Abstract

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.

中文翻译：

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HOLO算法求解运输方程的双紧凑型方案中的边界条件

摘要

本文考虑了HOLO算法的双紧方案来求解输运方程。为了加速散射迭代的收敛，不仅使用关于高阶（HO）的分布函数的传递方程的解，而且​​使用低阶（LO）的准扩散方程。对于这两个动力学方程组，构造了具有空间近似四阶的半离散双紧方案。随时间推移的积分可以任何近似的顺序进行。本文使用对角隐式三阶近似Runge-Kutta方法。每个阶段都可以简化为隐式Euler方法。详细描述了准扩散方程的离散化。考虑了LO部分的边界条件的两种变体：一种经典的方法是使用分数线性函数，另一种是直接根据HO部分的传输方程的解来设置辐射密度的条件。结果表明，准扩散方程本振系统的经典边界条件使该方案在时间上收敛为二阶。在输运方程的解下设置边界条件，我们保留了时间的三阶收敛性，但是HOLO算法降低了迭代的效率。结果表明，准扩散方程的本振系统的经典边界条件使该方案在时间上收敛为二阶。在输运方程的解下设置边界条件，我们保留了时间的三阶收敛性，但是HOLO算法降低了迭代的效率。结果表明，准扩散方程本振系统的经典边界条件使该方案在时间上收敛为二阶。在运输方程的解中设置边界条件，我们保留了时间的三阶收敛性，但是HOLO算法降低了迭代效率。