# Multimaterial Heat Flow Verification

## TL;DR

This paper presents a rigorous verification of the FLAG hydrocode's heat flow modeling using new exact solutions for multimaterial heat diffusion, focusing on grid resolution and model accuracy.

## Contribution

It introduces a set of generalized 2D heat flow solutions for code verification and applies ExactPack for convergence analysis of multimaterial heat diffusion models.

## Key findings

- Verification of FLAG code with new exact solutions
- Comparison of naive averaging models and sophisticated methods
- Convergence studies demonstrating model accuracy

## Abstract

Multimaterial heat diffusion can be a challenging numerical problem when the material boundaries are misaligned with the numerical grid. Even when the boundaries start out aligned, they typically become misaligned through hydrodynamic motion. There are usually a number of methods for handling multimaterial cells in any given hydro code. One of the simplest methods is to replace the multimaterial cell by an average single-material cell whose heat capacity and conductivity are averages over the constituent materials. One can further refine this model by using either the arithmetic or harmonic averages, thereby providing two distinct (albeit naive) multimaterial models for the arithmetic and harmonic averages. More sophisticated models typically involve a surrogate mesh of some kind, as with the thin mesh and static condensation methods. In this paper, we perform rigorous code verification of the multiphysics hydrocode FLAG, including grid resolution studies. We employ a number of newly constructed 2D heat flow solutions that generalize the standard {\em planar sandwich} solution, and this paper offers a smorgasbord of exact solutions for heat flow verification. To perform the analyses and to produce the corresponding convergence plots, we employ the code verification tool ExactPack.