Time-adaptive PIROCK method with error control for multi-fluid and single-fluid magnetohydrodynamics systems

¹ Lockheed Martin Solar & Astrophysics Laboratory, 3251 Hanover St, Palo Alto, CA 94304, USA
² Bay Area Environmental Research Institute, NASA Research Park, Moffett Field, CA 94035, USA
³ Section of Mathematics, University of Geneva, Switzerland
⁴ Rosseland Centre for Solar Physics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway
⁵ Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway
⁶ SETI Institute, 339 Bernardo Ave Mountain View, CA 94035, USA

^★ Corresponding author

Received: 23 September 2024
Accepted: 6 February 2025

Abstract

Context. The solar atmosphere is a complex environment characterized by numerous species with varying ionization states, which are particularly evident in the chromosphere, where the significant variations in ionization degree occur. This region transitions from highly collisional to weakly collisional states that exhibit diverse plasma state transitions influenced by varying magnetic strengths and collisional properties. The complexity of processes in the solar atmosphere introduces substantial numerical stiffness in multi-fluid models, leading to severe timestep restrictions in standard time integration methods.

Aims. To address the computational challenges, new numerical methods are essential. These methods must effectively manage the diverse timescales associated with multi-fluid and multi-physics models, including convection, dissipative effects, and reactions. The widely used time operator splitting technique provides a straightforward approach but necessitates careful timestep management to prevent stability issues and errors. Despite studies on splitting errors, their impact on solar and stellar astrophysics has largely been overlooked.

Methods. We focus on a multi-fluid multi-species model, which poses significant challenges for time integration. We propose a second-order Partitioned Implicit-Explicit Orthogonal Runge–Kutta (PIROCK) method. This method combines efficient explicit stabilized and implicit integration techniques while employing variable time-stepping with error control.

Results. Compared to a standard third-order explicit time integration method and a first-order Lie splitting approach as considered recently, the PIROCK method demonstrates robust advantages in terms of accuracy, numerical stability, and computational efficiency. For the first time, our results reveal PIROCK’s capability to effectively solve multi-fluid problems with unprecedented efficiency. Preliminary results on chemical fractionation, combined with this efficient method, represent a significant step toward understanding the well-known first-ionization-potential effect in the solar atmosphere.