Article
Mathematics, Applied
David A. Kopriva, Gregor J. Gassner, Jan Nordstrom
Summary: The paper discusses using the behavior of the L-2 norm to infer stability of discontinuous Galerkin spectral element methods for linear hyperbolic equations. By utilizing an upwind numerical flux that satisfies the Rankine-Hugoniot condition, the DGSEM is shown to have the same energy bound in the L-2 norm as the partial differential equation, with added dissipation depending on the approximation error.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Remi Abgrall, Philipp Oeffner, Hendrik Ranocha
Summary: This paper proposes an approach to construct entropy conservative/dissipative semidiscretizations in the general class of residual distribution (RD) schemes. The approach involves adding suitable correction terms characterized as solutions of certain optimization problems. The method is applied to the SBP- SAT framework and novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed. Explicit solutions are provided for all optimization problems, and a fully discrete entropy conservative/dissipative RD scheme is obtained using the deferred correction method for time integration.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Sergio Rojas, David Pardo, Pouria Behnoudfar, Victor M. Calo
Summary: This study presents a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization, which provides stable solutions on each mesh instance and minimizes errors by automatic mesh refinement.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Engineering, Multidisciplinary
Kerstin Weinberg, Christian Wieners
Summary: We propose a new numerical approach for wave induced dynamic fracture. The method combines a discontinuous Galerkin approximation of elastic waves and a phase-field approximation of brittle fracture. The algorithm is staggered in time and uses an implicit midpoint rule for wave propagation and an implicit Euler step for phase-field evolution. Examples in two and three dimensions demonstrate the advantages of this approach in computing crack growth and spalling initiated by reflected and superposed waves.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Environmental Sciences
Patricia Grassi, Matias Schadeck Netto, Sergio Luiz Jahn, Jordana Georgin, Dison S. P. Franco, Mika Sillanpaa, Lucas Meili, Luis F. O. Silva
Summary: The study investigated the potential use of leaf powder from Sansevieria trifasciato as an adsorbent for methylene blue removal. The leaf powder showed efficient adsorption capacity for different concentrations of methylene blue. The results of fixed bed experiments demonstrated that the leaf powder had good dye removal performance.
ENVIRONMENTAL SCIENCE AND POLLUTION RESEARCH
(2023)
Article
Nuclear Science & Technology
Rodolfo M. Ferrer
Summary: A stability analysis was conducted on CMFD, pCMFD, and lpCMFD acceleration schemes, showing similar stability behavior in LD and LC spatial discretization schemes.
ANNALS OF NUCLEAR ENERGY
(2021)
Article
Mathematics, Applied
Erik Burman
Summary: This article discusses the stability and accuracy of stabilized finite element methods for hyperbolic problems, focusing on the interaction between linear and nonlinear stabilizations. It is shown that a combination of linear and nonlinear stabilization can be designed to be invariant preserving. The proposed method allows for classical error estimates for smooth solutions and is demonstrated to accurately predict shock structures without polluting the smooth parts with high frequency content.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Bernard Kapidani, Lorenzo Codecasa, Joachim Schoeberl
Summary: The paper presents a new numerical method for solving the time-dependent Maxwell equations on unstructured meshes in two dimensions. The method incorporates arbitrary polynomial degrees and avoids the need for penalty parameters or numerical dissipation to achieve stability. It is shown to provide spurious-free solutions with expected convergence rates through numerical tests.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Huoyuan Duan, Junhua Ma, J. U. N. Zou
Summary: This paper presents a mixed finite element method for the Maxwell eigenproblem with modifications on the Kikuchi mixed formulation, resulting in efficient numerical solutions through discretization of the electric field and multiplier. In practical applications, the method shows superior performance on grids, with higher accuracy and reliability.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Francisco Fuica, Felipe Lepe, Enrique Otarola, Daniel Quero
Summary: In this study, error estimators for the Stokes system with singular sources in suitable function spaces were designed and analyzed. The error estimators were proven to be reliable and locally efficient in Lipschitz polytopal domains. Based on these estimators, a simple adaptive strategy was developed that achieved optimal rates of convergence in numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Ankit Chakraborty, Georg May
Summary: We propose an anisotropic hp-mesh adaptation strategy for discontinuous Petrov-Galerkin (DPG) finite element schemes using a continuous mesh model with optimal test functions, extending previous work on h-adaptation. The strategy utilizes the residual-based error estimator of the DPG discretization to compute the polynomial distribution and anisotropy of the mesh elements. Local problems on element patches are solved to predict the optimal order of approximation, making these computations highly parallelizable. The performance of the strategy is demonstrated through numerical examples on triangular grids.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
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
Computer Science, Interdisciplinary Applications
Thomas Wiltshire, Robert E. Bird, William M. Coombs, Stefano Giani
Summary: Open source codes are crucial for enhancing research integrity and accountability in computational science and engineering. However, many existing open source codes lack consideration for the ease of modifying the base code. This paper presents an open source finite element code written in MATLAB, which is designed to facilitate user understanding and implementation of new ideas within the core code.
ADVANCES IN ENGINEERING SOFTWARE
(2022)
Article
Mathematics, Applied
Jichun Li, Chi-Wang Shu, Wei Yang
Summary: This paper focuses on the time-domain carpet cloak model and proposes two new finite element schemes to address the numerical stability issue of previous schemes. The unconditional stability of the Crank-Nicolson scheme and the conditional stability of the leap-frog scheme are proved, both inheriting the exact form of the continuous stability.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Ingeborg G. Gjerde, Kundan Kumar, Jan M. Nordbotten
Summary: In this work, we analyze the Poisson equation with a line source using a dual-mixed variational formulation. By making assumptions on the problem parameters, we split the solution into higher- and lower-regularity terms, and propose a singularity removal-based mixed finite element method to approximate the higher-regularity terms, which significantly improves the convergence rate compared to approximating the full solution.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Remi Abgrall, Elise Le Meledo, Philipp Offner
Summary: The research introduces a class of discretization spaces and H(div)-conformal elements that can be applied to any polytope, combining flexibility of Virtual Element spaces with divergence properties of Raviart-Thomas elements. This design allows for a wide range of H(div)-conformal discretizations, easily adaptable to desired properties of approximated quantities. Additionally, a specific restriction of this general setting shows properties similar to classical Raviart-Thomas elements at each interface, for any order and polytopal shape.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Computer Science, Interdisciplinary Applications
Ronit Kumar, Lidong Cheng, Yunong Xiong, Bin Xie, Remi Abgrall, Feng Xiao
Summary: The THINC-scaling scheme unifies the VOF and level set methods by maintaining a high-quality reconstruction function, preserving the advantages of both methods, and allowing representation of interfaces with high-order polynomials. The scheme provides high-fidelity solutions comparable to other advanced methods and can resolve sub-grid filament structures if the interface is represented by a polynomial higher than second order.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Sixtine Michel, Davide Torlo, Mario Ricchiuto, Remi Abgrall
Summary: The paper studies continuous finite element dicretizations for one dimensional hyperbolic partial differential equations, providing a fully discrete spectral analysis and suggesting optimal values of the CFL number and stabilization parameters. Different choices for finite element space and time stepping strategies are compared to determine the most promising combinations for accuracy and stability, with suggestions for optimal discretization parameters.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Jianfang Lin, Yupeng Ren, Remi Abgrall, Jianxian Qiu
Summary: In this paper, a high order residual distribution (RD) method is developed for solving steady state conservation laws using a novel Hermite weighted essentially non-oscillatory (HWENO) framework. The proposed method has advantages in terms of computational efficiency and accuracy compared to traditional methods. Extensive numerical experiments confirm the high order accuracy and good quality of the scheme.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Paola Bacigaluppi, Julien Carlier, Marica Pelanti, Pietro Marco Congedo, Remi Abgrall
Summary: This work presents the formulation of a four-equation model for simulating unsteady two-phase mixtures with phase transition and strong discontinuities. The proposed method uses a non-conservative formulation to avoid oscillations obtained by many approaches and relies on a finite element based residual distribution scheme.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Barbara Re, Remi Abgrall
Summary: Within the framework of diffuse interface methods, a pressure-based Baer-Nunziato type model is derived for weakly compressible multiphase flows. The model can handle different equations of state and includes relaxation terms characterized by user-defined finite parameters. The solution strategy involves a semi-implicit finite-volume solver for the hyperbolic part and an ODE integrator for the relaxation processes. The developed simulation tool is validated through various tests, showing good agreement with analytical and reference results.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2022)
Article
Computer Science, Interdisciplinary Applications
Remi Abgrall, Philipp Oeffner, Hendrik Ranocha
Summary: This paper proposes an approach to construct entropy conservative/dissipative semidiscretizations in the general class of residual distribution (RD) schemes. The approach involves adding suitable correction terms characterized as solutions of certain optimization problems. The method is applied to the SBP- SAT framework and novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed. Explicit solutions are provided for all optimization problems, and a fully discrete entropy conservative/dissipative RD scheme is obtained using the deferred correction method for time integration.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Remi Abgrall, Fatemeh Nassajian Mojarrad
Summary: This paper presents schemes for the compressible Euler equations that focus on conserving angular momentum locally. A general framework is proposed, examples of schemes are described, and results are shown. These schemes can be of arbitrary order.
COMPUTERS & FLUIDS
(2022)
Article
Computer Science, Interdisciplinary Applications
Remi Abgrall, Pratik Rai, Florent Renac
Summary: In this work, a discretization method for simulating compressible multicomponent flows with shocks and material interfaces is proposed. The method is accurate, robust, and stable. By modifying the integrals over discretization elements, a scheme with a HLLC solver is designed to preserve material interfaces and satisfy minimum and maximum principles of entropy. Numerical experiments validate the stability, robustness, and accuracy of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Sixtine Michel, Davide Torlo, Mario Ricchiuto, Remi Abgrall
Summary: This study investigates various continuous finite element discretization methods for two-dimensional hyperbolic partial differential equations. The schemes are ranked based on efficiency, stability, and dispersion error, and the best CFL and stabilization coefficients are provided. Challenges in two dimensions include Fourier analysis and the introduction of high-order viscosity. The results suggest that combining Cubature elements with SSPRK and OSS stabilization yields the most promising combination.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Remi Abgrall, Saray Busto, Michael Dumbser
Summary: We present a simple and general framework for constructing thermodynamically compatible schemes for overdetermined hyperbolic PDE systems. The proposed algorithms solve the entropy inequality as a primary evolution equation, leading to total energy conservation as a consequence of the compatible discretization. We apply the framework to the construction of three different numerical methods and demonstrate their stability and accuracy through numerical experiments.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Remi Abgrall, Fatemeh Nassajian Mojarrad
Summary: We propose a class of fully explicit kinetic numerical methods in compressible fluid dynamics, which can achieve arbitrarily high order in both time and space. These methods, including the relaxation schemes by Jin and Xin, allow for the use of CFL number larger or equal to unity on regular Cartesian meshes for multi-dimensional problems. The methods depend on a small parameter that represents a Knudsen number and are asymptotic preserving in this parameter. The computational costs of the methods are comparable to fully explicit schemes. The extension of these methods to multi-dimensional systems has been assessed and proven to be robust and achieve the theoretically predicted high order of accuracy on smooth solutions.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
R. Abgrall
Summary: This article demonstrates a way to combine the conservative and non-conservative formulations of a hyperbolic system that has a conservative form. The solution is described using a combination of point values and average values, with different meanings for point-wise and cell average degrees of freedom. The article also presents a new method for nonlinear stability and provides results from various benchmark tests.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Remi Abgrall, Davide Torlo
Summary: This paper describes a method for constructing arbitrarily high order kinetic schemes on regular meshes and introduces a nonlinear stability method for simulating problems with discontinuities without sacrificing accuracy for smooth regular solutions.
COMMUNICATIONS IN MATHEMATICAL SCIENCES
(2022)
Article
Mathematics, Applied
R. Abgrall, J. Nordstrom, P. Oeffner, S. Tokareva
JOURNAL OF SCIENTIFIC COMPUTING
(2020)