Boundary Conditions in Bicompact Schemes for HOLO Algorithms to Solve Transport Equations

• Published in: Mathematical models and computer simulations Vol. 12; no. 3; pp. 271 - 281
• Main Authors: Aristova, E. N., Karavaeva, N. I.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.05.2020; Springer Nature B.V
• Subjects: Algorithms; Approximation; Boundary conditions; Convergence; Diffusion; Distribution functions; Kinetic equations; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Runge-Kutta method; Simulation and Modeling; Transport equations; diagonally implicit Runge–Kutta method; sweep method; transport equation; quasi-diffusion method; HOLO algorithms for solving the transport equation; bicompact scheme
• ISSN: 2070-0482; 2070-0490
• DOI: 10.1134/S2070048220030059

The paper considers bicompact schemes for HOLO algorithms to solve 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 a high order (HO) is used but also the quasi-diffusion equation of a low order (LO). For both systems of kinetic equations semidiscrete bicompact schemes with the fourth order of approximation in space are constructed. Integration over time can be carried out with any order of approximation. The diagonal-implicit third-order approximation Runge–Kutta method is used in the work; 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 and the one directly setting conditions for the radiation density from the solution of the transport equation from the HO part. 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 order. Setting the boundary conditions under the solution of the transport equation, we preserve the third order of convergence in time but the HOLO algorithms accelerate the iterations less efficiently.