17 research outputs found
Discontinuities in numerical radiative transfer
Observations and magnetohydrodynamic simulations of solar and stellar
atmospheres reveal an intermittent behavior or steep gradients in physical
parameters, such as magnetic field, temperature, and bulk velocities. The
numerical solution of the stationary radiative transfer equation is
particularly challenging in such situations, because standard numerical methods
may perform very inefficiently in the absence of local smoothness. However, a
rigorous investigation of the numerical treatment of the radiative transfer
equation in discontinuous media is still lacking. The aim of this work is to
expose the limitations of standard convergence analyses for this problem and to
identify the relevant issues. Moreover, specific numerical tests are performed.
These show that discontinuities in the atmospheric physical parameters
effectively induce first-order discontinuities in the radiative transfer
equation, reducing the accuracy of the solution and thwarting high-order
convergence. In addition, a survey of the existing numerical schemes for
discontinuous ordinary differential systems and interpolation techniques for
discontinuous discrete data is given, evaluating their applicability to the
radiative transfer problem
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Numerical solutions to linear transfer problems of polarized radiation III. Parallel preconditioned Krylov solver tailored for modeling PRD effects
Context. The polarization signals produced by the scattering of anistropic radiation in strong resonance lines encode important information about the elusive magnetic fields in the outer layers of the solar atmosphere. An accurate modeling of these signals is a very challenging problem from the computational point of view, in particular when partial frequency redistribution (PRD) effects in scattering processes are accounted for with a general angle-dependent treatment. Aims. We aim at solving the radiative transfer problem for polarized radiation in nonlocal thermodynamic equilibrium conditions, taking angle-dependent PRD effects into account. The problem is formulated for a two-level atomic model in the presence of arbitrary magnetic and bulk velocity fields. The polarization produced by scattering processes and the Zeeman effect is considered. Methods. The proposed solution strategy is based on an algebraic formulation of the problem and relies on a convenient physical assumption, which allows its linearization. We applied a nested matrix-free GMRES iterative method. Effective preconditioning is obtained in a multifidelity framework by considering the light-weight description of scattering processes in the limit of complete frequency redistribution (CRD). Results. Numerical experiments for a one-dimensional (1D) atmospheric model show near optimal strong and weak scaling of the proposed CRD-preconditioned GMRES method, which converges in few iterations, independently of the discretization parameters. A suitable parallelization strategy and high-performance computing tools lead to competitive run times, providing accurate solutions in a few minutes. Conclusions. The proposed solution strategy allows the fast systematic modeling of the scattering polarization signals of strong resonance lines, taking angle-dependent PRD effects into account together with the impact of arbitrary magnetic and bulk velocity fields. Almost optimal strong and weak scaling results suggest that this strategy is applicable to realistic 3D models. Moreover, the proposed strategy is general, and applications to more complex atomic models are possible
Assessment of the CRD approximation for the observer's frame RIII redistribution matrix
Approximated forms of the RII and RIII redistribution matrices are frequently
applied to simplify the numerical solution of the radiative transfer problem
for polarized radiation, taking partial frequency redistribution (PRD) effects
into account. A widely used approximation for RIII is to consider its
expression under the assumption of complete frequency redistribution (CRD) in
the observer frame (RIII CRD). The adequacy of this approximation for modeling
the intensity profiles has been firmly established. By contrast, its
suitability for modeling scattering polarization signals has only been analyzed
in a few studies, considering simplified settings.
In this work, we aim at quantitatively assessing the impact and the range of
validity of the RIII CRD approximation in the modeling of scattering
polarization. Methods. We first present an analytic comparison between RIII and
RIII CRD. We then compare the results of radiative transfer calculations, out
of local thermodynamic equilibrium, performed with RIII and RIII CRD in
realistic 1D atmospheric models. We focus on the chromospheric Ca i line at
4227 A and on the photospheric Sr i line at 4607 A
The impact of angle-dependent partial frequency redistribution on the scattering polarization of the solar Na i D lines
The long-standing paradox of the linear polarization signal of the Na i D1
line was recently resolved by accounting for the atom's hyperfine structure and
the detailed spectral structure of the incident radiation field. That modeling
relied on the simplifying angle-averaged (AA) approximation for partial
frequency redistribution (PRD) in scattering, which potentially neglects
important angle-frequency couplings. This work aims at evaluating the
suitability of a PRD-AA modeling for the D1 and D2 lines through comparisons
with general angle-dependent (AD) PRD calculations, both in the absence and
presence of magnetic fields. We solved the radiative transfer problem for
polarized radiation in a one-dimensional semi-empirical atmospheric model with
microturbulent and isotropic magnetic fields, accounting for PRD effects,
comparing PRD-AA and PRD-AD modelings. The D1 and D2 lines are modeled
separately as two-level atomic system with hyperfine structure. The numerical
results confirm that a spectrally structured radiation field induces linear
polarization in the D1 line. However, the PRD-AA approximation greatly impacts
the Q/I shape, producing an antisymmetric pattern instead of the more symmetric
PRD-AD one, while presenting a similar sensitivity to magnetic fields between
10 and 200 G. Under the PRD-AA approximation, the Q/I profile of the D2 line
presents an artificial dip in its core, which is not found for the PRD-AD case.
We conclude that accounting for PRD-AD effects is essential to suitably model
the scattering polarization of the Na i D lines. These results bring us closer
to exploiting the full diagnostic potential of these lines for the elusive
chromospheric magnetic fields
Formal solutions for polarized radiative transfer. III. Stiffness and instability
Efficient numerical approximation of the polarized radiative transfer equation is challenging because this system of ordinary differential equations exhibits stiff behavior, which potentially results in numerical instability. This negatively impacts the accuracy of formal solvers, and small step-sizes are often necessary to retrieve physical solutions. This work presents stability analyses of formal solvers for the radiative transfer equation of polarized light, identifies instability issues, and suggests practical remedies. In particular, the assumptions and the limitations of the stability analysis of Runge-Kutta methods play a crucial role. On this basis, a suitable and pragmatic formal solver is outlined and tested. An insightful comparison to the scalar radiative transfer equation is also presented
Numerical solutions to linear transfer problems of polarized radiation
Context. Numerical solutions to transfer problems of polarized radiation in solar and stellar atmospheres commonly rely on stationary iterative methods, which often perform poorly when applied to large problems. In recent times, stationary iterative methods have been replaced by state-of-the-art preconditioned Krylov iterative methods for many applications. However, a general description and a convergence analysis of Krylov methods in the polarized radiative transfer context are still lacking.
Aims. We describe the practical application of preconditioned Krylov methods to linear transfer problems of polarized radiation, possibly in a matrix-free context. The main aim is to clarify the advantages and drawbacks of various Krylov accelerators with respect to stationary iterative methods and direct solution strategies.
Methods. After a brief introduction to the concept of Krylov methods, we report the convergence rate and the run time of various Krylov-accelerated techniques combined with different formal solvers when applied to a 1D benchmark transfer problem of polarized radiation. In particular, we analyze the GMRES, BICGSTAB, and CGS Krylov methods, preconditioned with Jacobi, (S)SOR, or an incomplete LU factorization. Furthermore, specific numerical tests were performed to study the robustness of the various methods as the problem size grew.
Results. Krylov methods accelerate the convergence, reduce the run time, and improve the robustness (with respect to the problem size) of standard stationary iterative methods. Jacobi-preconditioned Krylov methods outperform SOR-preconditioned stationary iterations in all respects. In particular, the Jacobi-GMRES method offers the best overall performance for the problem setting in use.
Conclusions. Krylov methods can be more challenging to implement than stationary iterative methods. However, an algebraic formulation of the radiative transfer problem allows one to apply and study Krylov acceleration strategies with little effort. Furthermore, many available numerical libraries implement matrix-free Krylov routines, enabling an almost effortless transition to Krylov methods