INCREASING THE ALGEBRAIC ORDER OF ACCURACY FOR ESn–QUADRATURE
M. P. Pepelyaev, E. A. Irinichev
VANT. Ser.: Mat. Mod. Fiz. Proc 2021. Вып.2. С. 11-23.
When solving the particle transport problem in kinetic approximation using difference schemes one needs to develop quadrature formulas in angular variables on a sphere. The ESn-quadrature with equal weights is one of the commonly used for this purpose. The equality of weights reduces an error of the quadrature formula. However, the ESn-quadrature has a relatively low algebraic order of accuracy. The authors developed the way of how to improve the accuracy of the ESn-quadrature in angular variables to solve the particle transport equation. The quadrature still has equal weights, while the directional cosines of the azimuthal angle are corrected so that even moment conditions are satisfied. The system of equations for the calculation of the particle flight direction is linearized with Newton’s method and iteratively solved using Gaussian method at each iteration. The resultant ESn-quadrature has a higher order of accuracy, as compared to the ESn-quadrature. This is proved by results of numerical studies: the calculation of integrals of the given function over the sphere surface, the solution of a model problem with an anisotropic source, and the K. Kobayashi symmetrical test. Keywords: angular quadrature, algebraic order of accuracy, equal weights, the discrete ordinate method, even moment conditions, three-dimensional transport equation, Cartesian coordinate system.
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