# Automatic Differentiation using Operator Overloading (ADOO) for implicit resolution of hyperbolic single phase and two-phase flow models

## TL;DR

This paper introduces a flexible automatic differentiation method using operator overloading to simplify the derivation of implicit schemes for complex hyperbolic flow models in computational fluid dynamics.

## Contribution

It presents a novel, adaptable approach to automatic differentiation that eases the implicit resolution of complex flow equations in CFD.

## Key findings

- Successfully applied to compressible Euler equations
- Extended to two-phase flow models in equilibrium and disequilibrium
- Simplifies derivative computation for implicit schemes

## Abstract

Implicit time integration schemes are widely used in computational fluid dynamics numerical codes to speed-up computations. Indeed, implicit schemes usually allow for less stringent time-step stability constraints than their explicit counterpart. The derivation of an implicit scheme is however a challenging and time-consuming task, increasing substantially with the model equations complexity since this method usually requires a fairly accurate evaluation of the spatial scheme's matrix Jacobian. This article presents a flexible method to overcome the difficulties associated to the computation of the derivatives, based on the forward mode of automatic differentiation using operator overloading. Flexibility and simplicity of the method are illustrated through implicit resolution of various flow models of increasing complexity such as the compressible Euler equations, a two-phase flow model in full equilibrium (Le Martelot, et al., 2014) and a symmetric variant (Saurel , et al., 2003) of the two-phase flow model of (Baer \& Nunziato, 1986) dealing with mixtures in total disequilibrium.

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Source: https://tomesphere.com/paper/1904.02136