Article
Computer Science, Interdisciplinary Applications
Tak Shing Au Yeung, Ka Chun Cheung, Eric T. Chung, Shubin Fu, Jianliang Qian
Summary: We propose a deep learning approach to extract ray directions at discrete locations by analyzing wave fields. A deep neural network is trained to predict ray directions based on local plane-wave fields. The resulting network is then applied to solve the Helmholtz equations at higher frequencies. The numerical results demonstrate the efficiency and accuracy of the proposed scheme.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
David A. Kopriva, Gregor J. Gassner
Summary: A hybrid continuous and discontinuous Galerkin spectral element approximation is proposed, utilizing continuous Galerkin on interior element faces and discontinuous Galerkin at physical boundaries or faces with property jumps. The method employs split form equations and two-point fluxes for stability on unstructured curved element meshes, while maintaining conservation and constant state preservation. Spectral accuracy is achieved for examples involving wave scattering at discontinuous medium boundaries.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Ivy Weber, Gunilla Kreiss, Murtazo Nazarov
Summary: This paper investigates the stability of a numerical method for solving the wave equation using matrix eigenvalue analysis to calculate time-step restrictions. It is found that the time-step restriction for continuous Lagrange elements is independent of the nodal distribution, while the restriction for symmetric interior penalty DG schemes is tighter. The best time-step restriction is obtained for continuous Hermite finite elements.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Jan Nordstrom, Andrew R. Winters
Summary: This study proves the stability of the most common filtering procedure for nodal discontinuous Galerkin methods, utilizing polynomial basis functions and accurate quadrature methods. Theoretical discussions recontextualize stable filtering results from finite difference methods to the DG framework, with numerical tests verifying the theoretical findings.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Shinhoo Kang, Emil M. Constantinescu
Summary: In this paper, we propose entropy-preserving and entropy-stable partitioned Runge-Kutta methods that mitigate system stiffness and fully support entropy-preserving and entropy-stability properties at a discrete level. By adjusting the step completion and introducing a relaxation parameter, the proposed methods satisfy the discrete entropy condition. Numerical results demonstrate the effectiveness of these methods.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Engineering, Marine
Wenbin Wu, Yun-Long Liu, A-Man Zhang, Moubin Liu
Summary: This article presents a method for simulating the hydrodynamics of underwater explosion shock using the local discontinuous Galerkin method. By combining the mesh adaption technique with the LDG method, the shock wavefront can be accurately captured and computational efficiency can be improved.
Article
Engineering, Multidisciplinary
A. Bendali
Summary: This paper provides an analytical justification for the determination of an important parameter in the solution of the Helmholtz equation using the Interior Penalty Discontinuous Galerkin (IPDG) method proposed by Feng and Wu (2009). The parameter was determined through numerical tests conducted on a mesh consisting of equal equilateral triangles at a fixed frequency. It was found to play a crucial role in reducing the dispersion error for a first-degree polynomial approximation. The analytical results were further validated through numerical experiments, which demonstrated that the parameter determined for equal equilateral triangles also effectively eliminates dispersion for unstructured and well-smoothed meshes.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Christoph Erath, Lorenzo Mascotto, Jens M. Melenk, Ilaria Perugia, Alexander Rieder
Summary: In this paper, we present a coupling method that combines the discontinuous Galerkin finite element method with the boundary element method to solve the three-dimensional Helmholtz equation with variable coefficients. The coupling is achieved through a mortar variable related to an impedance trace on a smooth interface. The method has a block structure with nonsingular subblocks, and we prove the quasi-optimality of both the h and p versions of the scheme under certain conditions.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Dan Ling, Chi-Wang Shu, Wenjing Yan
Summary: The paper focuses on the design of numerical methods for the diffusive-viscous wave equations with variable coefficients and develops a local discontinuous Galerkin (LDG) method. Numerical experiments are provided to demonstrate the optimal convergence rate and effectiveness of the proposed LDG method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
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
Computer Science, Interdisciplinary Applications
Shashank Jaiswal
Summary: The kinetic equations describe the behavior of physical processes, but their computational complexity poses a challenge. This work presents an efficient scheme for studying these systems, including the construction of an entropy stable flux and coupling with a high-order discontinuous Galerkin discretization. Verification tests demonstrate the stability and accuracy of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Florian Kummer, Jens Weber, Martin Smuda
Summary: The software package BoSSS discretizes partial differential equations with discontinuous coefficients and/or time-dependent domains using an eXtended Discontinuous Galerkin (XDG) method. This work introduces the XDG method, develops a formal notation capturing important numerical details, and presents iterative solvers for extended DG systems.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Biswarup Biswas, Harish Kumar, Anshu Yadav
Summary: In this article, high order discontinuous Galerkin entropy stable schemes are proposed for ten-moment Gaussian closure equations, utilizing entropy conservative numerical flux and appropriate entropy stable numerical flux for stability. These schemes are extended to model plasma laser interaction source term and tested for stability, accuracy and robustness on several test cases using strong stability preserving methods.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Xinhui Wu, Jesse Chan
Summary: A high-order entropy stable discontinuous Galerkin method has been proposed for addressing nonlinear conservation laws on multi-dimensional domains and networks, using treatments of multi-dimensional interfaces and network junctions to maintain entropy stability when coupling entropy stable discretizations. Numerical experiments confirm the stability of the schemes and show the accuracy of junction treatments in comparisons with fully 2D implementations.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jinyang Lu, Yan Xu, Chao Zhang
Summary: This paper presents a uniform framework of local discontinuous Galerkin (LDG) methods for two-dimensional Camassa-Holm equations and two-dimensional mu-Camassa-Holm equations. The energy stability and semi-discrete error estimates are derived based on this framework. Numerical experiments demonstrate the accuracy and stability of these schemes.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
