Article
Mathematics, Applied
Jiming Yang, Jing Zhou, Yifan Su
Summary: This paper studies an incompressible miscible displacement problem in porous media and proposes a two-grid algorithm based on an interior penalty discontinuous Galerkin method. With this algorithm, solving a nonlinear system on the discontinuous finite element space is reduced to solving a nonlinear problem on a coarse grid and a linear problem on a fine grid. The error estimate for the concentration in H-1-norm and the error estimate for the velocity in L-2-norm are obtained, and numerical experiments are provided to confirm the theoretical analysis.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Jiming Yang, Yifan Su
Summary: An incompressible miscible displacement problem was investigated, and a two-grid algorithm for the problem was proposed. The analysis showed that the algorithm achieved asymptotically optimal approximation with less time spent, which was verified by numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jiming Yang, Jing Zhou, Cunyun Nie
Summary: This paper discusses the application of discontinuous Galerkin approximations to the compressible miscible displacement problem. A two-grid algorithm is proposed, consisting of one coarse grid space and one fine grid space. Error estimates for concentration and velocity are presented, showing that the two-grid method achieves optimal approximations under certain conditions. Numerical results confirm the effectiveness of the algorithm.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Jiming Yang, Jing Zhou
Summary: This paper proposes a two-grid algorithm based on the Newton iteration method for modeling a compressible miscible displacement problem in porous media, using a combined mixed finite element and discontinuous Galerkin approximation. Error estimates in the H-1 norm for concentration and the L-2 norm for velocity are derived, showing that an asymptotically optimal approximation rate can be achieved with the two-grid algorithm if h = O(H-2) is satisfied. Numerical experiments demonstrate the effectiveness of the algorithm, consistent with the theoretical analysis.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Jijing Zhao, Fuzheng Gao, Hongxing Rui
Summary: In this paper, a robust weak Galerkin discretization is proposed for the incompressible miscible displacement problem in porous media, which is a nonlinear time-dependent system. The method developed in this study is positive definite and flexible, and it can be implemented on arbitrarily shaped polygonal mesh without imposing extra stabilization conditions. Another important feature of this method is its independence from the ratio of time-step and spatial mesh size. Error estimates for concentration and pressure are derived in the L2 norm and energy norm, respectively. Extensive numerical experiments, including realistic test cases, are conducted to validate the theoretical results and efficiency of the proposed method.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Junpeng Song, Hongxing Rui
Summary: This paper establishes a reduced-order finite element (ROFE) method with very few degrees of freedom for the incompressible miscible displacement problem. By constructing a finite element (FE) method with second-order accuracy in time and applying the proper orthogonal decomposition (POD) technique, the method effectively reduces degrees of freedom and CPU time while proving optimal a priori error estimates for the solutions. Numerical examples verify the feasibility and effectiveness of the proposed method for solving the problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Giselle Sosa Jones, Loic Cappanera, Beatrice Riviere
Summary: This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. The algorithm is validated through theoretical analysis and numerical results, demonstrating a first-order convergence.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Caixia Nan, Huailing Song
Summary: This paper presents a numerical method for two-phase miscible flow in porous media, utilizing the local discontinuous Galerkin method and IMEX-RK method. The method achieves second-order time discretization and optimal convergence analysis for pressure and concentration in L2-norm, demonstrated through numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Jiansong Zhang, Rong Qin, Yun Yu, Jiang Zhu, Yue Yu
Summary: A new combined hybrid mixed finite element method is proposed to solve the propagation problem of incompressible wormholes. The method can maintain local mass balance while ensuring the boundedness of porosity. The convergence of the proposed method is analyzed and the optimal error estimate in L-2-norm is derived. Numerical examples are provided to verify the validity of the algorithm and the correctness of the theoretical results using the first-order backward Euler scheme in time.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Assyr Abdulle, Giacomo Rosilho de Souza
Summary: This paper introduces a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The method improves the accuracy of the solution by solving local elliptic problems in refined subdomains and provides an algorithm for the automatic identification of these subdomains. Numerical comparisons demonstrate the efficiency of the method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Chemical
Wenwen Xu, Hong Guo, Xindong Li, Yongqiang Ren
Summary: In this paper, a multi-point flux mixed-finite-element decoupled method was proposed for the compressible miscible displacement problem. A fully discrete backward Euler scheme was used to decouple the velocity and pressure equations using a multi-point flux MFE method. The concentration equation was solved using a standard FE method. Error analysis and numerical experiments were conducted to validate the method and simulate the miscible displacement problem.
Article
Mathematics, Applied
Qi Tao, Yan Xu, Chi-Wang Shu
Summary: This paper presents an ultra-weak local discontinuous Galerkin (UWLDG) method for a class of nonlinear fourth-order wave equations, designed and analyzed. The method demonstrates energy conserving properties and optimal error estimates, which are confirmed through numerical experiments. Compatible high order energy conserving time integrators are also proposed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Yao Cheng, Li Yan, Xuesong Wang, Yanhua Liu
Summary: In this study, the local discontinuous Galerkin (LDG) method with a generalized alternating numerical flux is proposed for a one-dimensional singularly perturbed convection-diffusion problem. The double-optimal local maximum-norm error estimate is derived on the quasi-uniform meshes for the first time. Additionally, the discrete maximum principle and global L1-error estimate established in the literature are improved.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Daxin Nie, Weihua Deng
Summary: This paper presents a framework for designing the local discontinuous Galerkin scheme for the integral fractional Laplacian (-Delta)(s) in two dimensions, where s is an element of (0, 1). The numerical stability and convergence of the scheme are theoretically proved and numerically verified, with the convergence rate no worse than O(h(k)(+1/2)) with k >= 1.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Leilei Wei, Huanhuan Wang
Summary: This paper presents an effective numerical method for multi-term variable-order time fractional diffusion equations. The method combines the local discontinuous Galerkin method and the finite difference method in the spatial and temporal directions, respectively. Numerical experiments are conducted to illustrate the effectiveness and applicability of the method.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Computer Science, Interdisciplinary Applications
Hui Guo, Xinyuan Liu, Yang Yang
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Computer Science, Interdisciplinary Applications
Ziyao Xu, Yang Yang
JOURNAL OF COMPUTATIONAL PHYSICS
(2020)
Article
Mathematics, Applied
Hui Guo, Rui Jia, Lulu Tian, Yang Yang
Summary: This paper applies two fully-discrete LDG methods to compressible wormhole propagation, proving stability and error estimates, introduces a new auxiliary variable and special time integration for porosity to obtain stability of the fully-discrete LDG methods, and demonstrates optimal error estimates for pressure, velocity, porosity, and concentration under weak temporal-spatial conditions.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Water Resources
Hui Guo, Wenjing Feng, Ziyao Xu, Yang Yang
Summary: This study improves the traditional discrete fracture model to be applicable on non-conforming meshes, and combines the interior penalty discontinuous Galerkin and enriched Galerkin methods to handle the pressure equation, ensuring local mass conservation. Numerical experiments in porous media demonstrate the effectiveness of the proposed methods.
ADVANCES IN WATER RESOURCES
(2021)
Article
Computer Science, Interdisciplinary Applications
Xiaofeng Cai, Jing-Mei Qiu, Yang Yang
Summary: The paper introduces a new method called ELDG, which incorporates a modified adjoint problem and integration of PDE over a space-time region partitioned by time-dependent linear functions. By introducing a new flux term to account for errors in characteristics approximation, the ELDG method combines the advantages of SL DG and classical Eulerian RK DG methods. The use of linear functions for characteristics approximation in the EL DG framework simplifies shapes of upstream cells and reduces time step constraints.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Ruize Yang, Yang Yang, Yulong Xing
Summary: In this paper, a family of second and third order temporal integration methods for stiff ordinary differential equations is proposed, combining traditional Runge-Kutta and exponential Runge-Kutta methods to preserve sign and steady-state properties. These methods are applied with well-balanced discontinuous Galerkin spatial discretization to solve nonlinear shallow water equations with friction terms. The fully discrete schemes are demonstrated to satisfy well-balanced, positivity-preserving, and sign-preserving properties simultaneously, showing good numerical results on various test cases.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Water Resources
Guosheng Fu, Yang Yang
Summary: We propose a hybrid-mixed finite element method for a novel hybrid-dimensional model of single-phase Darcy flow in fractured porous media. Our method distinguishes between conductive and blocking fractures and uses a combination of classical interface model and recent Dirac-delta function approach to handle them. The use of Dirac-delta function approach allows for nonconforming meshes with respect to the blocking fractures. Our numerical scheme produces locally conservative velocity approximations and leads to a symmetric positive definite linear system involving pressure degrees of freedom on the mesh skeleton only.
ADVANCES IN WATER RESOURCES
(2022)
Article
Mathematics, Applied
Lulu Tian, Hui Guo, Rui Jia, Yang Yang
Summary: This paper investigates the application of local discontinuous Galerkin methods to compressible wormhole propagation with a Darcy-Forchheimer model. By addressing various theoretical challenges, stability and error estimates of the scheme are proven, followed by numerical experiments to verify the results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Wenjing Feng, Hui Guo, Yue Kang, Yang Yang
Summary: In this paper, we introduce a novel SIPEC time marching method for the coupled system of two-component compressible miscible displacements. By incorporating a correction stage in each time step, we achieve second-order accuracy while maintaining bound preservation for the concentration equation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Jie Du, Yang Yang
Summary: This paper presents high-order bound-preserving discontinuous Galerkin (DG) methods for multicomponent chemically reacting flows. The proposed methods address the challenges of positivity preservation, ensuring the mass fractions sum up to 1, and handling stiff sources. The numerical experiments demonstrate the effectiveness of the proposed schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Hui Guo, Xueting Liang, Yang Yang
Summary: In this paper, numerical algorithms are investigated to capture the blow-up time for a class of convection-diffusion equations with blow-up solutions. The positivity-preserving technique is used to enforce stability and the L1-stability and L2-norm of numerical approximations are utilized to detect the blow-up phenomenon. Two methods for defining the numerical blow-up time are proposed and their convergence to the exact time is proven.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Ziyao Xu, Zhaoqin Huang, Yang Yang
Summary: In this paper, a novel discrete fracture model is proposed for flow simulation of fractured porous media with flow blocking barriers on non-conforming meshes. The traditional Darcy's law is modified into a hybrid-dimensional Darcy's law to represent the fractures and barriers using Dirac-delta functions in the permeability tensor and resistance tensor, respectively. The model accurately accounts for the influence of highly conductive fractures and blocking barriers on non-conforming meshes, and the local discontinuous Galerkin method is employed to handle the pressure/flux discontinuity.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Wenjing Feng, Hui Guo, Lulu Tian, Yang Yang
Summary: In this paper, we propose a sign-preserving second-order IMplicit Pressure Explicit Concentration (IMPEC) time method for generalized coupled non-Darcy flow and transport problems in petroleum engineering. The method utilizes interior penalty discontinuous Galerkin (IPDG) methods for spatial discretization and a bound-preserving technique to ensure physically relevant numerical approximations. The proposed method is different from previous algorithms as it linearizes the velocity equation and introduces a direct solver to solve for velocity, resulting in first-order accurate solutions and reduced computational cost.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Water Resources
Guosheng Fu, Yang Yang
Summary: We propose a new hybridizable discontinuous Galerkin (HDG) method on unfitted meshes for single-phase Darcy flow in a fractured porous medium. Our numerical scheme uses a Dirac-6 function approach for fractures, allowing for unfitted meshes with respect to the fractures. The scheme is simple and locally mass conservative.
ADVANCES IN WATER RESOURCES
(2023)
Article
Mathematics, Applied
Jie Du, Eric Chung, Yang Yang
Summary: This paper studies the classical Allen-Cahn equations and investigates the maximum-principle-preserving (MPP) techniques. It discusses the application of the local discontinuous Galerkin (LDG) method and the use of conservative modified exponential Runge-Kutta methods. Numerical experiments are used to demonstrate the effectiveness of the MPP LDG scheme.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
