A versatile and accurate treatment for the highly forward-peaked phase function in the three-dimensional (3D) radiative transfer equation (RTE) based on the discrete ordinates method (DOM) is crucial for biomedical optics. Our first objective was to compare the delta-Eddington (dE) and Galerkin quadrature (GQ) methods. The dE method decomposes the phase function into a purely forward-peaked component and the other component, and expands the other component by Legendre polynomials as well as the finite order Legendre expansion (FL) method does. The GQ method conducts the weighting procedure in addition to the Legendre expansion. Although it was reported that both methods can provide the accurate results for calculations of the RTE, the versatility of both methods is still unclear. The second objective was to examine a possibility of a conjunction of the GQ method with the dE method, called as the GQ-dE method, which has the advantages of both methods. We examined numerical errors in the moment conditions of the phase function using the FL, dE, GQ, and GQ-dE methods at various types and orders of the quadrature sets, mainly in the region of the errors induced by the angular discretization using the DOM. The errors were reduced by the dE method from those by the FL method, however the error reduction depended on the types and orders of the quadrature sets. Meanwhile, the errors were significantly reduced by the GQ and GQ-dE methods, regardless of the quadrature sets. We also verified the numerical calculations of the time-dependent 3D RTE by the analytical solution of the RTE for homogeneous media in the region of the scattering length scale, where the highly forward-peaked phase function strongly influences the RTE-results. The errors in the RTE-results were similar to those in the moment conditions. Our results suggest the higher versatility and accuracy of the GQ and GQ-dE methods than those of the FL and dE methods.

基于离散坐标法（DOM）的三维（3D）辐射传递方程（RTE）中高度前向相位函数的通用且准确的处理对于生物医学光学至关重要。我们的首要目标是比较delta-Eddington（dE）和Galerkin正交（GQ）方法。dE方法将相位函数分解为一个纯正向峰值的分量和另一个分量，并通过勒让德多项式和有限阶勒让德展开（FL）方法进行扩展。除Legendre展开外，GQ方法还执行加权过程。尽管据报道这两种方法都可以为RTE的计算提供准确的结果，但两种方法的通用性仍不清楚。第二个目标是研究将GQ方法与dE方法（称为GQ-dE方法）结合使用的可能性，该方法具有这两种方法的优点。我们使用FL，dE，GQ和GQ-dE方法在正交集的各种类型和阶次上检查了相位函数的矩条件下的数值误差，主要是在使用DOM进行角度离散化引起的误差区域内。dE方法的误差比FL方法的误差减小，但是误差的减小取决于正交集的类型和阶数。同时，无论正交集如何，通过GQ和GQ-dE方法均可显着降低误差。我们还通过散射长度尺度区域内均质介质的RTE解析解验证了随时间变化的3D RTE的数值计算，其中高度前向相位函数强烈影响RTE结果。RTE结果中的误差与当前情况下的误差相似。我们的结果表明，与FL和dE方法相比，GQ和GQ-dE方法具有更高的通用性和准确性。