Computational aspects of speed-dependent Voigt and Rautian profiles. (January 2021) - Schreier, Franz; Hochstaffl, Philipp
Highlights: Speed-dependent Voigt and Rautian given by difference of two complex error functions. A single rational approximation (Humlicek or Weideman) of the complex error function provides sufficient accuracy for SDV and SDR. Cancellation-safe implementation of difference defining complex error functions argument. Moderate time overhead compared to Voigt function. Abstract: For accurate line-by-line modeling of molecular cross sections several physical processes "beyond Voigt" have to be considered. For the speed-dependent Voigt and Rautian profiles (SDV, SDR) and the Hartmann-Tran profile the difference w ( i z − ) − w ( i z + ) of two complex error functions (essentially Voigt functions) has to be evaluated where the function arguments z  ±  are given by the sum and difference of two square roots. These two terms describing z  ±  can be huge and the default implementation of the difference can lead to large cancellation errors. First we demonstrate that these problems can be avoided by a simple reformulation of z − . Furthermore we show that a single rational approximation of the complex error function valid in the whole complex plane (e.g. by Humlíček, 1979 or Weideman, 1994) enables computation of the SDV and SDR with four significant digits or better. Our benchmarks indicate that the SDV and SDR functions are about a factor 2.2 slower compared to the Voigt function, but for evaluation of molecular cross sections this time lag does not significantly prolong the overall program execution because speed-dependent parameters are available only for a fraction of strong lines.