Article
Mathematics, Applied
Dongho Kim, Eun-Jae Park, Boyoon Seo
Summary: This paper focuses on error analysis of the semi-discrete and fully discrete approximations to the pseudostress-velocity formulation of the unsteady Stokes problem. The study proves the parabolic smoothing property of solution operators in the semi-discrete case and demonstrates the stability of backward Euler and Crank-Nicolson schemes in the fully discrete case.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Y. Rong, J. A. Fiordilino, F. Shi, Y. Cao
Summary: We study a modular Crank-Nicolson based Voigt regularization algorithm for the Navier-Stokes equations. The algorithm adds a minimally intrusive module that implements Voigt regularization and numerical dissipation, improving stability and accuracy in large-scale dynamics.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Yang Li, Yanhong Bai, Minfu Feng
Summary: This paper presents a stabilized virtual element method for unsteady Navier-Stokes problems on polygonal meshes. The method uses equal-order virtual elements in space and the Crank-Nicolson scheme in time, and provides a fully discrete formula. By introducing local-projection type stabilizations, the method is able to avoid the discrete inf-sup condition and control spurious oscillations caused by high Reynolds numbers. The stability and error estimates for velocity and pressure are analyzed, and error estimates independent of the negative powers of the viscosity are derived. Numerical experiments are conducted to validate the theoretical results.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Wei Li, Pengzhan Huang
Summary: "In this paper, a fully discrete finite element scheme with two-order temporal accuracy is proposed for solving the Navier-Stokes/Navier-Stokes equations. The scheme consists of two coupled Navier-Stokes equations and a linear interface condition. The paper presents the specific steps of the scheme and establishes the stability and error estimates. Numerical experiments verify the theoretical findings and efficiency of the scheme."
APPLIED NUMERICAL MATHEMATICS
(2023)
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
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
Nan Jiang, Ying Li, Huanhuan Yang
Summary: The CNLFAC method is proposed for solving the Stokes-Darcy equations, achieving high stability and convergence without the need to solve a Poisson equation for pressure. It reduces computational time significantly and maintains robustness in numerical experiments.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
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
Eric Ngondiep
Summary: This paper analyzes the stability of a two-level coupled explicit MacCormack/Crank-Nicolson method for solving the nonstationary mixed Stokes-Darcy model. The proposed method combines the explicit MacCormack technique and the implicit Crank-Nicolson discretization to achieve unconditional stability and reduced computational cost for solving time dependent mixed Stokes-Darcy problems efficiently.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Buyang Li, Shu Ma, Na Wang
Summary: This article discusses the numerical approximation of the two-dimensional nonstationary Navier-Stokes equations with H-1 initial data. By using special locally refined temporal stepsizes, it is proven that the linearly extrapolated Crank-Nicolson scheme, with the stabilized Taylor-Hood finite element method in space, can achieve second-order convergence in time and space. Numerical examples are provided to validate the theoretical analysis.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Huaijun Yang
Summary: This paper investigates two fully discrete schemes for the cubic Schrodinger equation and derives unconditionally optimal error estimates in the L-2 norm. The mass and energy stability of the schemes are rigorously proven, and the existence and uniqueness of the numerical solutions are presented. The obtained error estimates in L-2 norm are unconditionally optimal without any timestep restrictions, and numerical results are provided to validate the theoretical analysis.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Nick Fisher
Summary: A numerical method is proposed for solving a pressure Poisson reformulation of the Navier-Stokes equation in two space variables. The method discretizes the space using orthogonal spline collocation with splines of order r. The velocity terms are obtained through an alternating direction implicit extrapolated Crank-Nicolson scheme applied to a Burgers' type equation, while the pressure term is found using a matrix decomposition algorithm for a Poisson equation with non-homogeneous Neumann boundary conditions. The numerical results suggest that the scheme has convergence rates of order r in space and order 2 in time. The scheme is also applied to the lid-driven cavity problem and compared to standard benchmark values.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
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
Jad Doghman, Ludovic Goudenege
Summary: The primary focus of this research is the development of a finite element based space-time discretization method for solving the stochastic Lagrangian averaged Navier-Stokes equations in incompressible fluid turbulence. Convergence analysis shows that convergence to continuous strong solutions is achieved when alpha approaches zero or when alpha is fixed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Engineering, Multidisciplinary
Philsu Kim, Soyoon Bak
Summary: This paper proposes a novel trajectory-approximation technique as a time-integration scheme for solving advectional partial differential equations in engineering and physics, saving computational costs and achieving third-order accuracy. The method demonstrates superior performance in simulating benchmark test flows.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Zhongjie Lu, A. Cesmelioglu, J. J. W. Van der Vegt, Yan Xu
JOURNAL OF SCIENTIFIC COMPUTING
(2017)
Article
Mathematics, Applied
Aycil Cesmelioglu, Bernardo Cockburn, Weifeng Qiu
MATHEMATICS OF COMPUTATION
(2017)
Article
Mathematics, Applied
Aycil Cesmelioglu
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2017)
Article
Mathematics, Interdisciplinary Applications
A. Cesmelioglu, M. Song, D. Drignei
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2017)
Article
Mathematics, Applied
Susanne C. Brenner, Aycil Cesmelioglu, Jintao Cui, Li-Yeng Sung
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
Aycil Cesmelioglu, Vivette Girault, Beatrice Riviere
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2013)
Article
Mathematics, Applied
Aycil Cesmelioglu, Bernardo Cockburn, Ngoc Cuong Nguyen, Jaume Peraire
JOURNAL OF SCIENTIFIC COMPUTING
(2013)
Article
Mathematics, Applied
Aycil Cesmelioglu, Prince Chidyagwai
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2020)
Article
Mathematics, Applied
Aycil Cesmelioglu, Sander Rhebergen, Garth N. Wells
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Keegan L. A. Kirk, Aycil Cesmelioglu, Sander Rhebergen
Summary: This article proves the convergence of a space-time hybridized discontinuous Galerkin method for the evolutionary Navier-Stokes equations as the time step and mesh size approach zero, and further demonstrates that the weak solution satisfies the energy inequality. The analysis is carried out using discrete functional analysis tools and a discrete version of the Aubin-Lions-Simon theorem.
MATHEMATICS OF COMPUTATION
(2022)
Article
Mathematics, Applied
Aycil Cesmelioglu, Sander Rhebergen
Summary: We present and analyze a strongly conservative hybridizable discontinuous Galerkin finite element method for the coupled incompressible Navier-Stokes and Darcy problem with Beavers-Joseph-Saffman interface condition. An a priori error analysis shows that the velocity error does not depend on the pressure, and that velocity and pressure converge with optimal rates. These results are confirmed by numerical examples.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Aycil Cesmelioglu, Jeonghun J. Lee, Sander Rhebergen
Summary: We introduce and analyze a hybridizable discontinuous Galerkin finite element method for the coupled Stokes-Biot problem. The method has the property that the discrete velocities and displacements satisfy the compressibility equations pointwise on the elements. We prove well-posedness of the discretization and provide a priori error estimates that demonstrate the method is free of volumetric locking. Numerical examples further demonstrate optimal rates of convergence for all unknowns and locking-free discretization.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Aycil Cesmelioglu, Dinh Dong Pham, Sander Rhebergen
Summary: In this paper, a high-order hybridized discontinuous Galerkin (HDG) method is presented for the fully coupled time-dependent Stokes-Darcy-transport problem. The HDG method ensures compatibility between the discrete flow equations and the discrete transport equation. It also guarantees strong mass conservation and handles the interface conditions between the Stokes and Darcy regions effectively. A linearizing decoupling strategy is employed to sequentially solve the Stokes/Darcy and transport equations. Well-posedness and optimal a priori error estimates are proven for the velocity and concentration. Numerical examples demonstrate the compatibility and robustness of the discrete solution.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Aycil Cesmelioglu, Sander Rhebergen
Summary: This study presents a stability and error analysis of an embedded-hybridized discontinuous Galerkin finite element method for coupled Stokes-Darcy flow and transport. The compatibility and stability of the coupled flow and transport discretization are demonstrated, along with the existence and uniqueness of the semi-discrete transport problem and optimal a priori error estimates. Numerical examples illustrate the theoretical results, showing that compatible discretization eliminates spurious oscillations in the solution to the transport problem caused by incompatible discretization.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
K. L. A. Kirk, T. L. Horvath, A. Cesmelioglu, S. Rhebergen
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2019)
