The proposed discretization scheme, which is designed for the strongly anisotropic heat transport equation in magnetized plasmas, can be extended to handle more complex magnetic field topologies, including those with stochastic field lines of infinite length, by incorporating several key strategies.
Generalization of Magnetic Field Representation: The current scheme relies on a well-defined magnetic field structure, typically represented by a flux function. To accommodate stochastic fields, the discretization can be adapted to utilize a more flexible representation of the magnetic field, such as a vector potential formulation or a direct representation of the magnetic field lines. This would allow the scheme to account for the complex geometry and dynamics of stochastic fields.
Adaptive Mesh Refinement: Implementing adaptive mesh refinement techniques can enhance the scheme's ability to resolve regions of high magnetic field complexity. By dynamically adjusting the grid resolution based on the local characteristics of the magnetic field, the discretization can maintain accuracy in areas where field lines are densely packed or exhibit chaotic behavior.
Asymptotic-Preserving (AP) Methods: While the current scheme is not asymptotic-preserving, integrating AP methods could help manage the numerical challenges associated with varying anisotropies. These methods ensure that the numerical solution remains stable and convergent as the anisotropy ratio changes, which is particularly important in stochastic magnetic fields where the transport characteristics can vary significantly.
Nonlinear Solver Adaptations: The nonlinear solver strategy, particularly the Jacobian-free Newton-Krylov (JFNK) method, can be enhanced to better handle the complexities introduced by stochastic fields. This could involve developing specialized preconditioners that are tailored to the specific characteristics of the magnetic topology, ensuring efficient convergence even in challenging scenarios.
By implementing these strategies, the discretization scheme can be made more robust and versatile, allowing it to effectively simulate heat transport in plasmas with complex magnetic field topologies.