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
Mathematics, Applied
Biswarup Biswas, Harish Kumar, Deepak Bhoriya
Summary: This article presents entropy stable discontinuous Galerkin numerical schemes for special relativistic hydrodynamics equations with the ideal equation of state. Entropy stability is achieved by using two types of numerical fluxes and time discretization is performed using SSP Runge-Kutta methods. Several numerical test cases validate the accuracy and stability of the proposed schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
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
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
Yong Liu, Jianfang Lu, Chi-Wang Shu
Summary: In this paper, an essentially oscillation-free discontinuous Galerkin method is developed for systems of hyperbolic conservation laws. The method introduces numerical damping terms to control spurious oscillations. Both classical Runge-Kutta method and modified exponential Runge-Kutta method are used in time discretization. Extensive numerical experiments demonstrate the robustness and effectiveness of the algorithm.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Tristan Montoya, David W. Zingg
Summary: In this paper, we propose a unified framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. We use the multidimensional summation-by-parts (SBP) property to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties. Numerical experiments are conducted to validate the theoretical analysis.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Krishna Dutt, Lilia Krivodonova
Summary: The proposed method introduces a moment limiter of arbitrary high order for unstructured triangular meshes, which hierarchically limits solution coefficients to maintain stability and convergence of the numerical solution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Erica R. Johnson, James A. Rossmanith, Christine Vaughan
Summary: The HyQMOM variant of QMOM is proven to have moment-invertibility over a convex region in solution space. A high-order discontinuous Galerkin (DG) scheme is developed to solve the resulting fluid system, with novel limiters introduced to guarantee the system's hyperbolicity. The scheme is also extended to include a BGK collision operator, which is shown to be asymptotic-preserving in the high-collision limit.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Maciej Waruszewski, Jeremy E. Kozdon, Lucas C. Wilcox, Thomas H. Gibson, Francis X. Giraldo
Summary: This work examines a non-conservative balance law formulation that incorporates the rotating, compressible Euler equations for dry atmospheric flows. A semi-discretely entropy stable discontinuous Galerkin method is developed on curvilinear meshes using a generalization of flux differencing for numerical fluxes in fluctuation form. The method utilizes the skew-hybridized formulation of the element operators to ensure entropy stability, even on curvilinear meshes with under-integration. Various atmospheric flow test cases in different dimensions confirm the theoretical entropy stability results and demonstrate the high-order accuracy and robustness of the method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Bawfeh K. Kometa, Antoine Tambue, Naveed Iqbal
Summary: This article presents a new class of semi-Lagrangian methods for the numerical approximation of hyperbolic conservation laws. The methods combine the advantages of SLDG methods and RKDG methods, resulting in high-order accuracy and local conservative properties, suitable for convection-dominated convection-diffusion problems.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2022)
Article
Mathematics, Applied
S. R. Siva Prasad Kochi, M. Ramakrishna
Summary: A new CSWENO limiter is proposed for solving hyperbolic conservation laws, showing slightly better performance than the original WENO limiter for higher orders according to accuracy tests.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
Jianfang Lu, Yong Liu, Chi-Wang Shu
Summary: In this paper, a novel discontinuous Galerkin (DG) method is proposed to control spurious oscillations in solving scalar hyperbolic conservation laws, by introducing a numerical damping term. The proposed DG method retains many good properties of the classic DG method and demonstrates good performance in numerical examples, validating the theoretical results.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Computer Science, Interdisciplinary Applications
Hendrik Ranocha
Summary: Nishikawa (2007) proposed a reformulation of the classical Poisson equation as a steady state problem for a linear hyperbolic system, which provides optimal error estimates for the solution of the elliptic equation and its gradient. However, it hinders the use of well-known solvers for elliptic problems. We establish connections to a discontinuous Galerkin (DG) method studied by Cockburn, Guzman, and Wang (2009) that is generally difficult to implement. Additionally, we demonstrate the efficient implementation of this method using summation by parts (SBP) operators, particularly in the context of SBP DG methods like the DG spectral element method (DGSEM). The resulting scheme combines desirable properties from both the hyperbolic and the elliptic perspective, offering a higher order of convergence for the gradients than what is typically expected from DG methods for elliptic problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Shashank Jaiswal
Summary: Adaptivity is crucial for addressing practical challenges, especially in computational fluid dynamics workflow. The mixed non-conforming discontinuous Galerkin discretization method is introduced for the full Boltzmann equation, providing optimal convergence for non-linear kinetic systems on non-orthogonal grids. The method allows for analysis of complex problems on massively parallel scales and is applicable to a wide range of rarefied flows. The computational overhead for solving kinetic equations on non-conforming structured/unstructured domains is negligible compared to conforming domains.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
S. R. Siva Prasad Kochi, M. Ramakrishna
Summary: In this paper, the compact subcell weighted essentially non oscillatory (CSWENO) limiting strategy developed for structured meshes is generalized to unstructured triangular meshes. The performance of this strategy is demonstrated through accuracy tests and results for two-dimensional Burgers' equation and two-dimensional Euler equations.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Hardware & Architecture
Marcin Rogowski, Lisandro Dalcin, Matteo Parsani, David E. Keyes
Summary: Recently, new global and local relaxation Runge-Kutta methods have been developed to ensure certain properties of general convex functionals, which have a wide range of applications when it comes to dynamics-consistent problems. Solving scalar nonlinear algebraic equations to find the relaxation parameter can be computationally significant and technically challenging.
INTERNATIONAL JOURNAL OF HIGH PERFORMANCE COMPUTING APPLICATIONS
(2022)
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
Mathematics, Applied
R. Al Jahdali, L. Dalcin, R. Boukharfane, I. R. Nolasco, D. E. Keyes, M. Parsani
Summary: This study proposes new optimized explicit Runge-Kutta schemes for the integration of systems of ordinary differential equations arising from high-order entropy stable collocated discontinuous Galerkin methods. By optimizing the stability region of the time integration schemes, the efficiency and robustness of computational fluid dynamics simulations can be improved, leading to significant time and resource savings.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
Jan Nordstrom, Andrew R. Winters
Summary: Boundary conditions and estimates for systems of the nonlinear shallow water equations in two spatial dimensions are derived based on energy and entropy analysis. It is found that the energy method provides more detailed information and is consistent with the entropy analysis. The nonlinear energy analysis reveals the differences between linear and nonlinear analysis and shows that the results from linear analysis may not hold in the nonlinear case. The nonlinear analysis generally requires a different minimal number of boundary conditions compared to the linear analysis, and the magnitude of the flow does not influence the number of required boundary conditions.
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
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
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
Rasha Al Jahdali, Lisandro Dalcin, Matteo Parsani
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Theory & Methods
R. Al Jahdali, S. Kortas, M. Shaikh, L. Dalcin, M. Parsani
Summary: Industrially relevant computational fluid dynamics simulations often require vast computational resources. This study compares the performance and cost of simulating compressible flows using an entropy stable collocated discontinuous Galerkin framework on an on-premises cluster and the Amazon Web Services Elastic Compute Cloud. The results show that the cloud provides better performance, while the on-premises cluster offers lower costs.
Article
Mathematics, Applied
Hendrik Ranocha, Andrew R. Winters, Hugo Guillermo Castro, Lisandro Dalcin, Michael Schlottke-Lakemper, Gregor J. Gassner, Matteo Parsani
Summary: We study the temporal step size control of explicit Runge-Kutta methods for compressible computational fluid dynamics. We compare error-based approaches to classical step size control based on a CFL number and demonstrate that error-based methods are more convenient and efficient in various applications. Our numerical examples show the effectiveness of error-based step size control in different scenarios.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Computer Science, Interdisciplinary Applications
Jan Michael Breuer, Samuel Leweke, Johannes Schmoelder, Gregor Gassner, Eric von Lieres
Summary: We have developed spatial arbitrary order DG methods for three commonly used liquid chromatography models, and implemented them in the open source software CADET, providing efficient implementations publicly for the first time. Validation and benchmarking against the original finite volume CADET code show great performance advantages for DG, depending on the problem size. For a four-component steric mass action GRM model, we achieve a speed-up of an order of magnitude within the typical range of engineering applications. We have also explored the performance of a collocation Legendre-Gauss-Lobatto quadrature DG method in comparison to an exact integration DG method, finding a slight advantage for the collocation DG method in performance benchmarks.
COMPUTERS & CHEMICAL ENGINEERING
(2023)
Article
Biochemical Research Methods
Nadezhda Briantceva, Lokendra Chouhan, Matteo Parsani, Mohamed-Slim Alouini
Summary: This study focuses on the sub-diffusion motion and evanescence process in biological cells, and presents a 3D molecular communication system. The system is simulated and its performance is investigated.
IEEE TRANSACTIONS ON NANOBIOSCIENCE
(2023)
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
Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David Ketcheson
Summary: The study developed error-control based time integration algorithms for compressible fluid dynamics (CFD) applications, demonstrating their efficiency and robustness. Focusing on discontinuous spectral element semidiscretizations, new controllers were designed for existing methods and some new embedded Runge-Kutta pairs. By comparing error-control based methods with the common CFL number approach, optimized methods showed improved performance and adaptability.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Hendrik Ranocha, Gregor J. Gassner
Summary: The study investigates the local linear stability issues of entropy-conserving/dissipative high-order split-form discontinuous Galerkin methods for the compressible Euler equations and examines the impact of pressure equilibrium preservation on these issues.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
