Article
Computer Science, Interdisciplinary Applications
Jiayin Li, Chi-Wang Shu, Jianxian Qiu
Summary: This paper presents a new type of high-order finite volume and finite difference multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes for solving hyperbolic conservation laws. The schemes utilize information defined on central spatial stencils without introducing equivalent multi-resolution representation, demonstrating robustness and good performance in numerical experiments. The spatial reconstruction is derived from the original HWENO schemes, using large stencils similar to classical HWENO schemes but narrower than classical WENO schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Yicheng Lin, Zhenming Wang, Jun Zhu
Summary: In this paper, new high-order finite difference and finite volume ALW-WENO schemes are proposed for hyperbolic conservation laws. These schemes utilize adaptive linear weights to achieve desired accuracy in smooth regions and non-oscillatory properties in regions with strong shocks. The simple structure of these schemes makes it easier to obtain high-order accuracy and solve multi-dimensional problems in large scale engineering applications, with improved computational efficiency.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
A. J. Kriel
Summary: This study introduces a general condition for numerical schemes to mimic the properties of exact solutions of scalar conservation laws. By applying this condition to various schemes, different CFL-like conditions are derived to ensure the accuracy and reliability of the numerical simulations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
U. L. R. I. K. S. FJORDHOLM, M. A. R. K. U. S. MUSCH, N. I. L. S. H. RISEBRO
Summary: We extend the analysis of nonlinear conservation laws on networks, which was initiated in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101-128], to a large class of flux functions that are neither monotone nor convex/concave. By utilizing the framework laid down in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101-128], we prove the existence and uniqueness of entropy solutions within a natural class through the convergence of an explicit finite volume method. This leads to the existence of a semigroup of solutions. The theoretical results are supported by numerical experiments, including an experimental order of convergence.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Arthur E. P. Veldman
Summary: It has been found advantageous to have additional (secondary) invariants in finite-volume discretizations of flow equations besides the (primary) invariants from the constituting conservation laws. This paper presents general (necessary and sufficient) requirements for a method to convectively preserve discrete kinetic energy, with a key ingredient being close discrete consistency between the convective term in the momentum equation and the terms in the other conservation equations (mass, internal energy). Examples include the discretization of Euler equations for subsonic (in)compressible flow using supraconservative finite-volume methods on structured and unstructured grids.
Article
Mathematics, Applied
Indra Wibisono, Yanuar, Engkos A. Kosasih
Summary: The TENO scheme presented in this study utilizes Hermite polynomials for efficient and targeted non-oscillatory reconstruction, incorporating compact reconstruction and low dissipation advantages. It introduces a new high-order global smoothness indicator and demonstrates improved shock-capturing performance in numerical tests.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Yanhui Zhou, Qingsong Zou
Summary: In this paper, a post-processing procedure for an eight-nodes-serendipity finite element solution for elliptic equations is proposed. By enlarging the finite element space and ensuring local conservation law, the post-process achieves optimal convergence to the exact solution under both H-1 and L-2 norms. The computational cost is proportional to the number of serendipity elements and can be implemented in a parallel computing environment.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2021)
Article
Computer Science, Interdisciplinary Applications
Ke Xu, Zhenxun Gao, Zhansen Qian, Chongwen Jiang, Chun-Hian Lee
Summary: This paper focuses on the discontinuities capturing problems in nonconservative and nonconvex conservative hyperbolic systems. It analyzes the numerical dissipation at discontinuous points in the simulation process for the Godunov scheme of nonconservative hyperbolic systems. The paper proposes a novel numerical path preserving (NPP) method to modify the original Godunov schemes for accurately capturing the discontinuous structures.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Kyungrok Lee, Jung-Il Choi, Jungho Yoon
Summary: In this study, a novel weighted essentially non-oscillatory (WENO) conservative finite-difference scheme is introduced to enhance the performance of third-order WENO methods. The scheme incorporates an interpolation method using exponential or trigonometric polynomials with an internal shape parameter to accurately approximate sharp gradients and high oscillations. A locally optimized parameter is proposed to ensure a higher order of accuracy (i.e., fourth-order) independent of critical points. Additionally, a new type of smoothness measures with exponential vanishing moments is presented, resulting in higher decay rates compared to traditional indicators. The study derives a range of parameters to guarantee the improved order of accuracy and demonstrates the shock-capturing abilities of the proposed WENO scheme through numerical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Hua Shen, Rasha Al Jahdali, Matteo Parsani
Summary: We propose a class of weighted compact central schemes for solving hyperbolic conservation laws. These schemes are able to achieve arbitrarily uniform high-order accuracy on a compact stencil and capture discontinuities using a tailor-made limiter.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
A. Mignone, L. Del Zanna
Summary: The research focuses on comparing and extending existing upwind constrained transport methods for maintaining robustness and accuracy in MHD simulations, while proposing a new flux formula applicable to the induction equation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Sebastien Boyaval, Sofiane Martel, Julien Reygner
Summary: This study focuses on the numerical approximation of the invariant measure for a viscous scalar conservation law in one-dimensional and periodic space. The equation is stochastically forced with a white-in-time but spatially correlated noise. The numerical scheme discretizes the stochastic partial differential equation (SPDE) using a finite-volume method in space and a split-step backward Euler method in time. The main result of the study is the convergence of the invariant measures of the discrete approximations to the invariant measure of the SPDE as the space and time steps approach zero, based on the second-order Wasserstein distance. The study investigates the convergence rates theoretically and numerically for the case of a globally Lipschitz continuous flux function with a small Lipschitz constant.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
S. Mousavi Yeganeh, J. Farzi
Summary: This paper utilizes MPP and PP parametrized flux limiters to achieve strict maximum principle and positivity-preserving property for solving hyperbolic conservation laws, demonstrating efficiency and effectiveness through high-order MPP RK-SV and PP RK-SV schemes. The proposed schemes maintain the maximum principle without additional time step restrictions and preserve the high-order accuracy for linear advection problems.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Remi Abgrall, Wasilij Barsukow
Summary: We propose a numerical method for conservation laws that achieves arbitrary high-order accuracy by using a continuous approximation of the solution. The method updates the point values at cell interfaces and moments of the solution inside the cell. The stability and accuracy of the resulting methods are analyzed.
APPLIED MATHEMATICS AND COMPUTATION
(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
Nina Aguillon, Frederic Lagoutiere, Nicolas Seguin
MATHEMATICS OF COMPUTATION
(2017)
Article
Mathematics
Boris Andreianov, Clement Cances, Ayman Moussa
JOURNAL OF FUNCTIONAL ANALYSIS
(2017)
Correction
Mathematics
Boris Andreianov, Marjeta Kramar Fijavz, Aljosa Peperko, Eszter Sikolya
Article
Automation & Control Systems
Boris P. Andreianov, Carlotta Donadello, Andrea Marson
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2017)
Article
Mathematics, Applied
Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano D. Rosini
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2018)
Article
Mathematics
Boris Andreianov, Matthieu Brassart
JOURNAL OF DIFFERENTIAL EQUATIONS
(2020)
Article
Mathematics, Applied
Nathael Alibaud, Boris Andreianov, Adama Ouedraogo
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2020)
Article
Mathematics, Applied
Boris Andreianov, Carlotta Doiladello, Massimiliano D. Rosini
Summary: The study investigates a macroscopic two-phase transition model for vehicular traffic flow subject to a point constraint on the density flux. A new definition of admissible solutions for the Cauchy problem is introduced, ensuring compatibility with the modeling assumption at the level of the Riemann solver.
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Boris Andreianov, Abraham Sylla
Summary: This study introduces a toy model for self-organized road traffic, taking into account the orderliness in drivers' behavior. By combining Kruzhkov and Panov methods to define the existence of acceptable solutions, a BV-stable finite volume numerical scheme is developed.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Boris Andreianov, Shyam Sundar Ghoshal, Konstantinos Koumatos
Summary: In this paper, we provide an answer to a quantitative variant of the controllability problem for the viscous Burgers equation with initial and terminal data. We investigate the influence of additional a priori bounds on the (nonlinear) operator associated with the terminal state. Our approach combines scaling and compactness arguments with observations on the non-controllability of the inviscid Burgers equation to identify wide sets of terminal states that cannot be reached from zero initial data. We prove that under certain conditions, a constant terminal state is not attainable by weak solutions of the viscous Burgers equation with a bounded amplification restriction.
JOURNAL OF EVOLUTION EQUATIONS
(2022)
Article
Mathematics
Boris Andreianov, El Houssaine Quenjel
Summary: This paper highlights the interest and limitations of the L-1-based Young measure technique for studying the convergence of numerical approximations for diffusion problems. CVFE and DDFV schemes are analyzed and convergence is proven in the case of a log-Holder continuous variable exponent. The paper also investigates the structural stability of weak solutions and describes situations where different solution notions are selected by approximation methods.
VIETNAM JOURNAL OF MATHEMATICS
(2023)
Article
Mathematics
Paola Goatin
Summary: This survey provides a comprehensive overview of macroscopic models of traffic flow, highlighting their key characteristics and potential limitations. The presentation includes visual illustrations of the models' features. Additionally, open problems and future research directions are presented to stimulate further exploration.
COMMUNICATIONS IN APPLIED AND INDUSTRIAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Boris Andreianov, Mohamed Maliki
Summary: This study explores well-posedness classes for degenerate elliptic problems in R-N, focusing on the L-infinity setting with locally uniformly continuous nonlinearities. Sufficient conditions for uniqueness and comparison properties of solutions are given in terms of the moduli of continuity of u bar arrow phi(x,u). The existence results for corresponding classes of solutions and data are deduced under additional restrictions on the dependency of phi on x.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2021)
Article
Mathematics, Applied
Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2017)
Article
Mathematics, Applied
Paola Goatin, Francesco Rossi
COMMUNICATIONS IN MATHEMATICAL SCIENCES
(2017)