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Journal articles on the topic 'Divergence-free method'

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Author: Grafiati

Published: 2 September 2023

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1

Huerta, Antonio, Yolanda Vidal, and Pierre Villon. "Pseudo-divergence-free element free Galerkin method for incompressible fluid flow." Computer Methods in Applied Mechanics and Engineering 193, no. 12-14 (March 2004): 1119–36. http://dx.doi.org/10.1016/j.cma.2003.12.010.

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2

Klingenberg, Christian, Frank Pörner, and Yinhua Xia. "An Efficient Implementation of the Divergence Free Constraint in a Discontinuous Galerkin Method for Magnetohydrodynamics on Unstructured Meshes." Communications in Computational Physics 21, no. 2 (February 2017): 423–42. http://dx.doi.org/10.4208/cicp.180515.230616a.

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AbstractIn this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint (∇·B=0) of its magnetic field B. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements [19], and another approach of the divergence cleaning technique given by Dedner et al. [15]. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

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3

Avdeeva, E. N., and V. V. Lukin. "Divergence-free finite-difference method for 2D ideal MHD." Journal of Physics: Conference Series 1336 (November 2019): 012026. http://dx.doi.org/10.1088/1742-6596/1336/1/012026.

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4

Fu, Guosheng. "An Explicit Divergence-Free DG Method for Incompressible Magnetohydrodynamics." Journal of Scientific Computing 79, no. 3 (January 14, 2019): 1737–52. http://dx.doi.org/10.1007/s10915-019-00909-2.

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5

Fu, Guosheng. "An explicit divergence-free DG method for incompressible flow." Computer Methods in Applied Mechanics and Engineering 345 (March 2019): 502–17. http://dx.doi.org/10.1016/j.cma.2018.11.012.

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6

Li, Shengtai. "A fourth-order divergence-free method for MHD flows." Journal of Computational Physics 229, no. 20 (October 2010): 7893–910. http://dx.doi.org/10.1016/j.jcp.2010.06.044.

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7

Zhang, Man, and Xueshang Feng. "A Three-Order, Divergence-Free Scheme for the Simulation of Solar Wind." Universe 8, no. 7 (July 5, 2022): 371. http://dx.doi.org/10.3390/universe8070371.

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In this paper, we present a three-order, divergence-free finite volume scheme to simulate the steady state solar wind ambient. The divergence-free condition of the magnetic field is preserved by the constrained transport (CT) method. The CT method can keep the magnetic fields divergence free if the magnetic fields is divergence free initially. Thus, a least-squares reconstruction of magnetic field with the divergence free constraints is used to make the magnetic fields global solenoidality initially. High order spatial accuracy is obtained through a non-oscillatory hierarchical reconstruction, while a high order time discretization is achieved using a three-order Runge–Kutta scheme. This new model of three order in space and time is validated by numerical results for Carrington rotation 2207. The numerical results show its capability for producing stable reliable results for structured solar wind. The high-order, divergence-free properties of this method make it an ideal tool for the simulations of coronal mass ejection in future.

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8

Deng, Yongbo, and Jan G. Korvink. "Topology optimization for three-dimensional electromagnetic waves using an edge element-based finite-element method." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2189 (May 2016): 20150835. http://dx.doi.org/10.1098/rspa.2015.0835.

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This paper develops a topology optimization procedure for three-dimensional electromagnetic waves with an edge element-based finite-element method. In contrast to the two-dimensional case, three-dimensional electromagnetic waves must include an additional divergence-free condition for the field variables. The edge element-based finite-element method is used to both discretize the wave equations and enforce the divergence-free condition. For wave propagation described in terms of the magnetic field in the widely used class of non-magnetic materials, the divergence-free condition is imposed on the magnetic field. This naturally leads to a nodal topology optimization method. When wave propagation is described using the electric field, the divergence-free condition must be imposed on the electric displacement. In this case, the material in the design domain is assumed to be piecewise homogeneous to impose the divergence-free condition on the electric field. This results in an element-wise topology optimization algorithm. The topology optimization problems are regularized using a Helmholtz filter and a threshold projection method and are analysed using a continuous adjoint method. In order to ensure the applicability of the filter in the element-wise topology optimization version, a regularization method is presented to project the nodal into an element-wise physical density variable.

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9

Qin, Si-xue. "A divergence-free method to extract observables from correlation functions." Physics Letters B 742 (March 2015): 358–62. http://dx.doi.org/10.1016/j.physletb.2015.02.009.

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10

Hiptmair, Ralf, Lingxiao Li, Shipeng Mao, and Weiying Zheng. "A fully divergence-free finite element method for magnetohydrodynamic equations." Mathematical Models and Methods in Applied Sciences 28, no. 04 (April 2018): 659–95. http://dx.doi.org/10.1142/s0218202518500173.

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We propose a finite element method for the three-dimensional transient incompressible magnetohydrodynamic equations that ensures exactly divergence-free approximations of the velocity and the magnetic induction. We employ second-order semi-implicit timestepping, for which we rigorously establish an energy law and, as a consequence, unconditional stability. We prove unique solvability of the linear systems of equations to be solved in every timestep. For those we design an efficient preconditioner so that the number of preconditioned GMRES iterations is uniformly bounded with respect to the number of degrees of freedom. As both meshwidth and timestep size tend to zero, we prove that the discrete solutions converge to a weak solution of the continuous problem. Finally, by several numerical experiments, we confirm the predictions of the theory and demonstrate the efficiency of the preconditioner.

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11

Zhang, Li, Minfu Feng, and Jian Zhang. "A globally divergence-free weak Galerkin method for Brinkman equations." Applied Numerical Mathematics 137 (March 2019): 213–29. http://dx.doi.org/10.1016/j.apnum.2018.11.002.

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12

Muldoon, Frank, and Sumanta Acharya. "A divergence-free interpolation scheme for the immersed boundary method." International Journal for Numerical Methods in Fluids 56, no. 10 (2008): 1845–84. http://dx.doi.org/10.1002/fld.1565.

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13

Yang, M., D. del-Castillo-Negrete, G. Zhang, and M. T. Beidler. "A divergence-free constrained magnetic field interpolation method for scattered data." Physics of Plasmas 30, no. 3 (March 2023): 033901. http://dx.doi.org/10.1063/5.0138905.

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An interpolation method to evaluate magnetic fields, given its unstructured and scattered magnetic data, is presented. The method is based on the reconstruction of the global magnetic field using a superposition of orthogonal functions. The coefficients of the expansion are obtained by minimizing a cost function defined as the L2 norm of the difference between the ground truth and the reconstructed magnetic field evaluated on the training data. The divergence-free condition is incorporated as a constraint in the cost function, allowing the method to achieve arbitrarily small errors in the magnetic field divergence. An exponential decay of the approximation error is observed and compared with the less favorable algebraic decay of local splines. Compared to local methods involving computationally expensive search algorithms, the proposed method exhibits a significant reduction of the computational complexity of the field evaluation, while maintaining a small error in the divergence even in the presence of magnetic islands and stochasticity. Applications to the computation of Poincaré sections using data obtained from numerical solutions of the magnetohydrodynamic equations in toroidal geometry are presented and compared with local methods currently in use.

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14

Yue, Zhu, Jiang Shengyao, Yang Xingtuan, and Duan Riqiang. "Study on divergence approximation formula for pressure calculation in particle method." Journal of Computational Multiphase Flows 10, no. 4 (August 16, 2018): 159–69. http://dx.doi.org/10.1177/1757482x18791896.

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The moving particle semi-implicit method is a meshless particle method for incompressible fluid and has proven useful in a wide variety of engineering applications of free-surface flows. Despite its wide applicability, the moving particle semi-implicit method has the defects of spurious unphysical pressure oscillation. Three various divergence approximation formulas, including basic divergence approximation formula, difference divergence approximation formula, and symmetric divergence approximation formula are proposed in this paper. The proposed three divergence approximation formulas are then applied for discretization of source term in pressure Poisson equation. Two numerical tests, including hydrostatic pressure problem and dam-breaking problem, are carried out to assess the performance of different formulas in enhancing and stabilizing the pressure calculation. The results demonstrate that the pressure calculated by basic divergence approximation formula and difference divergence approximation formula fluctuates severely. However, application of symmetric divergence approximation formula can result in a more accurate and stabilized pressure.

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15

Zhou, Chuan, Jianhua Li, Huaan Wang, Kailong Mu, and Lanhao Zhao. "A Divergence-Free Immersed Boundary Method and Its Finite Element Applications." Journal of Mechanics 36, no. 6 (August 6, 2020): 901–14. http://dx.doi.org/10.1017/jmech.2020.23.

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ABSTRACTIn order to maintain the no-slip condition and the divergence-free property simultaneously, an iterative scheme of immersed boundary method in the finite element framework is presented. In this method, the Characteristic-based Split scheme is employed to solve the momentum equations and the formulation for the pressure and the extra body force is derived according to the no-slip condition. The extra body force is divided into two divisions, one is in relation to the pressure and the other is irrelevant. Two corresponding independent iterations are set to solve the two sections. The novelty of this method lies in that the correction of the velocity increment is included in the calculation of the extra body force which is relevant to the pressure and the update of the force is incorporated into the iteration of the pressure. Hence, the divergence-free properties and no-slip conditions are ensured concurrently. In addition, the current method is validated with well-known benchmarks.

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16

Zheng, XiaoBo, Gang Chen, and XiaoPing Xie. "A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows." Science China Mathematics 60, no. 8 (April 25, 2017): 1515–28. http://dx.doi.org/10.1007/s11425-016-0354-8.

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17

Schräder, Daniela, and Holger Wendland. "A high-order, analytically divergence-free discretization method for Darcy’s problem." Mathematics of Computation 80, no. 273 (June 7, 2010): 263–77. http://dx.doi.org/10.1090/s0025-5718-2010-02388-9.

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18

Sheu, Tony W. H., and P. H. Chiu. "A divergence-free-condition compensated method for incompressible Navier–Stokes equations." Computer Methods in Applied Mechanics and Engineering 196, no. 45-48 (September 2007): 4479–94. http://dx.doi.org/10.1016/j.cma.2007.05.015.

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19

Drake, Kathryn P., Edward J. Fuselier, and Grady B. Wright. "A Partition of Unity Method for Divergence-Free or Curl-Free Radial Basis Function Approximation." SIAM Journal on Scientific Computing 43, no. 3 (January 2021): A1950—A1974. http://dx.doi.org/10.1137/20m1373505.

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20

Cai, Wei, Jun Hu, and Shangyou Zhang. "High Order Hierarchical Divergence-Free Constrained Transport H(div) Finite Element Method for Magnetic Induction Equation." Numerical Mathematics: Theory, Methods and Applications 10, no. 2 (May 2017): 243–54. http://dx.doi.org/10.4208/nmtma.2017.s03.

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AbstractIn this paper, we propose to use the interior functions of an hierarchical basis for high order BDMp elements to enforce the divergence-free condition of a magnetic field B approximated by the H(div)BDMp basis. The resulting constrained finite element method can be used to solve magnetic induction equation in MHD equations. The proposed procedure is based on the fact that the scalar (p–1)-th order polynomial space on each element can be decomposed as an orthogonal sum of the subspace defined by the divergence of the interior functions of the p-th order BDMp basis and the constant function. Therefore, the interior functions can be used to remove element-wise all higher order terms except the constant in the divergence error of the finite element solution of the B-field. The constant terms from each element can be then easily corrected using a first order H(div) basis globally. Numerical results for a 3-D magnetic induction equation show the effectiveness of the proposed method in enforcing divergence-free condition of the magnetic field.

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21

SUN, WEI-MIN, HONG-SHI ZONG, XIANG-SONG CHEN, and FAN WANG. "THE TENSOR CURRENT DIVERGENCE EQUATION IN U(1) GAUGE THEORIES IS FREE OF ANOMALIES." International Journal of Modern Physics A 19, no. 16 (June 30, 2004): 2705–12. http://dx.doi.org/10.1142/s0217751x04019275.

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The possible anomaly of the tensor current divergence equation in U(1) gauge theories is calculated by means of perturbative method. It is found that the tensor current divergence equation is free of anomalies.

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22

Shams, Sheyda, and Masoud Movahhedi. "Unconditionally Stable Divergence-Free Vector Meshless Method Based on Crank–Nicolson Scheme." IEEE Antennas and Wireless Propagation Letters 16 (2017): 2671–74. http://dx.doi.org/10.1109/lawp.2017.2740379.

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23

Matthies, Gunar, and Friedhelm Schieweck. "A Multigrid Method for Incompressible Flow Problems Using Quasi Divergence Free Functions." SIAM Journal on Scientific Computing 28, no. 1 (January 2006): 141–71. http://dx.doi.org/10.1137/04061814x.

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24

Kadri Harouna, Souleymane, and Valérie Perrier. "Divergence-Free Wavelet Projection Method for Incompressible Viscous Flow on the Square." Multiscale Modeling & Simulation 13, no. 1 (January 2015): 399–422. http://dx.doi.org/10.1137/140969208.

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25

Bao, Yuanxun, Aleksandar Donev, Boyce E. Griffith, David M. McQueen, and Charles S. Peskin. "An Immersed Boundary method with divergence-free velocity interpolation and force spreading." Journal of Computational Physics 347 (October 2017): 183–206. http://dx.doi.org/10.1016/j.jcp.2017.06.041.

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26

Oyarzua, R., T. Qin, and D. Schotzau. "An exactly divergence-free finite element method for a generalized Boussinesq problem." IMA Journal of Numerical Analysis 34, no. 3 (October 3, 2013): 1104–35. http://dx.doi.org/10.1093/imanum/drt043.

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27

Shams, Sheyda, Ali Ghafoorzadeh Yazdi, and Masoud Movahhedi. "Unconditionally Stable Divergence-Free Vector-Based Meshless Method for Transient Electromagnetic Analysis." IEEE Transactions on Microwave Theory and Techniques 65, no. 6 (June 2017): 1929–38. http://dx.doi.org/10.1109/tmtt.2016.2646680.

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28

Li, Ruo, Zhiyuan Sun, and Zhijian Yang. "A discontinuous Galerkin method for Stokes equation by divergence‐free patch reconstruction." Numerical Methods for Partial Differential Equations 36, no. 4 (November 30, 2019): 756–71. http://dx.doi.org/10.1002/num.22449.

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29

Zhang, Man, Xueshang Feng, Xiaojing Liu, and Liping Yang. "A Provably Positive, Divergence-free Constrained Transport Scheme for the Simulation of Solar Wind." Astrophysical Journal Supplement Series 257, no. 2 (November 12, 2021): 32. http://dx.doi.org/10.3847/1538-4365/ac1e29.

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Abstract In this paper, we present a provably positive, divergence-free constrained transport (CT) scheme to simulate the steady-state solar wind ambient with the three-dimensional magnetohydrodynamics numerical model. The positivity can be lost in two ways: one way is in the reconstruction process, and the other is in the updating process when the variables are advanced to the next time step. We adopt a self-adjusting strategy to bring the density and pressure into the permitted range in the reconstruction process, and use modified wave speeds in the Harten–Lax–van Leer flux to ensure the positivity in the updating process. The CT method can keep the magnetic fields divergence-free if the magnetic fields are divergence-free initially. Thus, we combine the least-squares reconstruction of the magnetic fields with the divergence-free constraints to make the magnetic fields globally solenoidal initially. Furthermore, we adopt a radial basis function method to interpolate variables at boundaries that can keep the magnetic field locally divergence-free. To verify the capability of the model in producing structured solar wind, the modeled results are compared with Parker Solar Probe (PSP) in situ observations during its first two encounters, as well as Wind observations at 1 au. Additionally, a solar maximum solar wind background is simulated to show the property of the model’s ability to preserve the positivity. The results show that the model can provide a relatively satisfactory comparison with PSP or Wind observations, and the divergence error is about 10−10 for all of the tests in this paper.

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30

Hyouguchi, T., R. Seto, and S. Adachi. "Overlooked Degree of Freedom in Steepest Descent Method: Steepest Descent Method Corresponding to Divergence-Free WKB Method." Progress of Theoretical Physics 122, no. 6 (December 1, 2009): 1347–76. http://dx.doi.org/10.1143/ptp.122.1347.

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31

Allendes, Alejandro, Gabriel R. Barrenechea, and Julia Novo. "A Divergence-Free Stabilized Finite Element Method for the Evolutionary Navier--Stokes Equations." SIAM Journal on Scientific Computing 43, no. 6 (January 2021): A3809—A3836. http://dx.doi.org/10.1137/21m1394709.

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32

Li, Di, Zhiyuan Sun, Fengru Wang null, and Jerry Zhijian Yang. "The Discontinuous Galerkin Method by Divergence-Free Patch Reconstruction for Stokes Eigenvalue Problems." Numerical Mathematics: Theory, Methods and Applications 15, no. 2 (June 2022): 484–509. http://dx.doi.org/10.4208/nmtma.oa-2021-0085.

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33

Park, Chunjae. "P1-nonconforming divergence-free finite element method on square meshes for Stokes equations." Journal of Numerical Mathematics 28, no. 4 (December 16, 2020): 247–61. http://dx.doi.org/10.1515/jnma-2019-0056.

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AbstractRecently, the P1-nonconforming finite element space over square meshes has been proved stable to solve Stokes equations with the piecewise constant space for velocity and pressure, respectively. In this paper, we will introduce its locally divergence-free subspace to solve the elliptic problem for the velocity only decoupled from the Stokes equation. The concerning system of linear equations is much smaller compared to the Stokes equations. Furthermore, it is split into two smaller ones. After solving the velocity first, the pressure in the Stokes problem can be obtained by an explicit method very rapidly.

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34

Huang, Jianguo, and Shangyou Zhang. "A divergence-free finite element method for a type of 3D Maxwell equations." Applied Numerical Mathematics 62, no. 6 (June 2012): 802–13. http://dx.doi.org/10.1016/j.apnum.2011.06.009.

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35

Mu, Lin, Junping Wang, Xiu Ye, and Shangyou Zhang. "A discrete divergence free weak Galerkin finite element method for the Stokes equations." Applied Numerical Mathematics 125 (March 2018): 172–82. http://dx.doi.org/10.1016/j.apnum.2017.11.006.

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36

Brenner, Susanne C., Fengyan Li, and Li-Yeng Sung. "A Locally Divergence-Free Interior Penalty Method for Two-Dimensional Curl-Curl Problems." SIAM Journal on Numerical Analysis 46, no. 3 (January 2008): 1190–211. http://dx.doi.org/10.1137/060671760.

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37

Kreeft, Jasper, and Marc Gerritsma. "Mixed mimetic spectral element method for Stokes flow: A pointwise divergence-free solution." Journal of Computational Physics 240 (May 2013): 284–309. http://dx.doi.org/10.1016/j.jcp.2012.10.043.

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38

Kawai, Soshi. "Divergence-free-preserving high-order schemes for magnetohydrodynamics: An artificial magnetic resistivity method." Journal of Computational Physics 251 (October 2013): 292–318. http://dx.doi.org/10.1016/j.jcp.2013.05.033.

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39

Greif, Chen, Dan Li, Dominik Schötzau, and Xiaoxi Wei. "A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics." Computer Methods in Applied Mechanics and Engineering 199, no. 45-48 (November 2010): 2840–55. http://dx.doi.org/10.1016/j.cma.2010.05.007.

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40

Cervone, Antonio, Sandro Manservisi, and Ruben Scardovelli. "An optimal constrained approach for divergence-free velocity interpolation and multilevel VOF method." Computers & Fluids 47, no. 1 (August 2011): 101–14. http://dx.doi.org/10.1016/j.compfluid.2011.02.014.

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41

Fragile, P. Chris, Daniel Nemergut, Payden L. Shaw, and Peter Anninos. "Divergence-free magnetohydrodynamics on conformally moving, adaptive meshes using a vector potential method." Journal of Computational Physics: X 2 (March 2019): 100020. http://dx.doi.org/10.1016/j.jcpx.2019.100020.

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42

Pahar, Gourabananda, and Anirban Dhar. "Numerical Modelling of Free-Surface Flow-Porous Media Interaction Using Divergence-Free Moving Particle Semi-Implicit Method." Transport in Porous Media 118, no. 2 (March 31, 2017): 157–75. http://dx.doi.org/10.1007/s11242-017-0852-x.

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43

Voulis, Igor, and Arnold Reusken. "A time dependent Stokes interface problem: well-posedness and space-time finite element discretization." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 6 (November 2018): 2187–213. http://dx.doi.org/10.1051/m2an/2018053.

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In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of two-phase flows is studied. This Stokes interface problem has discontinuous density and viscosity coefficients and a pressure solution that is discontinuous across an evolving interface. This strongly simplified two-phase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for two-phase flow problems. In view of theunfitted finite element methods that are often used for two-phase flow simulations, we are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Euclidean setting. Such well-posed weak formulations, which are not known in the literature, are the main results of this paper. Different variants are considered, namely one with suitable spaces of divergence free functions, a discrete-in-time version of it, and variants in which the divergence free constraint in the solution space is treated by a pressure Lagrange multiplier. The discrete-in-time variational formulation involving the pressure variable for the divergence free constraint is a natural starting point for a space-time finite element discretization. Such a method is introduced and results of numerical experiments with this method are presented.

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44

Singh, Manoj Kumar, and Arvind K. Singh. "A derivative free globally convergent method and its deformations." Arabian Journal of Mathematics 10, no. 2 (May 1, 2021): 481–96. http://dx.doi.org/10.1007/s40065-021-00323-3.

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AbstractThe motive of the present work is to introduce and investigate the quadratically convergent Newton’s like method for solving the non-linear equations. We have studied some new properties of a Newton’s like method with examples and obtained a derivative-free globally convergent Newton’s like method using forward difference operator and bisection method. Finally, we have used various numerical test functions along with their fractal patterns to show the utility of the proposed method. These patterns support the numerical results and explain the compactness regarding the convergence, divergence and stability of the methods to different roots.

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45

Xu, Zhiliang, Dinshaw S. Balsara, and Huijing Du. "Divergence-Free WENO Reconstruction-Based Finite Volume Scheme for Solving Ideal MHD Equations on Triangular Meshes." Communications in Computational Physics 19, no. 4 (April 2016): 841–80. http://dx.doi.org/10.4208/cicp.050814.040915a.

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AbstractIn this paper, we introduce a high-order accurate constrained transport type finite volume method to solve ideal magnetohydrodynamic equations on two-dimensional triangular meshes. A new divergence-free WENO-based reconstruction method is developed to maintain exactly divergence-free evolution of the numerical magnetic field. In this formulation, the normal component of the magnetic field at each face of a triangle is reconstructed uniquely and with the desired order of accuracy. Additionally, a new weighted flux interpolation approach is also developed to compute the z-component of the electric field at vertices of grid cells. We also present numerical examples to demonstrate the accuracy and robustness of the proposed scheme.

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46

O’Neill, Larry W., Tracy Haack, and Theodore Durland. "Estimation of Time-Averaged Surface Divergence and Vorticity from Satellite Ocean Vector Winds." Journal of Climate 28, no. 19 (September 29, 2015): 7596–620. http://dx.doi.org/10.1175/jcli-d-15-0119.1.

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Abstract Two methods of computing the time-mean divergence and vorticity from satellite vector winds in rain-free (RF) and all-weather (AW) conditions are investigated. Consequences of removing rain-contaminated winds on the mean divergence and vorticity depend strongly on the order in which the time-average and spatial derivative operations are applied. Taking derivatives first and averages second (DFAS_RF) incorporates only those RF winds measured at the same time into the spatial derivatives. While preferable mathematically, this produces mean fields biased relative to their AW counterparts because of the exclusion of convergence and cyclonic vorticity often associated with rain. Conversely, taking averages first and derivatives second (AFDS_RF) incorporates all RF winds into the time-mean spatial derivatives, even those not measured coincidentally. While questionable, the AFDS_RF divergence and vorticity surprisingly appears qualitatively consistent with the AW means, despite using only RF winds. The analysis addresses whether the AFDS_RF method accurately estimates the AW mean divergence and vorticity. Model simulations indicate that the critical distinction between these two methods is the inclusion of typically convergent and cyclonic winds bordering rain patches in the AFDS_RF method. While this additional information removes some of the sampling bias in the DFAS_RF method, it is shown that the AFDS_RF method nonetheless provides only marginal estimates of the mean AW divergence and vorticity given sufficient time averaging and spatial smoothing. Use of the AFDS_RF method is thus not recommended.

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47

Sarin, V., and A. H. Sameh. "Hierarchical Divergence-Free Bases and Their Application to Particulate Flows." Journal of Applied Mechanics 70, no. 1 (January 1, 2003): 44–49. http://dx.doi.org/10.1115/1.1530633.

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The paper presents an algebraic scheme to construct hierarchical divergence-free basis for velocity in incompressible fluids. A reduced system of equations is solved in the corresponding subspace by an appropriate iterative method. The basis is constructed from the matrix representing the incompressibility constraints by computing algebraic decompositions of local constraint matrices. A recursive strategy leads to a hierarchical basis with desirable properties such as fast matrix-vector products, a well-conditioned reduced system, and efficient parallelization of the computation. The scheme has been extended to particulate flow problems in which the Navier-Stokes equations for fluid are coupled with equations of motion for rigid particles suspended in the fluid. Experimental results of particulate flow simulations have been reported for the SGI Origin 2000.

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48

Zhou, Yuanguo, Mingwei Zhuang, Linlin Shi, Guoxiong Cai, Na Liu, and Qing Huo Liu. "Spectral-Element Method With Divergence-Free Constraint for 2.5-D Marine CSEM Hydrocarbon Exploration." IEEE Geoscience and Remote Sensing Letters 14, no. 11 (November 2017): 1973–77. http://dx.doi.org/10.1109/lgrs.2017.2743781.

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Keim, Christopher, and Holger Wendland. "A High-Order, Analytically Divergence-Free Approximation Method for the Time-Dependent Stokes Problem." SIAM Journal on Numerical Analysis 54, no. 2 (January 2016): 1288–312. http://dx.doi.org/10.1137/151006196.

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Brenner, Susanne C., Fengyan Li, and Li-yeng Sung. "A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations." Mathematics of Computation 76, no. 258 (December 27, 2006): 573–95. http://dx.doi.org/10.1090/s0025-5718-06-01950-8.

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