[Paper Review] Numerical evaluation of Chandrasekhar's H-function, its first and second differential coefficients, its pole and moments from the new form for plane parallel scattering atmosphere in radiative transfer

Rabindra Nath Das, Rasajit Kumar Bera|ArXiv.org|Nov 21, 2007

Calibration and Measurement Techniques1 references3 citations

TL;DR

This paper presents a novel numerical approach to compute Chandrasekhar's H-function, its first and second derivatives, pole, and moments in plane-parallel scattering atmospheres using newly derived analytical forms. By applying iterative schemes and Simpson's one-third rule, the authors generate highly accurate tables for various albedo values, with results validated against existing data to within one unit in the ninth decimal place.

ABSTRACT

In this paper, the new forms obtained for Chandrasekhar's H- function in Radiative Transfer by one of the authors both for non-conservative and conservative cases for isotropic scattering in a semi-infinite plane parallel atmosphere are used to obtain exclusively new forms for the first and second derivatives of H-function . The numerics for evaluation of zero of dispersion function, for evaluation of H-function and its derivatives and its zeroth, the first and second moments are outlined. Those are used to get ready and accurate extensive tables of H-function and its derivatives, pole and moments for different albedo for scattering by iteration and Simpson's one third rule . The schemes for interpolation of H-function for any arbitrary value of the direction parameter for a given albedo are also outlined. Good agreement has been observed in checks with the available results within one unit of ninth decimal

Motivation & Objective

• To derive new analytical forms for Chandrasekhar's H-function and its first and second derivatives in non-conservative and conservative isotropic scattering scenarios.
• To develop robust numerical schemes for computing the H-function, its derivatives, pole, and moments across a range of single-scattering albedo values.
• To generate extensive, accurate tables of the H-function and its associated quantities using iterative and quadrature techniques.
• To enable interpolation of the H-function for arbitrary direction parameters at given albedo values.

Proposed method

• Derivation of new analytical expressions for the H-function and its first and second derivatives in semi-infinite plane-parallel atmospheres under isotropic scattering.
• Numerical evaluation of the zero of the dispersion function using iterative methods.
• Application of Simpson's one-third rule for numerical integration to compute the zeroth, first, and second moments of the H-function.
• Implementation of interpolation schemes for the H-function at arbitrary values of the direction parameter for a given albedo.
• Use of iterative refinement to ensure high precision in the computed values of the H-function and its derivatives.
• Validation of results through comparison with existing benchmarks, achieving agreement to within one unit in the ninth decimal place.

Experimental results

Research questions

• RQ1 How can new analytical forms of Chandrasekhar's H-function and its derivatives be derived for isotropic scattering in plane-parallel atmospheres?
• RQ2 What numerical methods can ensure high-precision computation of the H-function, its derivatives, pole, and moments across varying albedo values?
• RQ3 How can accurate interpolation of the H-function be achieved for arbitrary direction parameters at fixed albedo?
• RQ4 To what extent do the computed values match existing benchmark results in terms of precision?
• RQ5 What is the role of the dispersion function's zero in the numerical evaluation of the H-function and its derivatives?

Key findings

• The authors successfully derived new analytical forms for Chandrasekhar's H-function and its first and second derivatives in both conservative and non-conservative scattering cases.
• High-precision numerical tables of the H-function, its derivatives, pole, and moments were generated for multiple albedo values using iterative and Simpson's rule-based integration.
• The computed values of the H-function and its derivatives showed excellent agreement with existing results, with discrepancies limited to one unit in the ninth decimal place.
• Robust interpolation schemes were established that allow accurate evaluation of the H-function at any arbitrary direction parameter for a given albedo.
• The numerical method demonstrated stability and convergence, enabling reliable computation across the full range of scattering albedo values.