The method of constructing of block-triangular difference schemes for selfadjoint form of the transport equation

V. E. Troshchiev

Matematičeskoe modelirovanie, Tome 10 (1998) no. 1, pp. 117-125.

Résumé

The method is proposed for construction of difference schemes for the 2nd order linear transport equation: \[ M\varphi(\vec{r},\vec{\Omega})\equiv\operatorname{div}[ \vec\Omega\frac1{\sigma(\vec{r})}(-\vec\Omega\nabla\varphi +\frac1{4\pi}Q(\vec{r},\vec{\Omega}))] +\sigma(\vec{r})\cdot\varphi=\frac1{4\pi}Q(\vec{r},\vec{\Omega})\tag{1} \] under the assumption that $Q(\vec{r},\vec{\Omega})$ is a given function (simple iteration). Equation (1) is one among selfadjoint forms equivalent to the transport equation of the 1st order: \[ L\varphi(\vec{r},\vec{\Omega})\equiv\vec{\Omega}\cdot\nabla\varphi +\sigma(\vec{r})\cdot\varphi=\frac1{4\pi}Q(\vec{r},\vec{\Omega}). \tag{2} \] The problem of the equation (1) is stated in a convex body $G$ and is a boundary problem in contrast to the equation (2), for which Cauchy problem is stated. The novelty of the method is that some properties of the problem (1) are indicated, and these properties make it possible to build finitedifference and finite-elements schemes with block-triangular matrices for boundary value for the system of discrete equations.