Towards efficient, resistive, multi-fluid merger simulations

• Alex Wright
• Ian Hawke

• github.com/AlexJamesWright
• orcid.org/0000-0002-5953-4221
• STAG, University of Southampton
• Next Generation Computational Modelling Group
• EPSRC grant EP/ L015382/1

• More realistic models can be stiff;
• Require implicit schemes for stability:
  • E.g. IMEX (Pareschi & Russo 2004).

$$ \begin{align} \partial_t q = \mathcal{F}(q) + \frac{1}{\epsilon} \mathcal{S}(q) \end{align} $$

$$ \begin{align} q^{(i)} = q^n + \Delta t G(q^n) + \Delta t H(q^{(i)}) \end{align} $$

Performance

• Performance depends on timescale, $\epsilon$;
• Expect $5 \times$ slow down.

Massively parallel processors

Seeing wide use in scientific software:

• $\sim 100 \times$ greater FLOPS than CPU;
• Common on HPC clusters;
• FLOPS increasing faster than CPU.

GPU-capable codes

Some examples include:

• Wong (2011): MHD (non-relativistic);
• Zhang (2018): AMR-MHD (non-relativistic);
• Zink (2011): GRMHD (static spacetime).

  
  All examples use (R)MHD and explicit numerical methods.
  
  How will implicit schemes transfer?

METHOD:

• Lightweight, multi-physics MHD code;

• Ideal and resistive single and two-fluid models;
• Explicit or implicit integration;
• GPU capable;
• Open source through GitHub.
github.com/AlexJamesWright/METHOD

Porting to GPU

• Time integration is computationally expensive;
• Primitive recovery in each IMEX step;
• Demands correct use of shared memory.

Performance

• Parallel speed up of $21\times$ versus CPU
• Further optimisations are possible/necessary
• More details in Wright & Hawke (2019) - 'Resistive and multi-fluid RMHD on graphics processing units'.

Model extensions

REGIME:

A resistive extension to ideal MHD^*

^*in preparation

Chapman-Enskog expansion:

• Start from resistive MHD, $\overline{q}$ and $q$ are stiff and non-stiff:
  
  $$ \begin{align} \partial_t q + \partial_x f(q, \overline{q}) = s(q, \overline{q}) \end{align} $$

• Expand equations around equilibrium (ideal) solution
  
  $$ \begin{align} \overline{q} = \overline{q}_0 + \epsilon \overline{q}_1 + \mathcal{O}(\epsilon^2) \end{align} $$

Extension

To first order in $\epsilon$:

$$ \begin{align} \partial_t q + \partial_x f = s + \epsilon \partial_x D \end{align} $$

Results:

• Extremely good agreement with resistive MHD;
• Expected convergence with conductivity.

Results:

• Many factors faster than full model.

Whats next?

• Finalizing 'A resistive extension to ideal MHD';
• Apply to resistive GRMHD;
• Explore application to multi-fluid equations.