Article
Mathematics, Applied
Igor Tominec, Murtazo Nazarov
Summary: This paper presents a radial basis function generated finite difference method for solving nonlinear conservation laws, addressing the issue of numerical instabilities through the introduction of a residual-based artificial viscosity stabilization framework. Computational tests confirm the reliability and accuracy of the proposed stabilized methods in solving both scalar conservation laws and conservation law systems.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Engineering, Multidisciplinary
Dmitri Kuzmin, Hennes Hajduk, Andreas Rupp
Summary: The algebraic flux correction schemes presented in this work utilize limiters to enforce relevant maximum principles and entropy stability conditions on a standard continuous finite element discretization of nonlinear hyperbolic problems. These schemes can be applied to scalar hyperbolic equations and systems and impose entropy-conservative or entropy-dissipative bounds on entropy production rates, alongside developing two versions of fully discrete entropy fixes. The use of limiter-based entropy fixes is motivated by proving a finite element version of the Lax-Wendroff theorem and conducting numerical studies for various standard test problems.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Yuyu He, Xiaofeng Wang, Weizhong Dai, Yaqing Deng
Summary: In this paper, a fourth-order accurate conservative finite difference scheme for solving the coupled nonlinear Schrodinger equations is proposed, and its properties such as existence, uniqueness, convergence, and stability are proven. The experimental results support the theoretical analysis.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Jichun Li, Li Zhu, Todd Arbogast
Summary: In this paper, a new variational form is developed to simulate the propagation of surface plasmon polaritons on graphene sheets. Graphene is treated as a thin sheet of current with an effective conductivity, and modeled as a lower-dimensional interface. A novel time-domain finite element method is proposed to solve this graphene model, which couples an ordinary differential equation on the interface with Maxwell's equations in the physical domain. Discrete stability and error estimate are proved for the proposed method. Numerical results are presented to demonstrate the effectiveness of this graphene model for simulating the surface plasmon polaritons propagating on graphene sheets.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Automation & Control Systems
Yazhou Tian, Mengqian Liang, Yuangong Sun
Summary: This article primarily investigates the practical exponential stability of switched homogeneous positive nonlinear systems (SHPNSs) with partial unstable modes and perturbations. The max-separable Lyapunov function (MSLF) technique and a special pre-setting switching sequence are used to define significant stability conditions that ensure the convergence of state trajectories to a confined region in continuous-time and discrete-time domains. The key results include earlier findings as special cases and can be directly applied to general switched systems, regardless of their positivity. Two important instances are provided to further illustrate the validity of the theoretical findings.
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Carolin Mehlmann, Peter Korn
Summary: The study introduces a discretization method for sea-ice dynamics on triangular grids, utilizing the nonconforming Crouzeix-Raviart finite element and introducing an edge-based stabilization to address small scale noise in the velocity field. Through theoretical considerations and numerical experiments, it is shown that the stabilized Crouzeix-Raviart element provides a stable discretization for sea-ice dynamics relevant for modeling in ocean and climate science.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Yunqing Huang, Jichun Li, Wei Yang
Summary: This paper investigates a graphene-based absorber model, incorporating both interband and intraband conductivity. Energy identity and stability are established for the continuous model, and a finite element time-domain method is proposed for solving the model. Numerical stability and optimal error estimate are proven for the scheme, with numerical results justifying the error estimate and demonstrating the wave absorbing phenomenon by graphene slab.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics
Jie Xu
Summary: We present an iterative method to prove the existence and uniqueness of complex-valued nonlinear elliptic PDEs with specific boundary conditions on precompact domains and smooth, compact Riemannian manifolds. The method also provides a solution for an integral version of these PDEs using parametrix methods.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Shuhei Kitano
Summary: Holder estimates and Harnack inequalities are investigated for fully nonlinear integro-differential equations with variable order and critically close to 2 kernels.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Brittany Froese Hamfeldt, Jacob Lesniewski
Summary: We introduce a generalized finite difference method for solving a wide range of fully nonlinear elliptic partial differential equations in three dimensions. The method utilizes Cartesian grids and additional points along the boundary for high-resolution approximation. We also propose a least-squares approach for constructing consistent and monotone approximations of second directional derivatives. By discretizing orthogonal coordinate frames, we efficiently approximate functions of the eigenvalues of the Hessian. This method has been successfully applied to various challenging examples, demonstrating its effectiveness.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Automation & Control Systems
Boliang Lu, Ruili Song, Quanxin Zhu
Summary: This study focuses on highly nonlinear hybrid neutral stochastic differential equations with multiple time-dependent delays and different structures. The existence, uniqueness, and asymptotic boundedness of the global solution are proved, and novel criteria for pth moment and almost surely exponential stability are investigated. The study also establishes the convergence of the Euler-Maruyama approximate solution to the theoretical solution. A numerical example is presented to demonstrate the effectiveness of the main results.
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Robert Eymard, David Maltese
Summary: This work focuses on the approximation of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix using two nonlinear numerical methods. Both methods show convergence properties to a continuous solution in a weak sense. The nonlinear finite element approximation on any simplicial grid shows existence and convergence of a discrete solution, while a numerical scheme for entropy weak solution in the case where the right-hand side belongs to L-1 is also constructed.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Erik Burman
Summary: This article discusses the stability and accuracy of stabilized finite element methods for hyperbolic problems, focusing on the interaction between linear and nonlinear stabilizations. It is shown that a combination of linear and nonlinear stabilization can be designed to be invariant preserving. The proposed method allows for classical error estimates for smooth solutions and is demonstrated to accurately predict shock structures without polluting the smooth parts with high frequency content.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Bartosz Jaroszkowski, Max Jensen
Summary: In this paper, we investigate the strong uniform convergence of monotone P1 finite element methods for the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate diffusions. The discontinuity of boundary operators across face-boundaries and type changes is considered. Robin-type boundary conditions are discretized using a lower Dini derivative. In time, the Bellman equation is approximated using IMEX schemes. The existence and uniqueness of numerical solutions are proved using Howard's algorithm.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Ingeborg G. Gjerde, Kundan Kumar, Jan M. Nordbotten
Summary: In this work, we analyze the Poisson equation with a line source using a dual-mixed variational formulation. By making assumptions on the problem parameters, we split the solution into higher- and lower-regularity terms, and propose a singularity removal-based mixed finite element method to approximate the higher-regularity terms, which significantly improves the convergence rate compared to approximating the full solution.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
