Article
Computer Science, Interdisciplinary Applications
Zelalem Arega Worku, David W. Zingg
Summary: This study analyzes several types of SAT for diffusion problems discretized with diagonal-norm multidimensional SBP operators, presenting conditions for consistency, conservation, adjoint consistency, and energy stability. The theoretical results are supported by numerical experiments conducted on the two-dimensional Poisson problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Zelalem Arega Worku, David W. Zingg
Summary: In this study, we analyze the stability and functional superconvergence of discretizations of diffusion problems using narrow-stencil second-derivative generalized SBP operators coupled with SATs. The results show that linear functionals associated with the steady diffusion problem superconverge at a rate of 2p when a degree p + 1 narrow-stencil or a degree p wide-stencil generalized SBP operator is used for the spatial discretization. The conditions for stability of adjoint consistent discretizations with narrow-stencil generalized SBP operators are also presented.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Jason E. Hicken
Summary: This study shows that entropy-stable discretizations based on SBP operators and two-point entropy-conservative flux functions can accurately mimic the relationship between entropy variables and entropy adjoint. A detailed proof for first-order conservation laws semi-discretized using SBP operators with diagonal norm and boundary operators is provided, with further generalizations to second-order conservation laws, temporal discretizations, and non-diagonal SBP operators described and verified.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
David A. Craig Penner, David W. Zingg
Summary: The goal of this paper is to outline the requirements for obtaining accurate solutions and functionals from high-order tensor-product generalized summation-by-parts discretizations. Two procedures for constructing high-order grids are outlined. For the linear convection and Euler equations, several discretizations are derived and characterized. The paper demonstrates the numerical consistency of the schemes and outlines the requirements for achieving functional superconvergence.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Irving E. Reyna Nolasco, Aimad Er-Raiy, Radouan Boukharfane, Anwar A. Aldhafeeri, Lisandro Dalcin, Matteo Parsani
Summary: Guided by von Neumann and non-modal analyses, this study investigates the dispersion and diffusion properties of collocated discontinuous Galerkin methods. The analysis includes varying the spatial discretization order, the Peclet number, and the effect of the upwind term. The results show that the spatial discretization is stable for all flow regimes and independent of the solution polynomial degree. Non-modal analysis is used to compute short-term diffusion and analyze energy decay based on all eigenmodes, with results validated against turbulence simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Ge Yan, Sharanjeet Kaur, Jeffrey W. Banks, Jason E. Hicken
Summary: The paper presents the construction of entropy-stable discontinuous Galerkin difference (DGD) discretizations for hyperbolic conservation laws on unstructured grids. By utilizing existing theory for entropy-stable (diagonal-norm) summation-by-parts (SBP) discretizations, the paper demonstrates how to construct linear and nonlinear DGD discretizations by defining the SBP trial and test functions in terms of interpolated DGD degrees of freedom. The paper also shows that DGD matrix operators for the first derivative are dense-norm SBP operators, and presents numerical results to verify the entropy-stability and accuracy of the DGD discretization.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Johannes Kromer, Dieter Bothe
Summary: This paper presents a new method for efficiently and accurately computing volume fractions on unstructured polyhedral meshes. The method utilizes a principal coordinate system to approximate the phase boundary as the graph of an osculating paraboloid within each mesh cell. By applying the GAUSSIAN divergence theorem recursively, volume integrals are analytically transformed into curve integrals associated with polyhedron faces, which can be numerically approximated using standard GAUSS-LEGENDRE quadrature. This face-based formulation enables the application of the method to unstructured meshes and simplifies the numerical procedure significantly for three-dimensional applications.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Xinhui Wu, Ethan J. Kubatko, Jesse Chan
Summary: The study introduces a high-order entropy stable discontinuous Galerkin method for two dimensional shallow water equations on curved triangular meshes, maintaining a semi-discrete entropy inequality and balance for continuous bathymetry profiles. Numerical experiments confirm the high-order accuracy and theoretical properties of the scheme, comparing it to other entropy stable schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Philippe G. LeFloch, Hendrik Ranocha
Summary: This study investigates numerical methods for nonlinear hyperbolic conservation laws with non-convex flux, computing kinetic functions to characterize macro-scale dynamics. It demonstrates that entropy stability does not guarantee uniqueness of numerical solutions, and designs entropy-dissipative schemes for systems with delta shocks.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Niklas Behringer, Boris Vexler, Dmitriy Leykekhman
Summary: This article presents an optimal best-approximation-type result for fully discrete approximations of the transient Stokes problem, using the discontinuous Galerkin method for time discretization and standard finite elements for spatial discretization satisfying the discrete inf-sup condition. The analysis employs the technique of discrete maximal parabolic regularity, with results only requiring natural assumptions on the data without assuming additional smoothness of the solutions.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Engineering, Electrical & Electronic
Yuhui Wang, Langran Deng, Hanhong Liu, Zhizhang Chen, Shunchuan Yang
Summary: A provably stable SBP-SAT FDTD subgridding method with arbitrary grid ratios is proposed to accurately and flexibly solve 2-D electromagnetic problems. By carefully designing an interpolation matrix that satisfies norm-compatible conditions and the SBP property, long-term stability is guaranteed. The interpolation matrix with an arbitrary grid ratio is derived by decomposing it into two interpolation matrices and then combining them, making it flexible to model complex structures.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES
(2023)
Proceedings Paper
Computer Science, Artificial Intelligence
Dana El-Rushaidar, Raine Yeh, Xavier M. Tricoche
Summary: This paper presents a technique for accurately approximating large unstructured datasets onto modified rectilinear grids with boundary handling capabilities. This technique addresses the limitations of unstructured grids while retaining the advantages of rectilinear grids.
2022 IEEE 15TH PACIFIC VISUALIZATION SYMPOSIUM (PACIFICVIS 2022)
(2022)
Article
Mathematics
Djaber chemseddine Benchettah
Summary: This paper extends and generalizes previous results on the finite element approximation of non-coercive system of parabolic quasi-variational inequalities. It focuses on studying the management of energy production problem and proves optimal L-8 asymptotic behavior of the system with nonlinear source terms using finite element spatial approximation and the subsolutions method.
KRAGUJEVAC JOURNAL OF MATHEMATICS
(2023)
Article
Mathematics, Applied
Longfei Gao, David C. Del Rey Fernandez, Mark Carpenter, David Keyes
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Lucas Friedrich, Gero Schnuecke, Andrew R. Winters, David C. Del Rey Fernandez, Gregor J. Gassner, Mark H. Carpenter
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Computer Science, Interdisciplinary Applications
David C. Del Rey Fernandez, Jared Crean, Mark H. Carpenter, Jason E. Hicken
JOURNAL OF COMPUTATIONAL PHYSICS
(2019)
Article
Mathematics, Applied
David C. Del Rey Fernandez, Pieter D. Boom, Mark H. Carpenter, David W. Zingg
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Computer Science, Interdisciplinary Applications
Lisandro Dalcin, Diego Rojas, Stefano Zampini, David C. Del Rey Fernandez, Mark H. Carpenter, Matteo Parsani
JOURNAL OF COMPUTATIONAL PHYSICS
(2019)
Article
Computer Science, Interdisciplinary Applications
Nail K. Yamaleev, David C. Del Rey Fernandez, Jialin Lou, Mark H. Carpenter
JOURNAL OF COMPUTATIONAL PHYSICS
(2019)
Article
Computer Science, Interdisciplinary Applications
Irving Reyna Nolasco, Lisandro Dalcin, David C. Del Rey Fernandez, Stefano Zampini, Matteo Parsani
COMPUTERS & FLUIDS
(2020)
Article
Computer Science, Interdisciplinary Applications
David C. Del Rey Fernandez, Mark H. Carpenter, Lisandro Dalcin, Lucas Fredrich, Andrew R. Winters, Gregor J. Gassner, Matteo Parsani
COMPUTERS & FLUIDS
(2020)
Article
Computer Science, Interdisciplinary Applications
Diego Rojas, Radouan Boukharfane, Lisandro Dalcin, David C. Del Rey Fernandez, Hendrik Ranocha, David E. Keyes, Matteo Parsani
Summary: In computational fluid dynamics, transformational advances in individual components of future solvers are needed to meet the demand for reliable simulations in increasingly multidisciplinary fields. While hardware compatibility and efficiency are crucial, algorithmic robustness with minimal user intervention is equally important for viability.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Matteo Parsani, Radouan Boukharfane, Irving Reyna Nolasco, David C. Del Rey Fernandez, Stefano Zampini, Bilel Hadri, Lisandro Dalcin
Summary: This study presents a fully-discrete hp-adaptive entropy stable discontinuous collocated Galerkin method for the compressible Navier-Stokes equations, utilizing the SSDC framework. The method demonstrates high-order numerical performance and systematic design, showcasing its potential as a base scheme for future unstructured computational fluid dynamics tools. Results indicate efficient scaling of the parallel SSDC solver over 100,000 processes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Jesse Chan, Mario J. Bencomo, David C. Del Rey Fernandez
Summary: The study extends entropy-stable Gauss collocation schemes to non-conforming meshes, introducing a friction-based treatment of non-conforming interfaces with a face-local correction term for high-order accuracy. Numerical experiments for the compressible Euler equations confirm the stability and accuracy of this approach in two and three dimensions.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Alexander Cicchino, Siva Nadarajah, David C. Del Rey Fernandez
Summary: The flux reconstruction (FR) method is widely used to recover high-order methods, especially energy stable FR schemes, on unstructured grids. This paper presents a novel split form of the FR method that enables nonlinear stability proofs on different volume and surface cubature nodes, and demonstrates its effectiveness through numerical experiments.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Alexander Cicchino, David C. Del Rey Fernandez, Siva Nadarajah, Jesse Chan, Mark H. Carpenter
Summary: Provably stable flux reconstruction (FR) schemes for partial differential equations in curvilinear coordinates are derived. The analysis shows that the split form is essential for developing stable DG schemes and motivates the construction of metric dependent ESFR correction functions. The proposed FR schemes differ from previous schemes by incorporating the correction functions on the full split form of equations. Numerical verification demonstrates stability and optimal orders of convergence of the proposed FR schemes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Automation & Control Systems
David C. C. Del Rey Fernandez, Luis A. A. Mora, Kirsten Morris
Summary: In this letter, a first-order mixed finite element method is used to approximate a one-dimensional wave equation with a partially reflective boundary. The multiplier method is applied to prove that the approximated systems are exponentially stable, with a decay rate independent of the mesh size. Upper bounds on the exponential decay are obtained in terms of the physical parameters.
IEEE CONTROL SYSTEMS LETTERS
(2023)
Article
Mathematics, Applied
Jesse Chan, David C. Del Rey Fernandez, Mark H. Carpenter
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2019)
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)
