| 1. |
- Bonito, Andrea, et al.
(författare)
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Numerical Simulations of Surface Quasi-Geostrophic Flows on Periodic Domains
- 2021
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 43:2, s. B405-B430
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Tidskriftsartikel (refereegranskat)abstract
- We propose a novel algorithm for the approximation of surface quasi-geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an explicit strong-stability-preserving three-stage Runge-Kutta method while a flux-corrected-transport (FCT) method coupled with Dunford-Taylor representations of fractional operators is advocated for the space discretization. Standard continuous piecewise linear finite elements are employed, and the algorithm does not have restrictions on the mesh structure or on the computational domain. In the inviscid case, we show that the resulting scheme satisfies a discrete maximum principle property under a standard CFL condition and observe, in practice, its second order accuracy in space. The algorithm successfully approximates several benchmarks with sharp transitions and fine structures typical of SQG flows. In addition, theoretical Kolmogorov energy decay rates are observed on a freely decaying atmospheric turbulence simulation.
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| 2. |
- Bouderlique, Thibault, et al.
(författare)
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Surface flow for colonial integration in reef-building corals
- 2022
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Ingår i: Current Biology. - : Elsevier. - 0960-9822 .- 1879-0445. ; 32:12, s. 2596-2609
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Tidskriftsartikel (refereegranskat)abstract
- Reef-building corals are endangered animals with a complex colonial organization. Physiological mechanisms connecting multiple polyps and integrating them into a coral colony are still enigmatic. Using live imaging, particle tracking, and mathematical modeling, we reveal how corals connect individual polyps and form integrated polyp groups via species-specific, complex, and stable networks of currents at their surface. These currents involve surface mucus of different concentrations, which regulate joint feeding of the colony. Inside the coral, within the gastrovascular system, we expose the complexity of bidirectional branching streams that connect individual polyps. This system of canals extends the surface area by 4-fold and might improve communication, nutrient supply, and symbiont transfer. Thus, individual polyps integrate via complex liquid dynamics on the surface and inside the colony.
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| 3. |
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| 4. |
- Dao, Tuan Anh, et al.
(författare)
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A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations
- 2022
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Ingår i: Journal of Scientific Computing. - : Springer Nature. - 0885-7474 .- 1573-7691. ; 92:3
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Tidskriftsartikel (refereegranskat)abstract
- We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spacial dimensions. Sharp shocks and discontinuity resolutions are obtained.
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| 5. |
- Dao, Tuan Anh, et al.
(författare)
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A nodal based high order nonlinear stabilization for finite element approximation of Magnetohydrodynamics
- 2024
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Ingår i: Journal of Computational Physics. - 0021-9991 .- 1090-2716. ; 512, s. 113146-113146
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Tidskriftsartikel (refereegranskat)abstract
- We present a novel high-order nodal artificial viscosity approach designed for solving Magnetohydrodynamics (MHD) equations. Unlike conventional methods, our approach eliminates the need for ad hoc parameters. The viscosity is mesh-dependent, yet explicit definition of the mesh size is unnecessary. Our method employs a multimesh strategy: the viscosity coefficient is constructed from a linear polynomial space constructed on the fine mesh, corresponding to the nodal values of the finite element approximation space. The residual of MHD is utilized to introduce high-order viscosity in a localized fashion near shocks and discontinuities. This approach is designed to precisely capture and resolve shocks. Then, high-order Runge-Kutta methods are employed to discretize the temporal domain. Through a comprehensive set of challenging test problems, we validate the robustness and high-order accuracy of our proposed approach for solving MHD equations.
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| 6. |
- Dao, Tuan Anh, et al.
(författare)
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A structure preserving numerical method for the ideal compressible MHD system
- 2024
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Ingår i: Journal of Computational Physics. - 0021-9991 .- 1090-2716. ; 508, s. 113009-113009
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.
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| 7. |
- Dao, Tuan Anh, et al.
(författare)
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Energy stable and accurate coupling of finite element methods and finite difference methods
- 2022
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 449
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a hybrid method to couple continuous Galerkin finite element methods and high-order finite difference methods in a nonconforming multiblock fashion. The aim is to optimize computational efficiency when complex geometries are present. The proposed coupling technique requires minimal changes in the existing schemes while maintaining strict stability, accuracy, and energy conservation. Results are demonstrated on linear and nonlinear scalar conservation laws in two spatial dimensions.
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| 8. |
- Dao, Tuan Anh, 1994-
(författare)
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Invariant domain preserving schemes for magnetohydrodynamics
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Magnetohydrodynamics (MHD) studies the behaviors of ionized gases, such as plasmas, in the presence of a magnetic field. MHD is used in many applications, such as geophysics, space physics, and nuclear fusion.Despite intensive research in recent decades, many physical and numerical aspects of MHD are not well understood. The challenges inherent in solving MHD stem from the obstacles encountered in ordinary hydrodynamics, such as those described by the compressible Euler/Navier-Stokes equations, along with the intricacies arising from electromagnetism. A characteristic of compressible flows is their tendency to develop shocks/discontinuities over time. This often leads to unphysical traits in numerical approximations if the capturing scheme is not constructed properly. By physical laws, the magnetic field is solenoidal. However, in practice, numerical schemes seldom ensure this property precisely, which may lead to instability and convergence to wrong solutions. In numerical simulation of many applications, positive physical quantities such as density and pressure can easily become negative. On the whole, preserving the physical relevance of the numerical solutions poses a significant challenge in MHD.This thesis presents several numerical schemes based on Galerkin approximations to solve MHD. The schemes rely on viscous regularization, a technique to remove mathematical singularities by adding a vanishing viscosity term to the MHD equations. At the continuous level, we propose several choices of viscous regularization and rigorously show that they are consistent with thermodynamics. Based on these choices, we construct numerical schemes of which robustness is confirmed through many challenging benchmarks. Finally, we propose a nonconventional algorithm that simultaneously preserves many desirable physical properties, including positivity of density and internal energy, conservation of total energy, minimum entropy principle, and zero magnetic divergence.
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| 9. |
- Dao, Tuan Anh, et al.
(författare)
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Monolithic parabolic regularization of the MHD equations and entropy principles
- 2022
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 398
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Tidskriftsartikel (refereegranskat)abstract
- We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge–Kutta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.
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| 10. |
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