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
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
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
Mathematics, Applied
Kailiang Wu, Haili Jiang, Chi-Wang Shu
Summary: This paper focuses on the importance of ensuring positive pressure and density in the numerical simulation of ideal magnetohydrodynamics. The study presents high-order positive preserving Galerkin schemes based on the central discontinuous Galerkin (CDG) methods. In one dimension, the standard CDG method satisfies the positive preserving property, but in multidimensional cases, it does not fulfill the property due to the discrete divergence-free condition. To address this issue, new locally divergence-free CDG schemes are proposed.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Leonardo Colombo, Manuela Gamonal Fernandez, David Martin de Diego
Summary: Numerical methods that preserve geometric invariants are called geometric integrators, and variational integrators are an important class of such methods. This paper introduces variational integrators with fixed time step for time-dependent Lagrangian systems modeling an important class of autonomous dissipative systems. These integrators are derived using a series of discrete Lagrangian functions, allowing for the preservation properties of variational integrators for autonomous Lagrangian systems. The paper also presents a discrete Noether theorem for this class of systems.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
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
Guanyu Zhou
Summary: In this study, we analyze two finite volume schemes for the chemotaxis system in a two-dimensional domain, demonstrating mass conservation, positivity, and well-posedness without the need for the CFL condition. We investigate the stability of equilibrium and local stability, and apply discrete semigroup theory for error analysis, achieving a convergence rate O(tau+h) in Lp norm. The theoretical results are validated through numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(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
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
Elena Gaburro, Philipp oeffner, Mario Ricchiuto, Davide Torlo
Summary: In this paper, a fully discrete entropy preserving ADER-DG method is developed by introducing entropy correction terms and applying the relaxation approach to maintain entropy precision. The theoretical results are verified through numerical simulations.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Computer Science, Interdisciplinary Applications
Andrew J. Christlieb, William A. Sands, Hyoseon Yang
Summary: In this paper, an approximation method utilizing radial basis functions (RBFs) is introduced for enhancing the order of accuracy in the numerical solution of conservation laws. The development of high order finite volume (FV) weighted essentially non-oscillatory (WENO) methods, using RBF approximations to obtain required data at cell interfaces, is of particular interest. The paper addresses the practical elements of the approach, including evaluations of shape parameters and a hybrid implementation, and demonstrates notable improvements in accuracy compared to existing WENO schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Dorian Hillebrand, Simon-Christian Klein, Philipp Oeffner
Summary: The construction of high-order structure-preserving numerical schemes for hyperbolic conservation laws has been extensively studied. This paper compares different approaches, including deep neural networks, limiters, and polynomial annihilation, for constructing high-order accurate shock capturing FD/FV schemes. The analytical and numerical properties of these schemes are further analyzed. The investigation of these strategies aims to enhance the understanding of these techniques and can be applied to other numerical methods with similar ideas.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Igor Leite Freire, Rodrigo da Silva Tito
Summary: This study demonstrates that an equation discovered by V. Novikov describes pseudo-spherical surfaces and is geometrically integrable, resulting in an infinite hierarchy of conservation laws. The problem of local isometric immersions is also examined in the context of this equation.
STUDIES IN APPLIED MATHEMATICS
(2022)
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, Software Engineering
Jan Giesselmann, Tristan Pryer
BIT NUMERICAL MATHEMATICS
(2016)
Article
Mathematics, Applied
Jan Giesselmann, Corrado Lattanzio, Athanasios E. Tzavaras
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2017)
Article
Mathematics, Applied
Jan Giesselmann, Tristan Pryer
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2017)
Article
Mathematics, Applied
Fabian Meyer, Christian Rohde, Jan Giesselmann
IMA JOURNAL OF NUMERICAL ANALYSIS
(2020)
Article
Mathematics, Applied
Jan Giesselmann, Philippe G. LeFloch
NUMERISCHE MATHEMATIK
(2020)
Article
Computer Science, Software Engineering
Jan Giesselmann, Fabian Meyer, Christian Rohde
BIT NUMERICAL MATHEMATICS
(2020)
Article
Mathematics, Applied
Jan Giesselmann, Fabian Meyer, Christian Rohde
Summary: The paper introduces a new error estimation method in the Wasserstein distance between dissipative statistical solutions and numerical approximations, and splits the error estimator into deterministic and stochastic parts. Numerical experiments conducted verify the scaling properties of the residuals and the accuracy of the splitting.
Article
Automation & Control Systems
Martin Gugat, Jan Giesselmann
Summary: This paper analyzes the behavior of a semilinear gas flow model on a star-shaped network and presents boundary feedback laws that stabilize the system state exponentially fast under suitable coupling conditions and sufficiently small initial data. Numerical examples demonstrate a comparison between the semilinear model and the quasilinear system.
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
(2021)
Article
Automation & Control Systems
Martin Gugat, Jan Giesselmann, Teresa Kunkel
Summary: Gas flow through network pipes can be modeled using a combination of hyperbolic systems of partial differential equations and algebraic conditions. An observer system can be used to accurately approximate the complete system state based on nodal observations, with the state converging exponentially fast to the original state, as confirmed by numerical experiments.
IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION
(2021)
Article
Mathematics, Applied
Andreas Dedner, Jan Giesselmann, Tristan Pryer, Jennifer K. Ryan
Summary: This work examines a posteriori error control for post-processed approximations to elliptic boundary value problems, introducing a new class of post-processing operator to optimize various reconstruction operators. Extensive numerical tests are conducted to validate the analytical findings of the study.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
H. Egger, J. Giesselmann
Summary: This article investigates the transportation of gas in long pipes and pipeline networks, where the dynamics are primarily influenced by friction at the pipe walls. By employing nonlinear analysis, the governing equations are formulated as an abstract dissipative Hamiltonian system, enabling us to derive perturbation bounds through relative energy estimates. Consequently, stability estimates with respect to initial conditions and model parameters are proven, and a quantitative asymptotic analysis is conducted in the high friction limit. Initially, the results are established for flow in a single pipe, and then the analysis is extended to pipe networks in the energy-based port-Hamiltonian modeling framework.
NUMERISCHE MATHEMATIK
(2023)
Article
Mathematics, Applied
Neeraj Sarna, Jan Giesselmann, Manuel Torrilhon
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2020)
Article
Mathematics, Applied
Jan Giesselmann, Niklas Kolbe, Maria Lukacova-Medvidova, Nikolaos Sfakianakis
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2018)
Proceedings Paper
Mathematics, Applied
Jan Giesselmann, Tristan Pryer
FINITE VOLUMES FOR COMPLEX APPLICATIONS VIII-METHODS AND THEORETICAL ASPECTS, FVCA 8
(2017)
Article
Mathematics, Applied
Jan Giesselmann, Athanasios E. Tzavaras
APPLICABLE ANALYSIS
(2017)