# 2. Theory

PMR solves the radiative transport equation (RTE) using the method of discrete ordinates with user-selectable quadrature rules. The quadrature rule and order defines a set of discrete directions in which intensity is solved. Scattering is not currently supported, so this results in a decoupled set of linear PDEs in each ordinate direction.

For each ordinate direction (), a weight is provided () by the quadrature rule selected. For each ordinate direction, the ordinate intensity is solved for and overall scalar flux () and radiative heat flux () can then be defined by summing over ordinate directions as

and

The continuous form of the PDE evaluated for intensity (excluding scattering terms) in a given ordinate direction is

where the radiative source term () and absorption coefficient () are calculated by the fluid code and transferred to PMR. The radiative source term is usually similar to the form below, but may include subgrid effects where relevant:

On the domain boundaries, the following condition is enforced for intensity

where is the irradiation (sum of incoming intensities on the boundary) and the boundary source () is sent from the fluids code and usually has a form similar to the following equation:

In the case where the boundary is partially reflective and the problem becomes nonlinear and coupled across ordinate directions through . When this is the case, multiple nonlinear iterations should be used to achieve a converged solution. The irradiation is calculated on all boundaries as

where is the outward facing normal vector.

The primary numerical choices when solving this equation are what type of stabilization to use for the advection-like term (upwind or SUCV) and whether to use an edge-based or element-based stencil.

## 2.1. Quadrature Rules

### 2.1.1. Thurgood

The Thurgood [1] quadrature rule partitions space into ordinate directions, where is the specified quadrature order. The order must be at least 2 for this quadrature rule.

### 2.1.2. Level-Symmetric

The Level-Symmetric [2] quadrature rule partitions space into ordinate directions, where is the specified quadrature order. The order must be one of: 2, 4, 6, 8, 12, or 16. No other orders are valid for this quadrature rule.

### 2.1.3. PNTN

The Legendre-Chebyshev (PN-TN) [3] quadrature rule partitions space into ordinate directions, where is the specified quadrature order. The order must be even for this quadrature rule.