An official website of the United States government
In this paper, we propose and study first- and second-order (in time) stabilized linear finite element schemes for the incompressible Navier-Stokes (NS) equations. The energy, momentum, and angular momentum conserving (EMAC) formulation has emerged as a promising approach for conserving energy, momentum, and angular momentum of the NS equations, while the exponential scalar auxiliary variable (ESAV) has become a popular technique for designing linear energy-stable numerical schemes. Our method leverages the EMAC formulation and the Taylor-Hood element with grad-div stabilization for spatial discretization. We adopt the implicit-explicit backward differential formulas (BDFs) coupled with a novel stabilized ESAV approach for time stepping. For the solution process, we develop an efficient decoupling technique for the resulting fully-discrete systems so that only one linear Stokes solve is needed at each time step, which is similar to the cost of classic implicit-explicit BDF schemes for the NS equations. Robust optimal error estimates are successfully derived for both velocity and pressure for the two proposed schemes, with Gronwall constants that are particularly independent of the viscosity. Furthermore, it is rigorously shown that the grad-div stabilization term can greatly alleviate the viscosity-dependence of the mesh size constraint, which is required for error estimation when such a term is not present in the schemes. Various numerical experiments are conducted to verify the theoretical results and demonstrate the effectiveness and efficiency of the grad-div and ESAV stabilization strategies and their combination in the proposed numerical schemes, especially for problems with high Reynolds numbers.
more »
« less
An energy, momentum, and angular momentum conserving scheme for a regularization model of incompressible flow
Abstract We introduce a new regularization model for incompressible fluid flow, which is a regularization of the EMAC (energy, momentum, and angular momentum conserving) formulation of the Navier–Stokes equations (NSE) that we call EMAC-Reg. The EMAC formulation has proved to be a useful formulation because it conserves energy, momentum, and angular momentum even when the divergence constraint is only weakly enforced. However, it is still a NSE formulation and so cannot resolve higher Reynolds number flows without very fine meshes. By carefully introducing regularization into the EMAC formulation, we create a model more suitable for coarser mesh computations but that still conserves the same quantities as EMAC, i.e., energy, momentum, and angular momentum. We show that EMAC-Reg, when semi-discretized with a finite element spatial discretization is well-posed and optimally accurate. Numerical results are provided that show EMAC-Reg is a robust coarse mesh model.
more »
« less
- Award ID(s):
- 2011490
- PAR ID:
- 10327787
- Date Published:
- Journal Name:
- Journal of Numerical Mathematics
- Volume:
- 30
- Issue:
- 1
- ISSN:
- 1570-2820
- Page Range / eLocation ID:
- 1 to 22
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Total angular momentum conservation in Ehrenfest dynamics with a truncated basis of adiabatic statesWe show that standard Ehrenfest dynamics does not conserve linear and angular momentum when using a basis of truncated adiabatic states. However, we also show that previously proposed effective Ehrenfest equations of motion [M. Amano and K. Takatsuka, “Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions,” J. Chem. Phys. 122, 084113 (2005) and V. Krishna, “Path integral formulation for quantum nonadiabatic dynamics and the mixed quantum classical limit,” J. Chem. Phys. 126, 134107 (2007)] involving the non-Abelian Berry force do maintain momentum conservation. As a numerical example, we investigate the Kramers doublet of the methoxy radical using generalized Hartree–Fock with spin–orbit coupling and confirm that angular momentum is conserved with the proper equations of motion. Our work makes clear some of the limitations of the Born–Oppenheimer approximation when using ab initio electronic structure theory to treat systems with unpaired electronic spin degrees of freedom, and we demonstrate that Ehrenfest dynamics can offer much improved, qualitatively correct results.more » « less
-
Abstract In this letter, we investigate Bondi-like accretion flows with zero or low specific angular momentum by performing 3D general relativistic magnetohydrodynamic simulations. In order to check if relativistic jets can be launched magnetically from such flows, we insert a large-scale poloidal magnetic field into the accretion flow and consider a rapidly spinning black hole. We demonstrate that under such conditions the accretion flow needs to initially have specific angular momentum above a certain threshold to eventually reach and robustly sustain the magnetically arrested disk state. If the flow can reach such a state, it can launch very powerful jets at ≳100% energy efficiency. Interestingly, we also find that even when the accretion flow has initial specific angular momentum below the threshold, it can still launch episodic jets with an average energy efficiency of ∼10%. However, the accretion flow has nontypical behaviors such as having different rotation directions at different inclinations and exhibiting persistent outflows along the midplane even in the inner disk region. Our results give plausible explanations as to why jets can be produced from various astrophysical systems that likely lack large gas specific angular momenta, such as Sgr A*, wind-fed X-ray binaries, tidal disruption events, and long-duration gamma-ray bursts.more » « less
-
For a system without spin–orbit coupling, the (i) nuclear plus electronic linear momentum and (ii) nuclear plus orbital electronic angular momentum are good quantum numbers. Thus, when a molecular system undergoes a nonadiabatic transition, there should be no change in the total linear or angular momentum. Now, the standard surface hopping algorithm ignores the electronic momentum and indirectly equates the momentum of the nuclear degrees of freedom to the total momentum. However, even with this simplification, the algorithm still does not conserve either the nuclear linear or the nuclear angular momenta. Here, we show that one way to address these failures is to dress the derivative couplings (i.e., the hopping directions) in two ways: (i) we disallow changes in the nuclear linear momentum by working in a translating basis (which is well known and leads to electron translation factors) and (ii) we disallow changes in the nuclear angular momentum by working in a basis that rotates around the center of mass [which is not well-known and leads to a novel, rotationally removable component of the derivative coupling that we will call electron rotation factors below, cf. Eq. (96)]. The present findings should be helpful in the short term as far as interpreting surface hopping calculations for singlet systems (without spin) and then developing the new surface hopping algorithm in the long term for systems where one cannot ignore the electronic orbital and/or spin angular momentum.more » « less
-
Dholakia, Kishan; Spalding, Gabriel C. (Ed.)There is an unsettled problem in choosing the correct expressions for the local momentum density and angular momentum density of electromagnetic fields (or indeed, of any non-scalar field). If one only examines plane waves, the problem is moot, as the known possible expressions all give the same result. The momentum and angular momentum density expressions are generally obtained from the energy-momentum tensor, in turn obtained from a Lagrangian. The electrodynamic expressions obtained by the canonical procedure are not the same as the symmetric Belinfante reworking. For the interaction of matter with structured light, for example, twisted photons, this is important; there are drastically different predictions for forces and angular momenta induced on small test objects. We show situations where the two predictions can be checked, with numerical estimates of the size of the effects.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/jnma-2020-0080
