Article
Mathematics, Applied
Imre Fekete, Lajos Loczi
Summary: This paper studies the application of the classical Richardson extrapolation technique to accelerate the convergence of sequences resulting from linear multistep methods for solving initial-value problems of systems of ordinary differential equations numerically. The LMM-RE approach has the advantage of higher order and favorable linear stability properties, without any modification to existing LMM codes.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Zhaoyi LI, Hong-Lin Liao
Summary: This paper proves the stability of the two-step backward differentiation formula (BDF) method on arbitrary time grids and relaxes the mesh restrictions in previous literature. It also provides a new understanding of the stability conditions for the variable-step three-step backward differentiation formula scheme. The main tools used in the proof include discrete orthogonal convolution kernels and an elliptic type matrix norm.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Gui-Lai Zhang
Summary: This paper defines impulsive one-step numerical methods, proves their convergence and order, introduces another equivalent form, and validates their advantages through numerical experiments.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Boris Faleichik
Summary: This work introduces a new method called MRMS for solving stiff systems of ODEs, which reduces the number of unknowns on each step by adaptively choosing coefficients of an explicit linear multistep method to minimize the norm of the residual of an implicit BDF formula. The method shows promising results in numerical experiments with twodimensional non-autonomous heat equations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Moritz Schneider, Jens Lang, Rudiger Weiner
Summary: This paper explores dynamical systems with sub-processes evolving on different time scales and emphasizes the importance of automatic time step variation for efficient solution. The development of super-convergent IMEX-Peer methods, which combine the stability of implicit methods and the low computational costs of explicit methods, allows for effective solving of ordinary differential equations with stiff and non-stiff parts in the source term, while maintaining high order accuracy. New super-convergent IMEX-Peer methods are derived, showing potential when applied with local error control.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Yuwen Li, Ludmil Zikatanov
Summary: This framework relates preconditioning with a posteriori error estimates in finite element methods, using standard tools in subspace correction methods to obtain reliable and efficient error estimators, including recovering classical residual error estimators for second order elliptic equations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Wansheng Wang, Chengyu Jin, Yi Huang, Linhai Li, Chun Zhang
Summary: This paper compares the implicit-explicit (IMEX) and Newton linearized (NL) methods, which are two typical classes of time discretization methods for solving nonlinear differential equations. The stability of IMEX and NL two-step backward differentiation formula (BDF2) methods with variable step-size for solving semilinear parabolic differential equations is established under appropriate time-step ratio restriction using energy estimates and recent novel technique. Based on these stability results, a priori error bounds for these methods are derived. Numerical results not only demonstrate the feasibility of the proposed method for solving semilinear parabolic differential equations, but also show that the IMEX BDF2 method is more effective than the NL BDF2 method.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Luotang Ye, Yanmao Chen, Qixian Liu, Norbert Herencsar, Zhinan Peng, Xin Wang, Kaibo Shi, Xiao Cai
Summary: The fractional gradient method has received significant attention from researchers, who believed that it has a faster convergence rate compared to classical gradient methods. However, theoretical convergence analysis shows that the maximum convergence rate of the fractional-order gradient method is the same as that of the classical gradient method. This discovery suggests that the superiority of fractional gradients may not lie in achieving fast convergence rates. Building upon this discovery, a novel variable fractional-type gradient method is proposed with an emphasis on automatically adjusting the step size. Theoretical analysis confirms the convergence of the proposed method. Numerical experiments demonstrate that the proposed method can converge to the extremum point rapidly and accurately. Additionally, the Armijo criterion is introduced to ensure the selection of optimal step size in the proposed gradient methods. Results show that, despite having the same theoretical maximum convergence speed, the proposed method consistently demonstrates superior convergence stability and performance when applied to practical problems.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Yang Li, Qihang Sun, Naidan Feng, Jianjun Liu
Summary: A variable-step BDF2 time-stepping method is investigated for simulating the extended Fisher-Kolmogorov equation. The time-stepping scheme is shown to preserve a discrete energy dissipation law under certain conditions. Concise L-2 norm error estimates are proved for the first time, under specific step ratios constraint. The proposed numerical scheme is robust with respect to the variations of time steps.
JOURNAL OF FUNCTION SPACES
(2023)
Article
Mathematics, Applied
Hong-lin Liao, Tao Tang, Tao Zhou
Summary: This paper focuses on the discrete energy analysis of the variable-step BDF schemes, specifically the stability and convergence analysis of the third-order BDF schemes for linear diffusion equations. By establishing a discrete gradient structure, a discrete energy dissipation law is obtained, and mesh-robust stability and convergence analysis in the L2 norm are obtained. Numerical tests are also presented to support the theoretical results.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Rachael T. Keller, Qiang Du
Summary: This study investigates the application of linear multistep methods in learning dynamics, establishing a rigorous framework for convergence in the discovery problem and indicating convergence conditions for several M-step LMMs schemes. Additionally, numerical experiments are provided to support the theoretical analysis.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Engineering, Electrical & Electronic
Yu Wang, Zhen Qin, Jun Tao, Yili Xia
Summary: In this paper, an enhanced sparsity-aware recursive least squares (RLS) algorithm is proposed, which combines the proportionate updating (PU) and zero-attracting (ZA) mechanisms, and introduces a general convex regularization (CR) function and variable step-size (VSS) technique to improve performance.
Article
Mathematics, Applied
Rui Zhan, Weihong Chen, Xinji Chen, Runjie Zhang
Summary: This study investigates two exponential multistep methods for solving stiff delay differential equations and demonstrates their stiff convergent properties. Additionally, a new exponential multistep method based on the Rosenbrock method is introduced for solving nonlinear delay differential equations. The results show that these methods can effectively solve abstract stiff delay differential equations and can serve as time matching methods for delay partial differential equations.
Article
Mathematics, Applied
Carmen Arevalo, Gustaf Soderlind, Yiannis Hadjimichael, Imre Fekete
Summary: The paper discusses the importance of dynamic asymptotic error models and their estimation in multi-step methods, as well as the role of dynamically compensated step size controllers. The research shows that the new error models can improve the stability of controllers and produce more regular step size sequences.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
B. Saheya, T. Zhanlav, R. Mijiddorj
Summary: This paper proposes fifth- and sixth-order two-step schemes and further develops three-step iterative methods with ninth- and tenth-order of convergence. These methods are computationally efficient as they utilize only one matrix inversion per iteration.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Zahari Zlatev, Ivan Dimov, Istvan Farago, Krassimir Georgiev, Agnes Havasi
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2017)
Article
Mathematics, Applied
Istvan Farago, Gabriella Svantnerne Sebestyen
ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
(2018)
Editorial Material
Mathematics, Applied
Zahari Zlatev, Pasqua D'Ambra, Istvan Farago, Imre Fekete
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
Istvan Farago, Robert Horvath
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
Jozsef Csoka, Istvan Farago, Robert Horvath, Janos Karatson, Sergey Korotov
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2018)
Article
Mathematics, Applied
Robert Horvath, Istvan Farago, Janos Karatson
NUMERISCHE MATHEMATIK
(2019)
Article
Mathematics, Applied
Stefan M. Filipov, Ivan D. Gospodinov, Istvan Farago
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Istvan Farago, Dusan Repovs
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2019)
Article
Mathematics, Applied
Carmen Arevalo, Gustaf Soderlind, Yiannis Hadjimichael, Imre Fekete
Summary: The paper discusses the importance of dynamic asymptotic error models and their estimation in multi-step methods, as well as the role of dynamically compensated step size controllers. The research shows that the new error models can improve the stability of controllers and produce more regular step size sequences.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics
Balint Takacs, Robert Horvath, Istvan Farago
EUROPEAN JOURNAL OF MATHEMATICS
(2019)
Article
Meteorology & Atmospheric Sciences
Istvan Farago, Agnes Havasi, Zahari Zlatev
Article
Mathematics, Applied
Istvan Farago, Miklos Emil Mincsovics, Rahele Mosleh
APPLICATIONS OF MATHEMATICS
(2018)
Article
Mathematics, Applied
Stefan M. Filipov, Ivan D. Gospodinov, Istvan Farago
APPLIED MATHEMATICS LETTERS
(2017)