Monotonic difference schemes for transfer equation in plane layer

V. E. Troshchiev ; Yu. V. Troshchiev

Matematičeskoe modelirovanie, Tome 15 (2003) no. 1, pp. 3-13

Résumé

New finite-difference weighted schemes for the transport equation in plane-parallel geometry $$ LN(x,\mu)\equiv\mu\frac{\partial N(x,\mu)}{\partial x}+\alpha(x)N(x,\mu)=S(x,\mu),\qquad 0\le x\le H,\quad -1\le\mu\le1, $$ with the initial-value conditions $N(H,\mu0)=N_H(\mu)$, $N(0,\mu>0)=N_0(\mu)$ are constructed and investigated. The schemes are constructed in two ways: 1) as equivalent one to the classical three-point scheme for the self-adjoint transport equation of the second order \begin{gather*} -\mu^2\frac{\partial}{\partial x}\biggl[\frac1{\alpha(x)}\frac{\partial N(x,\mu)}{\partial x}\biggr]+\alpha(x)N(x,\mu)=S(x,\mu)-\mu\frac{\partial}{\partial x}\biggl(\frac{S(x,\mu)}{\alpha(x)}\biggr), \\ 0\le x\le H,\quad -1\le\mu\le1, \end{gather*} with the boundary-value conditions $N(H,\mu0)=N_H(\mu0)$, $LN(0,\mu0)=S(0,\mu0)$, $N(0,\mu>0)=N_0(\mu>0)$, $LN(H,\mu>0)=S(H,\mu>0)$; 2) as equivalent one to multi-point schemes for the first order transport equation. The constructed schemes are positive, monotonous, of the second order of accuracy and high-effective for numerical solution of transport problems. These theoretical and practical properties caused by special dependence of weights on the net interval.