The boundary conditions in the bicompact schemes for HOLO algorithms for solving the transport equation

E. N. Aristova ; N. I. Karavaeva

Matematičeskoe modelirovanie, Tome 31 (2019) no. 9, pp. 3-20.

Résumé

The paper considers bicompact schemes for HOLO algorithms for solving the transport equation. To accelerate the convergence of scattering iterations, not only the solution of the transfer equation with respect to the distribution function of high order (HO) is used, but also the quasidiffusion equation of low order (LO) is used. For both systems of kinet-ic equations semi-discrete bicompact schemes with the fourth order of approximation in space are constructed. Integration over time can be carried out with any order of approx-imation. The diagonal-implicit third order approximation method is used in the work, its each stage can be reduced to the implicit Euler method. The discretization of quasi-diffusion equations is described in detail. Two variants for the boundary conditions for the LO part are considered: the classical one using fractional-linear functionals for the flux and radiation density ratio, and also by the radiation density value from the HO part of the system. It is shown that the classical boundary conditions for the LO system of equations of quasi-diffusion reduces the order of convergence of the scheme in time to the second. Setting the boundary conditions under the solution of the transport equation preserves the third order of convergence in time, but worsens the efficiency of iteration acceleration in HOLO algorithm.

Détail

Mots-clés : transport equation, quasi-diffusion method
Keywords: bicompact scheme, HOLO algorithms for transport equation solving, sweep method, diagonally implicit Runge-Kutta method.

Bibliographie

[1] M. L. Adams, E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations”, Progress in Nuclear Energy, 40:1 (2002), 3–159 | DOI

[2] L. Chácon, G. Chen, D. A. Knoll, C. Newman, H. Park et al., “Multiscale high-order/low-order (HOLO) algorithms and applications”, J. Comp. Phys., 330 (2017), 21–45 | DOI | MR | Zbl

[3] W. A. Wiesequist, D. Y. Anistratov, J. E. Morel, “A cell-local finite difference discretization of the low order of the quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes”, J. Comp. Phys., 273 (2014), 343–357 | DOI | MR

[4] V. Ya. Gol'din, “A quasi-diffusion method of solving the kinetic equation”, USSR Computational Mathematics and Mathematical Physics, 4:6 (1964), 136–149 | DOI

[5] V. Ya. Gol'din, “O matematicheskom modelirovanii zadach sploshnoi sredy s neravnovesnym perenosom”, Sovremennye problemy matematicheskoi fiziki i vychislitelnoi matematiki, Nauka, 1982, 340 pp.

[6] E. N. Aristova, “Simulation of radiation transport in channel on the basis of quasi-diffusion method”, Transport Theory and Statistical Physics, 37:05–07 (2008), 483–503 | DOI | MR | Zbl

[7] E. N. Aristova, D. F. Baydin, “Efficiency of Quasi-Diffusion Method for Calculating Critical Parameters of a Fast Reactor”, Mathem. Mod. and Comp. Simul., 4:6 (2012), 568–573 | DOI | MR | Zbl

[8] E. N. Aristova, D. F. Baydin, “Implementation of the Quasi Diffusion Method for Calculating the Critical Parameters of a Fast Neutron Reactor in 3D Hexagonal Geometry”, Mathematical Models and Computer Simulations, 5:2 (2013), 145–155 | DOI | MR | Zbl

[9] B. V. Rogov, M. N. Mikhailovskaya, “Monotone Bicompact Schemes for a Linear Advection Equation”, Doklady Mathematics, 83:1 (2011), 121–125 | DOI | MR | Zbl

[10] B. V. Rogov, M. N. Mikhailovskaya, “Monotonic bicompact schemes for linear transport equations”, Mathematical Models and Computer Simulations, 4:1 (2012), 92–100 | DOI | MR | Zbl

[11] B. V. Rogov, M. N. Mikhailovskaya, “Fourth-Order Accurate Bicompact Schemes for Hyperbolic Equations”, Doklady Mathematics, 81:1 (2010), 146–150 | DOI | MR | Zbl

[12] B. V. Rogov, M. N. Mikhailovskaya, “On the convergence of compact difference schemes”, Mathematical Models and Computer Simulations, 1:1 (2009), 91–104 | DOI | MR | Zbl

[13] A. V. Chikitkin, B. V. Rogov, E. N. Aristova, “High-Order Accurate Bicompact Schemes for Solving the Multidimensional Inhomogeneous Transport Equation and Their Efficient Parallel Implementation”, Doklady Mathematics, 94:2 (2016), 516–521 | DOI | MR

[14] M. D. Bragin, B. V. Rogov, “Iterative Approximate Factorization for Difference Operators of High-Order Bicompact Schemes for Multidimensional Nonhomogeneous Hyperbolic Systems”, Doklady Mathematics, 95:2 (2017), 140–143 | DOI | DOI | MR | Zbl

[15] E. N. Aristova, B. V. Rogov, A. V. Chikitkin, “Optimal Monotonization of a High-Order Accurate Bicompact Scheme for the Nonstationary Multidimensional Transport Equation”, Computational Mathematics and Mathematical Physics, 56:6 (2016), 962–976 | DOI | MR | Zbl

[16] E. N. Aristova, B. V. Rogov, “Boundary conditions implementation in bicompact schemes for the linear transport equation”, Mathem. Mod. and Comp. Simul., 5:3 (2013), 199–208 | DOI | MR | Zbl

[17] E. A. Alshina, E. M. Zaks, N. N. Kalitkin, “Optimalnye parametry yavnyh shem Runge-Kutty nevysokih poryadkov”, Matematicheskoe modelirovanie, 18:2 (2006), 61–71 | MR | Zbl

[18] E. A. Alshina, E. M. Zaks, N. N. Kalitkin, “Optimal first- to sixth-order accurate Runge-Kutta schemes”, Computational Mathem. and Mathem. Physics, 48:3 (2008), 395–405 | DOI | MR | MR | Zbl

[19] E. N. Aristova, N. I. Karavaeva, “Bicompact Higt Order Schemes for Quasi-Diffusion Equations”, Preprint Keldysh Institute of Applied Mathematics of RAS, 2018, 045, 28 pp.