Article
Mathematics, Applied
Luoping Chen, Fanyun Wu, Guoyan Zeng
Summary: This paper investigates a two-grid weak Galerkin method for solving semilinear elliptic differential equations. The method involves solving the equation on a coarse mesh and then linearizing it on a fine mesh. Theoretical analysis shows that the method achieves the same convergence accuracy as the weak Galerkin method when certain conditions are met.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Juntao Huang, Yong Liu, Yuan Liu, Zhanjing Tao, Yingda Cheng
Summary: This paper proposes an adaptive multiresolution ultra-weak discontinuous Galerkin (UWDG) method for solving nonlinear dispersive wave equations, including the Korteweg-de Vries (KdV) equation and Zakharov-Kuznetsov (ZK) equation. The UWDG formulation for the KdV equation has been previously proposed, and a new UWDG formulation is developed for the ZK equation with mixed derivative terms. By achieving adaptivity based on multiresolution, the method is particularly effective for capturing solitary wave structures.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Nohan Joemon, Melpakkam Pradeep, Lokesh K. Rajulapati, Raghunathan Rengaswamy
Summary: This paper introduces a smoothing-based approach for discovering partial differential equations from noisy measurements. The method is data-driven and improves performance by incorporating first principles knowledge. The effectiveness of the algorithm is demonstrated in a real system using a new benchmark metric.
COMPUTERS & CHEMICAL ENGINEERING
(2024)
Article
Computer Science, Interdisciplinary Applications
Sandip Maji, Srinivasan Natesan
Summary: This article discusses an efficient numerical method for solving nonlinear time-fractional integro-partial differential initial-boundary-value problems. The non-linearity is tackled using the Newton linearization process. The non-symmetric interior penalty Galerkin method is applied for the spatial variable, and a semi-discrete problem is obtained in the time variable. By using L1-scheme and L2-scheme for the time-fractional derivative, and trapezoidal rule for the integral term, a fully-discrete scheme is derived.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Joerg Weber
Summary: The time evolution of collisionless plasma is modeled using the relativistic Vlasov-Maxwell system, coupling the Vlasov equation with Maxwell equations. Different particle species and external currents are considered, with weak solution concept introduced to prove the existence of global-in-time solutions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Engineering, Multidisciplinary
Yasen Wang, Huazhen Fang, Junyang Jin, Guijun Ma, Xin He, Xing Dai, Zuogong Yue, Cheng Cheng, Hai-Tao Zhang, Donglin Pu, Dongrui Wu, Ye Yuan, Jorge Goncalves, Juergen Kurths, Han Ding
Summary: This study presents a novel framework for identifying stochastic differential equations (SDEs) by leveraging sparse Bayesian learning (SBL) technique. The framework automatically searches for a parsimonious representation from the space of candidate basis functions using SBL and formulates the linear regression problem for the discovery of SDEs in an efficient way. The effectiveness of the proposed framework is demonstrated using real and simulated data.
Article
Mathematics, Applied
Zhen Wu, Bing Xie, Zhiyong Yu
Summary: This paper discusses the Sobolev type weak solutions of a certain class of second order quasilinear parabolic partial differential equations (PDEs). It provides a probabilistic interpretation for the weak solutions by using a family of coupled forward-backward stochastic differential equations (FBSDEs) that satisfy the monotonous assumption. The existence of weak solutions is proven based on the classical solutions of a family of PDEs approximating the weak solutions of the quasilinear PDEs. The uniqueness of the weak solutions is obtained through the principle of norm equivalence linking FBSDEs and PDEs.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Interdisciplinary Applications
Muhammad Bhatti, Md Habibur Rahman, Nicholas Dimakis
Summary: A multivariable technique using B-polynomials is employed to estimate solutions of NPDE, with coefficients determined using the Galerkin method before conversion to an operational matrix equation. The method provides higher-order precision compared to finite difference in solving NPDE equations and has potential for solving complex partial differential equations in multivariable problems.
FRACTAL AND FRACTIONAL
(2021)
Article
Mathematics, Applied
Maria Lukacova-Medvid'ova, Philipp Oeffner
Summary: This paper presents the convergence analysis of high-order finite element methods, with a focus on the discontinuous Galerkin scheme. By preserving structure properties and utilizing dissipative weak solutions, the convergence of the multidimensional high-order DG scheme is proven. Numerical simulations validate the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
M. Brio, J-G Caputo, H. Kravitz
Summary: This study introduces and compares three methods for solving linear PDEs on metric graphs: spectral method, finite difference method, and discontinuous Galerkin method. The spectral method can obtain eigenvalues and eigenvectors of arbitrary order with machine precision; the discontinuous Galerkin method provides approximations of arbitrary polynomial order; while the finite difference method requires additional treatments to maintain accuracy.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
William Anderson, Mohammad Farazmand
Summary: Reduced order nonlinear solutions (RONS) is a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent parameters. By minimizing the discrepancy between reduced dynamics and full PDE dynamics, explicit ordinary differential equations on the tangent bundle of the manifold can be obtained.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Khalil Ezzinbi, Mohamed Aziz Taoudi
Summary: This paper discusses the existence and attractiveness of periodic solutions for some partial functional differential equations in Banach spaces. By assuming a first linear part generates a strongly continuous semi-group and the delayed part is periodic, it is proved that the existence of a bounded solution implies the existence of a periodic solution. The analysis relies on a fixed point theorem and weak topology arguments, extending both new and classical results in a broad sense. An application to a transport equation with delay is also presented.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Feng Lu, Wenqi Bi
Summary: In this paper, we present the complete solutions of certain nonlinear partial differential equations, including the well-known PDE of tubular surfaces as a special case.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Omar Bazighifan, Areej A. Al-Moneef, Hasan Ali Ali, Thangaraj Raja, Kamsing Nonlaopon, Taher A. Nofal
Summary: The study aims to identify the necessary conditions for the oscillation of impulsive conformable partial differential equation systems with the Robin boundary condition, and the important findings are demonstrated with a robust example.
Article
Engineering, Mechanical
Jelena Karakasevic, Michael Oberguggenberger
Summary: In recent decades, there has been increasing interest in combining probability and interval analysis to model parameter uncertainty in engineering models. The theory of random sets provides a unified mathematical framework for describing the input and output of structural models using set-valued random variables. This paper aims to highlight the mathematical principles behind this approach and demonstrate its modeling and computational implications through prototypical partial differential equations in elastostatics and elastodynamics.
PROBABILISTIC ENGINEERING MECHANICS
(2022)
Article
Microbiology
Kristin A. Moore, Sabina Altus, Jian W. Tay, Janet B. Meehl, Evan B. Johnson, David M. Bortz, Jeffrey C. Cameron
NATURE MICROBIOLOGY
(2020)
Article
Multidisciplinary Sciences
Nicholas C. Hill, Jian Wei Tay, Sabina Altus, David M. Bortz, Jeffrey C. Cameron
Article
Multidisciplinary Sciences
Graycen E. Wheeler, Amrita Purkayastha, Eric N. Bunker, David M. Bortz, Xuedong Liu
Summary: This study describes a method for measuring and quantifying single-cell and bulk motility of HaCaT keratinocytes using a nuclear stain. The method includes a MATLAB script for analyzing TrackMate output files to calculate displacements, motility rates, and trajectory angles, allowing for quick, straightforward, and scalable analysis of cell motility rates.
JOVE-JOURNAL OF VISUALIZED EXPERIMENTS
(2021)
Article
Mathematics, Applied
Jacqueline Wentz, Jeffrey C. Cameron, David M. Bortz
Summary: This research presents an analytical singular value decomposition method for the stoichiometry matrix of a reaction-diffusion system to reveal spatial flux patterns. Approaches for approximate and exact singular value decompositions are studied for different reaction and diffusion scenarios. The study shows that the singular value decomposition depends on smaller matrix decompositions, providing efficient analysis of the stoichiometry matrix of reaction-diffusion systems.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Daniel A. Messenger, David M. Bortz
Summary: We have developed a weak-form sparse identification method for interacting particle systems (IPS), aiming to reduce computational complexity for large number of particles and offer robustness to noise. By combining mean-field theory of IPS with the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy), we provide a fast and reliable system identification scheme for recovering the governing stochastic differential equations for an IPS.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Multidisciplinary Sciences
Daniel A. Messenger, Graycen E. Wheeler, Xuedong Liu, David M. Bortz
Summary: Researchers have developed the WSINDy method to learn equations for communities of cells and classify them using a novel scheme. The method demonstrates efficiency and proficiency through various test scenarios.
JOURNAL OF THE ROYAL SOCIETY INTERFACE
(2022)
Article
Mathematics, Applied
Razvan C. Fetecau, Hui Huang, Daniel Messenger, Weiran Sun
Summary: We investigate the zero-diffusion limit for both continuous and discrete aggregation-diffusion models over convex and bounded domains. Our result relaxes the regularity assumptions on the interaction and external potentials and improves the convergence rate.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Interdisciplinary Applications
Daniel A. Messenger, David M. Bortz
Summary: A novel weak formulation and discretization method, called WSINDy, is proposed to discover governing equations from noisy measurement data. Compared to the standard SINDy algorithm, WSINDy allows for reliable model identification and reduces errors in data with large noise levels.
MULTISCALE MODELING & SIMULATION
(2021)
Article
Immunology
Andrea G. Buchwald, Jude Bayham, Jimi Adams, David Bortz, Kathryn Colborn, Olivia Zarella, Meghan Buran, Jonathan Samet, Debashis Ghosh, Rachel Herlihy, Elizabeth J. Carlton
Summary: The study conducted in Colorado showed that early policy measures related to COVID-19 significantly decreased mobility and reduced virus transmission, with the effective reproductive number falling below 1. Although mobility increased to near baseline levels after some restrictions were lifted, transmission remained low. As the model parameters were adjusted to better reflect the reality in Colorado over time, there were modest changes in estimates of intervention effects and more conservative long-term projections.
EMERGING INFECTIOUS DISEASES
(2021)
Article
Mathematics, Applied
Jacqueline M. Wentz, David M. Bortz
Summary: This study focuses on determining sufficient conditions to guarantee that the discrete reaction-diffusion system is bounded for all time, using a Lyapunov-like function on a 1D domain with homogeneous Neumann boundary conditions and nonnegative initial data and solutions. The existence of this function ensures the boundedness of the system, as illustrated in the context of four example systems.
SIAM JOURNAL ON APPLIED MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)