Article
Mathematics, Applied
Chaolong Jiang, Yushun Wang, Yuezheng Gong
Summary: A novel class of explicit high-order energy-preserving methods for general Hamiltonian partial differential equations with non-canonical structure matrix is proposed. By reformulating the original system into an equivalent form with a modified quadratic energy conservation law and discretizing it in time using explicit high-order Runge-Kutta methods with orthogonal projection techniques, the schemes are shown to share the order of explicit Runge-Kutta method and preserve energy while reaching high-order accuracy.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Zhaohui Fu, Tao Tang, Jiang Yang
Summary: This study aims to extend the strong stability preserving (SSP) theory to solve the nonlinear phase field equation that satisfies both the maximum bound property (MBP) and the energy dissipation law. By using the Runge-Kutta time discretization method, we derive a necessary and sufficient condition for satisfying MBP, and further provide a necessary condition for the MBP scheme to satisfy energy dissipation.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Changying Liu, Jiayin Li, Zhenqi Yang, Yumeng Tang, Kai Liu
Summary: In this paper, an energy-preserving collocation integrator is derived for solving hyperbolic Hamiltonian systems. Two specific high-order energy-preserving and symmetric integrators are presented by choosing collocation nodes as two and three Gauss-Legendre points, respectively. The convergence and symmetry of the constructed energy-preserving integrators are rigorously analyzed. Numerical results confirm the energy conservation property and accuracy of the proposed integrators.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
T. Dzanic, W. Trojak, F. D. Witherden
Summary: In this work, a modified explicit Runge-Kutta temporal integration scheme is proposed to guarantee the preservation of locally-defined quasiconvex set of bounds for the solution. The schemes use a bijective mapping to enforce bounds between the admissible set of solutions and the real domain. It is shown that these schemes can recover a wide range of methods, including positivity preserving, discrete maximum principle satisfying, entropy dissipative, and invariant domain preserving schemes. The additional computational cost is the evaluation of two nonlinear mappings which generally have closed-form solutions. The approach is demonstrated in numerical experiments using a pseudospectral spatial discretization without explicit shock capturing schemes for nonlinear hyperbolic problems with discontinuities.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
N. S. Hoang
Summary: This paper presents a class of modified collocation Runge-Kutta-Nystrom (RKN) methods for solving second-order initial value problems, demonstrating their higher accuracy and stability. The methods are applicable to a wider range of problems and show superconvergence under certain conditions. Numerical experiments confirm the advantages of these new methods.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Xueyu Qin, Zhenhua Jiang, Jian Yu, Lintao Huang, Chao Yan
Summary: In this study, explicit strong stability-preserving (SSP) three-derivative Runge-Kutta (ThDRK) methods are proposed and their order accuracy conditions are determined. The SSP theory is developed based on a new Taylor series condition for ThDRK methods, and the optimal SSP coefficient is found. Comparison with other methods shows that ThDRK methods have the highest effective SSP coefficient for order accuracy (3 = p = 5). Numerical experiments demonstrate that ThDRK methods maintain the desired order of convergence and have efficient computational cost.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Stephan Nuesslein, Hendrik Ranocha, David Ketcheson
Summary: This paper proposes a method to ensure the solution of differential equations remains positive or within a certain range by adjusting the weights of Runge-Kutta integration. The weights are chosen by solving a linear program and further constraints are considered for selecting the weights. Numerical examples demonstrate the effectiveness of this approach in tackling both stiff and non-stiff problems.
COMMUNICATIONS IN APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE
(2021)
Article
Mathematics
Janez Urevc, Miroslav Halilovic
Summary: This paper presents a new class of Runge-Kutta-type collocation methods for numerical integration of ordinary differential equations (ODEs), derived from the integral form of the differential equation. The approach enhances accuracy while keeping the same number of stages, leading to improved accuracy for methods like Gauss-Legendre and Lobatto IIIA. The methods are expressed in table form similar to Butcher tableaus and their performance is investigated on a variety of ODEs.
Article
Mathematics, Applied
Zachary J. Grant
Summary: This work explores a mixed precision method to accelerate the implementation of multi-stage methods. It demonstrates that properly designed Runge-Kutta methods can perform certain costly intermediate computations at lower precision without affecting the accuracy of the overall solution. Numerical studies confirm the effectiveness of this approach.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
M. Adams, J. Finden, P. Phoncharon, P. H. Muir
Summary: This paper presents the application of COLSYS/COLNEW collocation software package in numerically solving boundary value ODEs. It introduces the use of continuous Runge-Kutta methods to obtain superconvergent interpolants and generalizes the method for solving mixed order BVODE systems. The results show that the superconvergent interpolants are more accurate and cost-effective compared to the collocation solutions.
Article
Mathematics, Applied
Benjamin Yeager, Ethan Kubatko, Dylan Wood
Summary: In this work, the linear stability properties of discontinuous Galerkin spatial discretizations with strong-stability-preserving multistep Runge-Kutta methods are assessed. It is found that the constraint for linear stability is more strict than that for strong-stability-preservation. Through testing, an optimal time stepper that requires the fewest evaluations of the discontinuous Galerkin operator is selected, and all methods are found to converge under the stability constraints determined in the study.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Michelle Muniz, Matthias Ehrhardt, Michael Guenther, Renate Winkler
Summary: In this paper, numerical methods for solving nonlinear Ito stochastic differential equations on manifolds are presented. The strong convergence of the extended Runge-Kutta-Munthe-Kaas (RKMK) schemes for stochastic ordinary differential equations on manifolds is analysed. The effectiveness of these schemes is demonstrated by numerical results of applying them to a problem with an autonomous underwater vehicle.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
H. T. Huynh
Summary: A defining feature of the DG method is the jump discontinuity in the piecewise polynomial solution at each step. The method is derived in differential form using a correction function that approximates the jump and creates a continuous solution. The construction of implicit Runge-Kutta schemes (IRK-DG) is facilitated by the correction function, leading to different methods based on different quadratures. The correction function also helps establish accuracy conditions and unique properties of the IRK-DG methods.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Jingwei Sun, Hong Zhang, Xu Qian, Songhe Song
Summary: In this work, a class of up to eighth-order maximum-principle-preserving (MPP) methods for the Allen-Cahn equation is developed. The integrating factor two-step Runge-Kutta (IFTSRK) methods are extended and combined with the linear stabilization technique to derive sufficient conditions for the unconditional preservation of the discrete maximum principle. Numerical experiments demonstrate the high-order accuracy and MPP characteristic of the proposed methods.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
A. Moradi, A. Abdi, G. Hojjati
Summary: This paper discusses implicit-explicit (IMEX) methods for systems of ordinary differential equations. The explicit part of the methods has strong stability preserving (SSP) property, while the implicit part has Runge-Kutta stability property and A- or L-stability. The explicit part is treated by explicit second derivative general linear methods, and the implicit part is treated by implicit general linear methods. Various methods with different orders and combinations of orders are constructed, considering the interaction between the implicit and explicit parts. The performance of the proposed IMEX schemes is tested on one-dimensional linear and nonlinear problems, and the expected order of convergence is presented.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Luigi Brugnano, Felice Iavernaro, Paolo Zanzottera
Summary: The paper introduces a new forecast model that can account for the spatial and temporal heterogeneity of an epidemic, and successfully applies it to the spread of COVID-19 in Italy. This model is proven to be robust and reliable in predicting total and active cases, as well as simulating different scenarios to address various issues such as lockdown measures, actual attack rate estimation, and rapid screening tests for containing the epidemic.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Editorial Material
Mathematics, Applied
Luigi Brugnano, Yaroslav D. Sergeyev, Anatoly Zhigljavsky
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Editorial Material
Mathematics, Applied
Luigi Brugnano, Dmitry E. Kvasov, Yaroslav D. Sergeyev
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Arturo De Marinis, Felice Iavernaro, Francesca Mazzia
Summary: This article presents a new strategy for determining an unmanned aerial vehicle trajectory that minimizes flight time in the presence of avoidance areas and obstacles. The method combines classical results from optimal control theory with a continuation technique to adapt the solution curve dynamically to obstacles. Numerical illustrations for both two-dimensional and three-dimensional path planning problems show the efficiency of the approach.
NUMERICAL ALGORITHMS
(2022)
Article
Computer Science, Interdisciplinary Applications
Pierluigi Amodio, Luigi Brugnano, Filippo Scarselli
Summary: The study introduces new PaperRank and AuthorRank indices that can be implemented in the Scopus database to provide quantitative and qualitative information that traditional citation counts and h-index cannot. These indices can be cheaply updated and examples are provided to demonstrate their potential and possible extensions.
JOURNAL OF INFORMETRICS
(2021)
Article
Mathematics, Applied
Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro
Summary: This paper focuses on the energy-conserving methods for Poisson problems and proposes a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The practical implementation and the conservation of Casimirs are fully discussed. Numerical tests are conducted to evaluate the theoretical findings.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro
Summary: This paper focuses on numerical methods for one-sided event location in discontinuous differential problems with nonlinear event functions, particularly of polynomial type. The original problem is transformed into an equivalent Poisson problem, which is effectively solved using a recently devised class of energy conserving methods for Poisson systems. The implementation of the methods is fully discussed, with emphasis on the specific problem at hand. Numerical tests are conducted to assess the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Luigi Brugnano, Gianluca Frasca-Caccia, Felice Iavernaro, Vincenzo Vespri
Summary: This paper discusses a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework is based on an expansion of the vector field along an orthonormal basis, and relies on perturbation results for the considered problem. Initially devised for the approximation of ordinary differential equations, it is here further extended and, moreover, generalized to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro
Summary: In this note, we propose an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than the one based on the expansion of the polynomials with respect to the canonical basis. This algorithm is designed to solve corresponding fractional differential equations, and a few numerical illustrations are provided.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro
Summary: Recently, the efficient numerical solution of Hamiltonian problems has been achieved by using energy-conserving Runge-Kutta methods called Hamiltonian Boundary Value Methods (HBVMs). These methods are derived by expanding the vector field along the Legendre orthonormal basis. In this paper, we extend this approach to the Chebyshev polynomial basis and analyze the corresponding Runge-Kutta methods, along with their generalizations when used as spectral formulae in time.
NUMERICAL ALGORITHMS
(2023)
Review
Mathematics, Applied
Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro
Summary: Efficient numerical solutions for Hamiltonian problems have led to the development of energy-conserving Runge-Kutta methods called Hamiltonian Boundary Value Methods (HBVMs). These methods have an interesting interpretation in terms of continuous-stage Runge-Kutta methods and can be extended to higher-order differential problems.
Article
Mathematics, Applied
Hao Liu, Yuzhe Li
Summary: This paper investigates the finite-time stealthy covert attack on reference tracking systems with unknown-but-bounded noises. It proposes a novel finite-time covert attack method that can steer the system state into a target set within a finite time interval while being undetectable.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Nikolay A. Kudryashov, Aleksandr A. Kutukov, Sofia F. Lavrova
Summary: The Chavy-Waddy-Kolokolnikov model with dispersion is analyzed, and new properties of the model are studied. It is shown that dispersion can be used as a control mechanism for bacterial colonies.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Qiang Ma, Jianxin Lv, Lin Bi
Summary: This paper introduces a linear stability equation based on the Boltzmann equation and establishes the relationship between small perturbations and macroscopic variables. The numerical solutions of the linear stability equations based on the Boltzmann equation and the Navier-Stokes equations are the same under the continuum assumption, providing a theoretical foundation for stability research.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Samuel W. Akingbade, Marian Gidea, Matteo Manzi, Vahid Nateghi
Summary: This paper presents a heuristic argument for the capacity of Topological Data Analysis (TDA) to detect critical transitions in financial time series. The argument is based on the Log-Periodic Power Law Singularity (LPPLS) model, which characterizes financial bubbles as super-exponential growth (or decay) with increasing oscillations approaching a tipping point. The study shows that whenever the LPPLS model fits the data, TDA generates early warning signals. As an application, the approach is illustrated using positive and negative bubbles in the Bitcoin historical price.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Xavier Antoine, Jeremie Gaidamour, Emmanuel Lorin
Summary: This paper is interested in computing the ground state of nonlinear Schrodinger/Gross-Pitaevskii equations using gradient flow type methods. The authors derived and analyzed Fractional Normalized Gradient Flow methods, which involve fractional derivatives and generalize the well-known Normalized Gradient Flow method proposed by Bao and Du in 2004. Several experiments are proposed to illustrate the convergence properties of the developed algorithms.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Lianwen Wang, Xingyu Wang, Zhijun Liu, Yating Wang
Summary: This contribution presents a delayed diffusive SEIVS epidemic model that can predict and quantify the transmission dynamics of slowly progressive diseases. The model is applied to fit pulmonary tuberculosis case data in China and provides predictions of its spread trend and effectiveness of interventions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Shuangxi Huang, Feng-Fei Jin
Summary: This paper investigates the error feedback regulator problem for a 1-D wave equation with velocity recirculation. By introducing an invertible transformation and an adaptive error-based observer, an observer-based error feedback controller is constructed to regulate the tracking error to zero asymptotically and ensure bounded internal signals.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Weimin Liu, Shiqi Gao, Feng Xu, Yandong Zhao, Yuanqing Xia, Jinkun Liu
Summary: This paper studies the modeling and consensus control of flexible wings with bending and torsion deformation, considering the vibration suppression as well. Unlike most existing multi-agent control theories, the agent system in this study is a distributed parameter system. By considering the mutual coupling between the wing's deformation and rotation angle, the dynamics model of each agent is expressed using sets of partial differential equations (PDEs) and ordinary differential equations (ODEs). Boundary control algorithms are designed to achieve control objectives, and it is proven that the closed-loop system is asymptotically stable. Numerical simulation is conducted to demonstrate the effectiveness of the proposed control scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Gourav Mandal, Lakshmi Narayan Guin, Santabrata Chakravarty
Summary: The ecological framework investigates the dynamical complexity of a system influenced by prey refuge and alternative food sources for predators. This study provides a thorough investigation of the stability-instability phenomena, system parameters sensitivity, and the occurrence of bifurcations. The bubbling phenomenon, which indicates a change in the amplitudes of successive cycles, is observed in the current two-dimensional continuous system. The controlling system parameter for the bubbling phenomena is found to be the most sensitive. The prediction and identification of bifurcations in the dynamical system are crucial for theoretical and field researchers.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Damian Trofimowicz, Tomasz P. Stefanski, Jacek Gulgowski, Tomasz Talaska
Summary: This paper presents the application of control engineering methods in modeling and simulating signal propagation in time-fractional electrodynamics. By simulating signal propagation in electromagnetic media using Maxwell's equations with fractional-order constitutive relations in the time domain, the equations in time-fractional electrodynamics can be considered as a continuous-time system of state-space equations in control engineering. Analytical solutions are derived for electromagnetic-wave propagation in the time-fractional media based on state-transition matrices, and discrete time zero-order-hold equivalent models are developed and their analytical solutions are derived. The proposed models yield the same results as other reference methods, but are more flexible in terms of the number of simulation scenarios that can be tackled due to the application of the finite-difference scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Yuhao Zhao, Fanhao Guo, Deshui Xu
Summary: This study develops a vibration analysis model of a nonlinear coupling-layered soft-core beam system and finds that nonlinear coupling layers are responsible for the nonlinear phenomena in the system. By using reasonable parameters for the nonlinear coupling layers, vibrations in the resonance regions can be reduced and effective control of the vibration energy of the soft-core beam system can be achieved.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
S. Kumar, H. Roy, A. Mitra, K. Ganguly
Summary: This study investigates the nonlinear dynamic behavior of bidirectional functionally graded plates (BFG) and unidirectional functionally graded plates (UFG). Two different methods, namely the whole domain method and the finite element method, are used to formulate the dynamic problem. The results show that all three plates exhibit hardening type nonlinearity, with the effect of material gradation parameters being more pronounced in simply supported plates.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Isaac A. Garcia, Susanna Maza
Summary: This paper analyzes the role of non-autonomous inverse Jacobi multipliers in the problem of nonexistence, existence, localization, and hyperbolic nature of periodic orbits of planar vector fields. It extends and generalizes previous results that focused only on the autonomous or periodic case, providing novel applications of inverse Jacobi multipliers.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Yongjian Liu, Yasi Lu, Calogero Vetro
Summary: This paper introduces a new double phase elliptic inclusion problem (DPEI) involving a nonlinear and nonhomogeneous partial differential operator. It establishes the existence and extremality results to the elliptic inclusion problem and provides definitions for weak solutions, subsolutions, and supersolutions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Shangshuai Li, Da-jun Zhang
Summary: In this paper, the Cauchy matrix structure of the spin-1 Gross-Pitaevskii equations is investigated. A 2 x 2 matrix nonlinear Schrodinger equation is derived using the Cauchy matrix approach, serving as an unreduced model for the spin-1 BEC system with explicit solutions. Suitable constraints are provided to obtain reductions for the classical and nonlocal spin-1 GP equations and their solutions, including one-soliton solution, two-soliton solution, and double-pole solution.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
