Hyperbolic Conduction: A Fast, Physical Conduction Model Implemented in Smoothed Particle Hydrodynamics

TL;DR

This paper introduces a hyperbolic thermal conduction model implemented in smoothed particle hydrodynamics, enabling faster simulations by allowing larger time steps while maintaining physical accuracy.

Contribution

It presents the first implementation of hyperbolic conduction in SPH, including a novel dissipation scheme and validation through multiple test cases.

Findings

Hyperbolic conduction allows larger explicit time steps.

The scheme is stable and significantly faster than traditional methods.

Longer relaxation times recover hydrodynamical results.

Abstract

We present the first implementation of hyperbolic thermal conduction in smoothed particle hydrodynamics (SPH). Hyperbolic conduction is a physically-motivated alternative to traditional, parabolic conduction. It incorporates a relaxation time, which ensures that heat propagates no faster than a physical signal speed. This allows for larger, Courant like, time steps for explicit schemes. Numerical solutions of the hyperbolic conduction equations require added dissipation to remain stable at discontinuities and we present a novel scheme for this. Test cases include a simple step, the Sod shock tube, the Sedov-Taylor blast, and a super bubble. We demonstrate how longer relaxation times limit conduction, recovering the purely hydrodynamical results, while short relaxation times converge on the parabolic conduction result. We demonstrate that our scheme is stable with explicit Courant-like time steps and can be orders of magnitude faster than explicit parabolic conduction, depending on the application.