Article
Computer Science, Interdisciplinary Applications
Min Zhu, Aaron Towne
Summary: Spatial marching methods can generate low-cost models of flows with slowly varying direction. In this paper, we introduce a new variant of one-way Navier-Stokes equations that has a cost similar to parabolized stability equations while capturing contributions of all downstream-traveling modes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mechanics
C. Chan, P. Schlatter, R. C. Chin
Summary: The flow physics of turbulent boundary layers was investigated using spectral analysis based on the spanwise scale decomposition of the Reynolds stress transport equation with data obtained from direct numerical simulation. The study revealed evidence of inverse turbulent kinetic energy transfer occurring in the near-wall region, as well as inverse transfer of Reynolds shear stress transport across the entire boundary layer. Interactions between large-scale structures and the free stream flow were also observed at the edge of the boundary layer.
JOURNAL OF FLUID MECHANICS
(2021)
Article
Mathematics, Applied
Jin Zhao, Wen-An Yong
Summary: In this paper, a vectorial finite-difference-based lattice Boltzmann method (FDLBM) is proposed to solve the incompressible Navier-Stokes equations. The consistency, stability, and accuracy of the numerical schemes are analyzed, and a new boundary scheme is developed. Numerical experiments validate the feasibility of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
John Sebastian H. Simon, Hirofumi Notsu
Summary: This study investigates the convective boundary condition (CBC) on an outflow boundary for flow problems governed by the Navier-Stokes equations, and establishes the existence and uniqueness results of solutions. Numerical observations of different flow conditions are made by comparing with other known outflow boundary conditions. The CBC has the properties of being natural from a mathematical viewpoint, allow for discussions on the existence and uniqueness of solutions, and can be easily implemented with numerical methods such as the Newton method and the adjoint method.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Computer Science, Interdisciplinary Applications
Magnus Svard
Summary: This study focuses on deriving boundary conditions for the initial-boundary value Euler equations to establish an entropy bound for the physical (Navier-Stokes) entropy. The research begins by reviewing the entropy bound obtained for standard no-penetration wall boundary conditions and proposes a numerical implementation. The main results include deriving full-state boundary conditions and demonstrating that linear theory boundary conditions are unable to bound the entropy, requiring nonlinear bounds and additional boundary conditions. The theoretical findings are supported by numerical experiments.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mechanics
Chuong V. Tran, Xinwei Yu, David G. Dritschel
Summary: Incompressible fluid flows are characterized by high correlations between velocity and pressure, as well as between vorticity and pressure. This correlation plays a significant role in maintaining regularity in Navier-Stokes flows. The study suggests that as long as global pressure minimum (or minima) and velocity maximum (or maxima) are mutually exclusive, regularity is likely to persist.
JOURNAL OF FLUID MECHANICS
(2021)
Article
Physics, Mathematical
Masahiro Suzuki, Katherine Zhiyuan Zhang
Summary: In this paper, we investigate the compressible Navier-Stokes equation in a perturbed half-space with an outflow boundary condition and the supersonic condition. We demonstrate the unique existence of stationary solutions for the perturbed half-space, which exhibit multidirectional flow and are independent of the tangential directions. Additionally, we prove the asymptotic stability of these stationary solutions.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Yanqing Wang, Yulin Ye
Summary: In this paper, an energy conservation criterion is derived for weak solutions of both the incompressible and compressible Navier-Stokes equations. The criterion is based on a combination of velocity and its gradient. For the incompressible case, it extends known results on periodic domain, including the famous Lions' energy conservation criterion. For the compressible case, it improves recent results and extends criteria for energy conservation from incompressible to compressible flow.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Hugo Beirao da Veiga, Francesca Crispo
Summary: This paper investigates the convergence of solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations as the viscosity tends to zero. It focuses on the convergence under Navier slip-type boundary conditions after considering the Cauchy problem. It is shown that, in the presence of flat boundaries (such as the half-space case), convergence holds uniformly in time with respect to the initial data's norm. However, strong inviscid limit results are proven to be false in general domains corresponding to a large family of smooth initial data. A result in this direction is presented in Section 6.
ADVANCES IN NONLINEAR ANALYSIS
(2023)
Article
Computer Science, Interdisciplinary Applications
Hui Yao, Chuanju Xu, Mejdi Azaiez
Summary: In this paper, numerical solutions of an electrohydrodynamics model are studied. The model describes the electric convection dynamics arising from unipolar charge injection on the boundary of insulating liquid, which involves the Navier-Stokes equations, charge transfer equation, and potential energy equation. A class of stable numerical schemes is proposed and analyzed for this coupled equation system. The advantage of the proposed schemes is that they are unconditionally stable, allowing for flexibility in choosing the time step size for accuracy. Additionally, these schemes efficiently decouple the charge density and potential energy from the Navier-Stokes equations. The numerical examples provided demonstrate the expected convergence rate and accurate simulation of flow field and electric field changes induced by electrical convection. The schemes are extended to consider the case of variable density.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Interdisciplinary Applications
Renato Vacondio, Corrado Altomare, Matthieu De Leffe, Xiangyu Hu, David Le Touze, Steven Lind, Jean-Christophe Marongiu, Salvatore Marrone, Benedict D. Rogers, Antonio Souto-Iglesias
Summary: This paper provides a brief overview of the main challenges of Smoothed Particle Hydrodynamics (SPH) method. While SPH can simulate a wide range of applications, it still requires further development to address important issues. The SPHERIC SPH Grand Challenges aim to focus the attention of researchers and users worldwide on key development areas.
COMPUTATIONAL PARTICLE MECHANICS
(2021)
Article
Mathematics
P. Acevedo Tapia, C. Amrouche, C. Conca, A. Ghosh
Summary: This study proves the existence and uniqueness of weak and strong solutions in W-1,W-p(Ω) and W-2,W-p(Ω) with minimal regularity on the friction coefficient α. Additionally, uniform estimates are deduced for the solution with respect to α, allowing for analysis of the solution's behavior as α approaches infinity.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Physics, Mathematical
Hui Yao, Mejdi Azaiez, Chuanju Xu
Summary: This paper proposes efficient schemes for the Navier-Stokes equations, utilizing the Uzawa algorithm and modified algorithms to overcome existing shortcomings. Energy stability and error analysis are conducted for first- and second-order schemes, and simulations demonstrate the robustness and efficiency of the proposed schemes at high Reynolds numbers.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Alessio Falocchi, Filippo Gazzola
Summary: For the evolution of Navier-Stokes equations in bounded 3D domains, the uniqueness of the solution is determined by the existence of a regular solution. Under appropriate assumptions on the data and smoothness assumptions on the domain, this uniqueness can be obtained. Using a symmetrization technique, we prove these results for the case of Navier boundary conditions in a class of merely Lipschitz domains called sectors.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Juan Vicente Gutierrez-Santacreu, Marko Antonio Rojas-Medar
Summary: The Navier-Stokes-alpha equations are a type of LES models that aim to capture the influence of small scales on large ones without calculating the entire flow range. The parameter α represents the smallest resolvable scale by the model. When α=0, the classical Navier-Stokes equations for viscous, incompressible, Newtonian fluids are recovered. These equations can also be seen as a regularization of the Navier-Stokes equations, where α stands for the regularization parameter.
PHYSICA D-NONLINEAR PHENOMENA
(2023)
Article
Mathematics, Applied
Frances Y. Kuo, Giovanni Migliorati, Fabio Nobile, Dirk Nuyens
Summary: The study explores the connection between periodic Fourier space and non-periodic cosine space and Chebyshev space in the non-periodic settings, transferring known results using tent transform and cosine transform. Fast discrete cosine transform is applied for reconstruction, while a set of bi-orthogonal basis functions is used to reduce the size of the auxiliary index set in the component-by-component construction. New theory and efficient algorithmic strategies for CBC construction are provided, with results interpreted in the context of general function approximation and discrete least-squares approximation.
MATHEMATICS OF COMPUTATION
(2021)
Article
Mathematics, Applied
Matthieu Martin, Sebastian Krumscheid, Fabio Nobile
Summary: This study focuses on the numerical approximation of an optimal control problem for an elliptic Partial Differential Equation (PDE) with random coefficients. By investigating methods like gradient-type iterations and conjugate gradient methods, approximate optimal control solutions are obtained, with error and complexity analyses provided.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Mathematics, Applied
Fabio Nobile, Davide Pradovera
Summary: The proposed method tackles model order reduction for parametric dynamical systems by building a simplified model through frequency surrogates. It can be applied in high-dimensional settings by using locally-refined sparse grids in parameter space to alleviate the curse of dimensionality.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2021)
Article
Computer Science, Theory & Methods
Jonas Latz, Juan P. Madrigal-Cianci, Fabio Nobile, Raul Tempone
Summary: In this work, two generalizations of the Parallel Tempering algorithm are introduced, with state-dependent swapping rates inspired by a continuous time Infinite Swapping algorithm. The analysis of reversibility and ergodicity properties show that these generalized PT algorithms significantly improve sampling efficiency compared to more traditional sampling algorithms.
STATISTICS AND COMPUTING
(2021)
Article
Mathematics, Applied
Yoshihito Kazashi, Fabio Nobile, Eva Vidlickova
Summary: In this paper, we investigate the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Our schemes are shown to satisfy a discrete variational formulation and we establish their stability properties under certain conditions. Additionally, we demonstrate the relationship between our schemes and projector-splitting integrators, as well as the applicability of our stability analysis to related schemes proposed in previous literature.
NUMERISCHE MATHEMATIK
(2021)
Article
Mathematics, Applied
Vesa Kaarnioja, Yoshihito Kazashi, Frances Y. Kuo, Fabio Nobile, Ian H. Sloan
Summary: This paper discusses the kernel-based interpolation method for approximating multivariate periodic functions, particularly in the context of uncertainty quantification for elliptic partial differential equations with a diffusion coefficient given by a periodic random field. The paper includes a complete error analysis, lattice construction details, and numerical experiments supporting the proposed theory, ensuring a convergence rate and error bound independent of dimension.
NUMERISCHE MATHEMATIK
(2022)
Article
Statistics & Probability
Eleonora Arnone, Alois Kneip, Fabio Nobile, Laura M. Sangalli
Summary: This paper examines the consistency of the estimator in a spatial regression with partial differential equation (PDE) regularization. By using the finite-element method to obtain an approximate solution, the paper investigates the consistency, bias, and variance of the estimators with respect to sample size and the value of the smoothing parameter.
Article
Mathematics, Interdisciplinary Applications
Davide Pradovera, Fabio Nobile
Summary: This article introduces the minimal rational interpolation (MRI) method and its numerical instabilities when building a surrogate model for frequency response problems over a large range. To overcome these instabilities, the article proposes a strategy of replacing the unstable global MRI surrogate with a union of stable local rational models. This strategy includes automatically and adaptively partitioning the frequency range and selecting the sampled frequencies in each sub-range. The effectiveness of this method is verified through two numerical examples.
JOURNAL OF MATHEMATICS IN INDUSTRY
(2022)
Article
Mathematics
Giovanni Migliorati, Fabio Nobile
Summary: This paper proposes and analyzes a randomized cubature formula for efficient numerical integration. The formula is exact and stable on a finite-dimensional subspace, with error estimates in both preasymptotic and asymptotic regimes. It can be seen as a variance reduction technique for a Monte Carlo estimator, offering significant variance reduction and spectral convergence advantages.
JOURNAL OF APPROXIMATION THEORY
(2022)
Article
Computer Science, Interdisciplinary Applications
Christian Vergara, Simone Stella, Massimiliano Maines, Pasquale Claudio Africa, Domenico Catanzariti, Cristina Dematte, Maurizio Centonze, Fabio Nobile, Alfio Quarteroni, Maurizio Del Greco
Summary: This study assessed a computational tool for estimating electrical activation in the left ventricle of patients with left bundle branch block and possible myocardialfibrosis, with a focus on the latest electrically activated segment (LEAS). The results showed that the tool was able to accurately reproduce electrical activation maps and had excellent agreement in LEAS location.
MEDICAL & BIOLOGICAL ENGINEERING & COMPUTING
(2022)
Article
Mathematics, Applied
Fabio Nobile, Tommaso Vanzan
Summary: This manuscript investigates the discretization of robust quadratic optimal control problems under uncertainty. It proposes efficient preconditioners and estimates the dependence of the spectrum of the preconditioned system matrix on the statistical properties. Numerical experiments confirm the theoretical results.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2023)
Article
Engineering, Multidisciplinary
Dimitri Goutaudier, Fabio Nobile, Jurg Schiffmann
Summary: In this paper, an ordered reduced basis interpolation (ORBI) technique is proposed to improve the adaptation accuracy and computational efficiency of the POD basis. By considering more information in the construction of the interpolation operator, the proposed method is more accurate than the ITSGM method at a similar computation cost. Trained at only three points of the parameters space, the developed h-ROM performs much faster simulations with satisfactory accuracy even far from the training points.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2023)
Article
Mathematics, Applied
Yoshihito Kazashi, Fabio Nobile
Summary: The paper presents a kernel method for estimating a probability density function from an independent and identically distributed sample. The estimator is a linear combination of kernel functions with coefficients determined by a linear equation. An error analysis is conducted for the mean integrated squared error in a general reproducing kernel Hilbert space setting. The developed theory is then applied to estimate probability density functions in weighted Korobov spaces, achieving a dimension-independent convergence rate close to the optimal rate. Numerical results validate the theory.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Interdisciplinary Applications
Juan P. Madrigal-Cianci, Fabio Nobile, Raul Tempone
Summary: In this work, a class of multilevel Markov chain Monte Carlo (ML-MCMC) algorithms based on independent Metropolis-Hastings proposals is presented, analyzed, and implemented for Bayesian inverse problems. The algorithm aims to construct highly coupled Markov chains together with the standard multilevel Monte Carlo method to achieve better cost-tolerance complexity. The effectiveness of the proposed method is demonstrated through convergence analysis and numerical experiments on various academic examples.
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2023)
Article
Mathematics, Interdisciplinary Applications
Matthieu Martin, Fabio Nobile
Summary: This study focuses on the optimal control problem for a partial differential equation with random coefficients, proposing a numerical approximation method and using an importance sampling version of the SAGA algorithm for practical solution. A full error and complexity analysis of the proposed numerical scheme is provided, showing the superiority of the approach in terms of performance.
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2021)