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Flux-corrected transport techniques applied to the radiation transport equation discretized with continuous finite elements. (English) Zbl 1380.76038

Summary: The flux-corrected transport (FCT) algorithm is applied to the unsteady and steady-state particle transport equation. The proposed FCT method employs the following: (1) a low-order, positivity-preserving scheme, based on the application of M-matrix properties, (2) a high-order scheme based on the entropy viscosity method introduced by J.-L. Guermond et al. [ibid. 230, No. 11, 4248–4267 (2011; Zbl 1220.65134)], and (3) local, discrete solution bounds derived from the integral transport equation. The resulting scheme is second-order accurate in space, enforces an entropy inequality, mitigates the formation of spurious oscillations, and guarantees the absence of negativities. Space discretization is achieved using continuous finite elements. Time discretizations for unsteady problems include theta schemes such as explicit and implicit Euler, and strong-stability preserving Runge-Kutta (SSPRK) methods. The developed FCT scheme is shown to be robust with explicit time discretizations but may require damping in the nonlinear iterations for steady-state and implicit time discretizations.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76V05 Reaction effects in flows
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics

Citations:

Zbl 1220.65134

Software:

SHASTA
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Guermond, J.-L.; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 4248-4267 (2011) · Zbl 1220.65134
[2] Bell, G. I.; Glasstone, S., Nuclear Reactor Theory (1970), Litton Educational Publishing, Inc.
[3] Schäfer, M.; Frank, M.; Herty, M., Optimal treatment planning in radiotherapy based on Boltzmann transport calculations, Math. Models Methods Appl. Sci., 18, 4, 573-592 (2008) · Zbl 1143.76058
[4] Bodenheimer, P., Numerical Methods in Astrophysics: An Introduction (2006), CRC Press
[5] Lewis, E. E.; Miller, W. F., Computational Methods of Neutron Transport (1993), American Nuclear Society: American Nuclear Society La Grange Park, IL · Zbl 0594.65096
[6] Eidmann, K., Radiation transport and atomic physics modeling in high-energy-density laser-produced plasmas, Laser Part. Beams, 12, 2, 223-244 (1994)
[7] Duderstadt, J. J.; Martin, W. R., Transport Theory (1979), John Wiley & Sons · Zbl 0407.76001
[8] Lesaint, P.; Raviart, P. A., On a finite element method for solving the neutron transport equation, Publ. Math. Inf. Rennes, 4, 1-40 (1974)
[9] Reed, W.; Hill, T., Triangular Mesh Methods for the Neutron Transport Equation (1973), Los Alamos Scientific Laboratory, Tech. Rep. LA-UR-73-479
[10] Zingan, V.; Guermond, J.-L.; Morel, J.; Popov, B., Implementation of the entropy viscosity method with the discontinuous Galerkin method, Comput. Methods Appl. Mech. Eng., 253, 479-490 (2013) · Zbl 1297.76109
[11] Lathrop, K. D., Spatial differencing of the transport equation: positivity vs. accuracy, J. Comput. Phys., 4, 475-498 (1969) · Zbl 0199.50703
[12] Hamilton, S.; Benzi, M., Negative flux fixups in discontinuous finite element sn transport, (International Conference on Mathematics, Computational Methods & Reactor Physics (2009))
[13] Walters, W. F.; Wareing, T. A., An accurate, strictly-positive, nonlinear characteristic scheme for the discrete ordinates equations, Transp. Theory Stat. Phys., 25, 2, 197-215 (1996) · Zbl 0935.65146
[14] Wareing, T. A., An exponential discontinuous scheme for discrete-ordinate calculations in cartesian geometries, (Joint International Conference on Mathematical Methods and Supercomputing in Nuclear Applications. Joint International Conference on Mathematical Methods and Supercomputing in Nuclear Applications, Saratoga Springs, NY (1997))
[15] Maginot, P., A Nonlinear Positive Extension of the Linear Discontinuous Spatial Discretization of the Transport Equation (December 2010), Texas A&M University, Master’s Thesis
[16] Maginot, P., A non-negative, non-linear Petrov-Galerkin method for bilinear discontinuous differencing of the Sn equations, (Joint International Conference on Mathematics and Computation, Supercomputing in Nuclear Applications, and the Monte Carlo Method. Joint International Conference on Mathematics and Computation, Supercomputing in Nuclear Applications, and the Monte Carlo Method, M&C 2015, Nashville, TN (2015))
[17] Maginot, P. G.; Ragusa, J. C.; Morel, J. E., Nonnegative methods for bilinear discontinuous differencing of the sn equations on quadrilaterals, Nucl. Sci. Eng., 185, 1, 53-69 (2017)
[18] Boris, J. P.; Book, D. L., Flux-corrected transport, I: SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11, 38-69 (1973) · Zbl 0251.76004
[19] Zalesak, S. T., Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 335-362 (1979) · Zbl 0416.76002
[20] Parrott, A. K.; Christie, M. A., FCT applied to the 2-d finite element solution of tracer transport by single phase flow in a porous medium, (Proceedings on the ICFD Conference on Numerical Methods in Fluid Dynamics (1986), Oxford University Press), 609 · Zbl 0607.76093
[21] Löhner, R.; Morgan, K.; Peraire, J.; Vahdati, M., Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations, Int. J. Numer. Methods Fluids, 7, 1093-1109 (1987) · Zbl 0633.76070
[22] Kuzmin, D.; Löhner, R.; Turek, S., Flux-Corrected Transport (2005), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, Germany
[23] Kuzmin, D., On the design of general-purpose flux limiters for finite element schemes, I: scalar convection, J. Comput. Phys., 219, 513-531 (2006) · Zbl 1189.76342
[24] Möller, M.; Kuzmin, D.; Kourounis, D., Implicit FEM-FCT algorithms and discrete Newton methods for transient convection problems, Int. J. Numer. Methods Fluids, 57, 761-792 (2008) · Zbl 1264.76063
[25] Kuzmin, D.; Möller, M.; Shadid, J. N.; Shashkov, M., Failsafe flux limiting and constrained data projections for equations of gas dynamics, J. Comput. Phys., 761-792 (2010)
[26] Kuzmin, D.; Gorb, Y., A flux-corrected transport algorithm for handling the close-packing limit in dense suspensions, J. Comput. Appl. Math., 236, 4944-4951 (2012) · Zbl 1426.76280
[27] Guermond, J.-L.; Nazarov, M.; Popov, B.; Yang, Y., A second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations, SIAM J. Numer. Anal., 52, 2163-2182 (2014) · Zbl 1302.65225
[28] Guermond, J.-L.; Nazarov, M., A maximum-principle preserving \(C^0\) finite element method for scalar conservation equations, Comput. Methods Appl. Mech. Eng., 272, 198-213 (2014) · Zbl 1296.65133
[29] Gottlieb, S., On high order strong stability preserving Runge-Kutta and multi step time discretizations, J. Sci. Comput., 25, 1 (2005) · Zbl 1203.65166
[30] Macdonald, C. B., Constructing High-Order Runge-Kutta Methods with Embedded Strong-Stability-Preserving Pairs (August 2003), Acadia University, Master’s Thesis
[31] Plemmins, R. J., M-matrix characterizations, I: nonsingular m-matrices, Linear Algebra Appl., 18, 2, 175-188 (1977) · Zbl 0359.15005
[32] Delchini, M.-O., Entropy-based viscous regularization for the multi-dimensional Euler equations in low-Mach and transonic flows, Comput. Fluids, 118, 225-244 (2015) · Zbl 1390.76255
[33] Delchini, M.-O.; Ragusa, J. C.; Berry, R. A., Viscous regularization for the non-equilibrium seven-equation two-phase flow model, J. Sci. Comput., 69, 2, 764-804 (2016) · Zbl 1432.76156
[34] LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems (2002), Cambridge University Press · Zbl 1010.65040
[35] Schär, C.; Smolarkiewicz, P. K., A synchronous and iterative flux-correction formalism for coupled transport equations, J. Comput. Phys., 128, 1, 101-120 (1996) · Zbl 0861.76054
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