational velocity correction projection scheme and extrapolation algorithm to handle the hydrodynamic equation and the magnetic equation nonlinear terms respectively. The unconditional energy stability of the proposed scheme is proven using the Gauge-Uzawa format. Numerical experiments including accuracy test, stability test, flow around a cylinder test and driven cavity flow problem are conducted to verify the effectiveness of the presented scheme.
COMPUTERS & FLUIDS
(2022)
Article
Computer Science, Interdisciplinary Applications
Fangxiong Cheng, Hui Xu, Xinlong Feng
Summary: In this paper, a non-intrusive data-driven model order reduction method called (r)PODANNs is proposed. This method performs (r)POD priori dimension reduction on high-fidelity data set and implicitly constructs the mapping relationship using neural network. The method reduces the time cost of the reduced order model and is suitable for high-fidelity models with larger degrees of freedom and higher complexity.
COMPUTERS & FLUIDS
(2022)
Article
Computer Science, Interdisciplinary Applications
Jiali Xu, Xinlong Feng, Haiyan Su
Summary: This paper presents a two-level Newton iterative method based on nonconforming finite element discretization for solving 2D/3D stationary incompressible magneto-hydrodynamics equations. The method uses Newton iterations on a coarse mesh and correction by Stokes iteration on a fine mesh, resulting in higher computational efficiency and convergence rate compared to the one-level method.
COMPUTERS & FLUIDS
(2022)
Article
Physics, Multidisciplinary
Haiyan Su, Jiali Xu, Xinlong Feng
Summary: Several two-level iterative methods based on nonconforming finite element methods are studied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. The proposed methods, motivated by applying iterations and correction on coarse and fine grids, are proven to be stable and effective through rigorous analysis on stability and error estimate.
Article
Computer Science, Interdisciplinary Applications
Xufeng Xiao, Xinlong Feng
Summary: This paper presents a highly efficient space-time operator splitting finite element method for solving the two- and three-dimensional Allen-Cahn equations. The method reduces the storage requirements and complexity of high-dimensional computations by splitting the problem into one-dimensional subproblems. It is space-time second-order and can be performed in parallel.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Mathematics, Applied
Yan Wang, Xufeng Xiao, Xinlong Feng
Summary: This paper presents a novel numerical algorithm for efficient modeling of three-dimensional shape transformation governed by the modified Allen-Cahn equation. The proposed method achieves high precision and high efficiency through operator splitting, temporal p-adaptive strategy, and parallel least distance modification technique.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Huili Zhang, Xinlong Feng, Kun Wang
Summary: This article focuses on anisotropic immersed finite element (AIFE) methods for the elliptic interface system. The existence and uniqueness of the weak solution are established, and a linear AIFE method is constructed to solve the interface system. Unlike the scalar problem, the jump conditions for each unknown function are coupled together in the anisotropic problem considered here. By defining suitable interpolations, error estimates in L2-$$ {L}<^>2\hbox{-} $$ and H1-norms$$ {H}<^>1\hbox{-} \mathrm{norms} $$ are derived for the proposed AIFE method. Furthermore, numerical experiments are conducted to verify the theoretical predictions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Physics, Multidisciplinary
Zhe Zhang, Haiyan Su, Xinlong Feng
Summary: In this paper, we propose an effective decoupling algorithm for unsteady thermally coupled magneto-hydrodynamic equations. The algorithm improves computational efficiency and accuracy of the pressure field. The energy stability and proofs of the algorithm are provided.
Article
Mathematics, Applied
Wei Wu, Xinlong Feng, Hui Xu
Summary: An improved neural networks method based on domain decomposition is proposed in this paper to solve partial differential equations, which is an extension of the physics informed neural networks (PINNs). The experimental results show that this method outperforms the classical PINNs in terms of training effectiveness, computational accuracy, and computational cost.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Xiaoli Lu, Pengzhan Huang, Xinlong Feng, Yinnian He
Summary: This paper presents a stabilized difference finite element (SDFE) method for the 3D steady incompressible Navier-Stokes equations, using the Oseen iterative method to handle the nonlinear term. The method overcomes the difficulty of 3D space discretization and retains the symmetry of the original equations. It is proven to be stable and has optimal convergence through rigorous stability analysis and error estimation.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Yan Wang, Xufeng Xiao, Xinlong Feng
Summary: This study proposes a fast, accurate, and stable numerical algorithm for solving the anisotropic phase-field dendritic crystal growth model. The algorithm combines the first-order direction splitting method and the linear stabilization technique to ensure energy stability and fast computing. Two post-processing methods are developed to control the boundedness of numerical solution. The effectiveness of the algorithm is demonstrated through numerical examples.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
