A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface (Q2268182)

The following scaled linear transport equation is considered \[ \mu\partial_z\Psi+ \frac{\sigma_T}{\varepsilon}\Psi= \left(\frac{\sigma_T}{\varepsilon}-\varepsilon\sigma_a\right) \frac{1}{2} \int_{-1}^{1}\Psi(z,\mu')\,d\mu'+ \varepsilon q, \tag{1a} \] with boundary conditions \[ \Psi(z_L,\mu)= \Psi_L(\mu),\quad \mu>0; \qquad \Psi(z_R,\mu)=\Psi_R(\mu), \quad \mu<0. \tag{1b} \] Here \(z_L\) and \(z_R\) are the left and right boundaries respectively; the phase space density \(\Psi(z,\mu)\) is defined over \([z_L,z_R]\times[-1,1]\); \(\sigma_T\), \(\sigma_a\), \(q\) are the piecewise smooth total scattering and absorption coefficients and the source respectively. The authors propose a uniformly second order numerical method for the discrete-ordinates approximation to (1a)--(1b) in the slab geometry in the diffusive regimes with interfaces. At the interfaces, the scattering coefficients have discontinuities, so suitable conditions are needed to define the unique solution. The authors first approximate the scattering coefficients by piecewise constants determined by their cell averages, and then obtain the analytic solution at each cell, using which to piece together the numerical solution with the neighboring cells by the interface condition. It is also shown that this method is asymptotic-preserving, which preserves the discrete diffusion limit with the correct interface condition. Numerical examples are presented to justify the uniform convergence.