Article
Mathematics, Applied
Y. I. N. I. N. G. Cao, X. I. A. O. M. I. N. G. Wang
Summary: This study proves that the difference between the solutions to the Stokes-Darcy system derived using the Beavers-Joseph or Beavers-Joseph-Saffman-Jones interfacial conditions is proportional to the Darcy number when the Reynolds number is below a certain threshold. Therefore, the Beavers-Joseph-Saffman-Jones interface boundary condition is an excellent approximation of the classical Beavers-Joseph interface boundary condition in the regime of small Darcy numbers.
SIAM JOURNAL ON APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Zhipeng Yang, Ju Ming, Changxin Qiu, Maojun Li, Xiaoming He
Summary: A multigrid multilevel Monte Carlo (MGMLMC) method is developed for the stochastic Stokes-Darcy interface model with random hydraulic conductivity. The method aims to efficiently solve the stochastic model, especially focusing on the interface and the random Beavers-Joseph interface condition.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Luling Cao, Yinnian He, Jian Li, Di Yang
Summary: This paper develops the numerical theory of decoupled modified characteristic FEMs for the fully evolutionary Navier-Stokes-Darcy model with the Beavers-Joseph interface condition. The optimal L-2-norm error convergence order of the solutions is proved by mathematical induction, implying the uniform L-2-boundedness of the fully discrete velocity solution. High efficiency of this method is demonstrated through numerical tests.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Antoine Michael Diego Jost, Stephane Glockner
Summary: The article proposes linear/quadratic square shifting methods to improve the accuracy and convergence of ghost-cell immersed boundary methods for Cartesian grids. The methods aim to increase the order of convergence while maintaining a maximum stencil size of 2, and are evaluated through a comprehensive verification and validation process.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Yingzhi Liu, Yinnian He, Xuejian Li, Xiaoming He
Summary: This paper demonstrates the convergence analysis of the Robin-Robin domain decomposition method for the Stokes-Darcy system with Beavers-Joseph interface condition, focusing on the case of convergence for small viscosity and hydraulic conductivity. Utilizing discrete techniques, the almost optimal geometric convergence rate for gamma(f) > gamma(p) is obtained. The results provide a general guideline for choosing parameters to achieve convergence and geometric convergence rate, which is confirmed by numerical simulations.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Eduard Marusic-Paloka
Summary: An effective boundary condition for a porous wall is derived based on basic principles of mechanics. The study focuses on the Stokes system governing the flow of viscous fluid through a reservoir with small pores on the boundary. Through rigorous asymptotic analysis, a macroscopic model is obtained. Assuming periodicity of the pores, a Darcy-type effective boundary condition is derived using homogenization and boundary layer techniques. Further asymptotic analysis with respect to the porosity reveals a sequence of recursive boundary value problems indicating a significant pressure jump on the boundary.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Xinhui Wang, Guangzhi Du, Yi Li
Summary: In this paper, a modified local and parallel finite element method (MLPFEM) is proposed for the coupled Stokes-Darcy problem. The method combines a partition of unity with backtracking technique to improve computational efficiency while maintaining global continuity, and achieves optimal error bounds.
NUMERICAL ALGORITHMS
(2023)
Article
Mechanics
Guo-Qing Chen, A-Man Zhang, Nian-Nian Liu, Yan Wang
Summary: This paper develops a fluid-solid coupling model to study contact line dynamics between two fluids with high density ratio. The model uses the multiphase lattice Boltzmann flux solver and the immersed boundary method, and its robustness is successfully verified through several test cases.
Article
Mechanics
H. Chen, P. Yu, C. Shu
Summary: A novel numerical method, UIB-LBFS, is proposed for simulating incompressible flows past homogeneous porous bodies. The method introduces a diffuse layer to unify the governing equations in porous and pure-fluid domains, and employs a fractional-step technique to split the computational procedure. The accuracy and reliability of the method are proven through numerical validations.
Article
Mathematics, Applied
Guangzhi Du, Shilin Mi, Xinhui Wang
Summary: This paper provides and studies some local and parallel finite element methods based on two-grid discretizations for the non-stationary Stokes-Darcy model with the Beavers-Joseph interface condition. Two local algorithms, the semi-discrete and fully discrete finite element algorithms, are introduced and related error estimates are derived. Two fully discrete parallel algorithms are subsequently developed based on the fully discrete local algorithm. The validity of the algorithms is illustrated through numerical results.
NUMERICAL ALGORITHMS
(2023)
Article
Computer Science, Interdisciplinary Applications
Xinjie Ji, James Gabbard, Wim M. van Rees
Summary: This paper introduces a sharp-interface approach based on the immersed interface method for handling the one- and two-way coupling between an incompressible flow and rigid bodies using the vorticity-velocity Navier-Stokes equations. The authors develop a moving boundary treatment and a two-way coupling methodology that do not require the pressure field. Extensive testing shows that the resulting solver achieves second-order accuracy and provides efficiency benefits compared to a representative first-order approach.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Engineering, Multidisciplinary
Haifeng Ji, Feng Wang, Jinru Chen, Zhilin Li
Summary: This paper presents an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. The method modifies the traditional finite element near the interface according to the interface jump conditions and proves the unisolvent property and optimal approximation capabilities of the IFE method. The stability and optimal error estimates are also derived rigorously.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Derrick Jones, Xu Zhang
Summary: This paper introduces a class of lowest-order nonconforming immersed finite element methods for solving two-dimensional Stokes interface problems, which do not require the solution mesh to align with the fluid interface and can use triangular or rectangular meshes. The new vector-valued IFE functions are constructed to approximate the interface jump conditions, and the approximation capabilities of these new IFE spaces for the Stokes interface problems are examined through numerical examples, showing optimal convergence rates.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Brian Turnquist, Mark Owkes
Summary: Solving the pressure Poisson equation in the incompressible form of Navier-Stokes is often the most costly aspect of computation, especially in multiphase flow simulations. Researchers have proposed a decomposed pressure correction method to improve computational cost, and extended it to intrusive uncertainty quantification of multiphase flows. By modifying the estimated pressure field, the new method shows improved accuracy and reduction in computational cost in various test cases.
COMPUTERS & FLUIDS
(2021)
Article
Mechanics
Raghav Singhal, Jiten C. Kalita
Summary: The new method shows high accuracy and stability in solving two-dimensional elliptic problems with singular sources and discontinuous coefficients, especially maintaining high order compactness at irregular points. By embedding circular and star-shaped interfaces in a rectangular region to validate the algorithm, it also demonstrates superiority in multi-body fluid flow problems.
Article
Mathematics, Applied
Rui Li, Zhi-Lin Li, Jun-Feng Yin
Summary: Inspired by the modulus-based Newton method for linear complementarity problems, a generalized modulus-based Newton method was proposed in this study to solve a class of non-linear complementarity problems with P-matrices. A sufficient condition for convergence was obtained and numerical experiments showed that the proposed method is efficient and outperforms existing methods.
NUMERICAL ALGORITHMS
(2022)
Article
Computer Science, Interdisciplinary Applications
Kejia Pan, Xiaoxin Wu, Hongling Hu, Yunlong Yu, Zhilin Li
Summary: This paper introduces a new extrapolation cascadic multigrid method for solving 3D anisotropic diffusion equations, avoiding the costly solution of local linear systems and demonstrating its efficiency and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Jiali Xu, Haiyan Su, Zhilin Li
Summary: This paper focuses on analyzing nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. It introduces a locally stabilization term to address the instability issue in the finite element method for velocity field and pressure, along with proposing three effective iterative methods. The theoretical analysis is verified through numerical experiments in the end.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Swarn Singh, Suruchi Singh, Zhilin Li
Summary: This paper presents a new technique based on cubic spline interpolation for solving second order elliptic equations with irregularities in one dimension and two dimensions. The proposed method provides second order accurate solutions at the interface and regular points, as well as for the first and second order derivatives. Numerical experiments demonstrate the effectiveness and accuracy of the method for 2D problems.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Rui Li, Jun-Feng Yin, Zhi-Lin Li
Summary: We propose a variant modified skew-normal splitting iterative method to solve a class of large sparse non-Hermitian positive definite linear systems. The preconditioned version of the proposed method is also constructed using the preconditioning technique. Theoretical analysis shows the unconditional convergence of the proposed method. Numerical experiments demonstrate the efficiency and superior performance of the proposed method compared to other methods.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Zhilin Li, Hayk Mikayelyan
Summary: This paper deals with the challenging obstacle-like minimization problem in a cylindrical domain, where the non-local nature of the obstacle makes it difficult to apply standard optimization techniques. The paper introduces a new algorithm that can compute the global minimum and provides numerical testing and construction of exact solutions.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Engineering, Multidisciplinary
Shuanggui Hu, Kejia Pan, Xiaoxin Wu, Yongbin Ge, Zhilin Li
Summary: This paper presents an efficient multigrid method combined with a high-order compact finite difference scheme on nonuniform rectilinear grids for solving 3D diagonal anisotropic convection-diffusion problems with boundary/interior layers. The method involves a fourth-order compact finite difference scheme, novel multigrid prolongation operators based on quintic Lagrange interpolation and completed Richardson extrapolation, and a SSOR-preconditioned biconjugate gradient stabilized smoother. Numerical experiments demonstrate fourth-order accuracy and improved efficiency compared to the state-of-the-art algebraic multigrid method for large linear systems arising from second order elliptic PDE discretization.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Sergio Amat, Zhilin Li, Juan Ruiz-Alvarez, Concepcion Solano, Juan C. Trillo
Summary: This paper presents a new technique for computing accurate first order derivatives of a piecewise smooth function close to singularities using a Hermite spline. The technique corrects the system of equations to achieve O(h(4)) accuracy even near singularities. Theoretical proofs and numerical experiments confirm the effectiveness of the proposed method in eliminating oscillations and smearing of singularities.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Kejia Pan, Kang Fu, Jin Li, Hongling Hu, Zhilin Li
Summary: Some new sixth-order compact finite difference schemes for Poisson/ Helmholtz equations on rectangular domains in both two-and three-dimensions are developed and analyzed. The finite difference and weight coefficients of the new methods have analytic simple expressions, different from a few schemes in the literature. The new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6, and the coefficient matrices are M-matrices for Helmholtz equations, guaranteeing the discrete maximum principle and convergence.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Zhilin Li, Kejia Pan
Summary: In this paper, new fourth order compact schemes for Robin and Neumann boundary conditions are developed for boundary value problems of elliptic PDEs in two and three dimensions. These schemes utilize carefully designed undetermined coefficient methods and can be applied to various elliptic PDEs including both flux and linear boundary conditions. The developed schemes are versatile and generally have M-matrices as coefficient matrices, ensuring well-posed problems and convergence of the methods. Examples with large wave numbers and oscillatory solutions are presented to demonstrate the performance of the new schemes.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Sergio Amat, Zhilin Li, Juan Ruiz-Alvarez, Concepcion Solano, Juan C. Trillo
Summary: This work introduces a novel nonlinear technique for accurate numerical integration of any order, even with data containing discontinuities and known only at grid points. The technique includes correction terms that depend on the size of jumps and derivatives of the function at discontinuities. These correction terms allow recovering the accuracy of classical integration formulas near the discontinuities and can be added during integration or as post-processing. Numerical experiments validate the theoretical conclusions of this article.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Zhilin Li, Kejia Pan, Juan Ruiz-Alvarez
Summary: This paper presents alternative approaches for interface and internal layer problems using non-matching grids. The developed methods achieve high order accuracy by using two mesh sizes near and away from the interface or internal layer. For problems with straight interfaces or boundary layers parallel to one axis, a fourth order compact finite difference scheme is constructed at border grid points. For problems with curved interfaces or internal layers, a level set representation is utilized to build a fine mesh within a tube around the interface. A new super-third seven-point discretization is developed for hanging nodes to guarantee the discrete maximum principle.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Jin Li, Zhilin Li, Kejia Pan
Summary: In this study, new high order compact finite difference schemes are developed to accurately approximate the derivatives of the solutions to some elliptic partial differential equations (PDEs). The convergence analysis shows that the accuracy of the computed derivatives is the same as that of the solution. The developed schemes take into account the partial differential equations, including the source term and/or the boundary conditions, and have important applications in solving incompressible Stokes equations with periodic boundary conditions.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Haifeng Ji, Feng Wang, Jinru Chen, Zhilin Li
Summary: In this paper, a significant discovery has been made regarding nonconforming immersed finite element (IFE) methods for solving elliptic interface problems. It has been shown that IFE methods without penalties may not converge optimally in the presence of non-zero tangential derivative of the exact solution and coefficient jump on the interface. To address this issue, a new parameter-free nonconforming IFE method with additional terms on interface edges has been developed to recover the optimal convergence rates. The method has been shown to have unisolvent IFE basis functions on arbitrary triangles.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Baiying Dong, Xiufeng Feng, Zhilin Li
Summary: A second order accurate method is proposed for general three dimensional anisotropic elliptic interface problems. The method handles finite jumps across smooth interfaces by deriving 3D interface relations and using different discretizations. The convergence analysis is shown by splitting errors and using various numerical examples.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2022)
