# Euler time integrator for unsteady flows

**URL:** <https://pyfr.discourse.group/t/euler-time-integrator-for-unsteady-flows/712>\
**Category:** Cases\
**Created:** [24 October 2022 00:54 UTC](https://pyfr.discourse.group/t/euler-time-integrator-for-unsteady-flows/712 "2022-10-24T00:54:24Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![YoonSuKyung](https://avatars.discourse-cdn.com/v4/letter/y/5e9695/32.png) [@YoonSuKyung](https://pyfr.discourse.group/u/YoonSuKyung)\
**Post date:** [24 October 2022 00:54 UTC](https://pyfr.discourse.group/t/euler-time-integrator-for-unsteady-flows/712/1 "2022-10-24T00:54:24Z")

</div>

I used dual time integrator (backward-euler / euler method).  
Then i compared time step effect using 3 time steps.

If three results was scaled in time, the results were (almost) the same. Therefore, it seemed like the result of a steady state simulation.  
Is it correct that the **backward-euler(euler) method is working in unsteady simulation?**

 ![dt](https://global.discourse-cdn.com/free1/uploads/pyfr/original/1X/f9fa6de7819ab9408ebe9236d80abb98945d79dc.jpeg)

Most of the papers use the Runge-kutta method, is there a reason not to use the Euler method? or does the **Euler method exist for initialization?**

And when i simulate compressible N-S / incompressible N-S, **are dt and pseudo-dt physical time ?** or only determine cfl number?

---

<div class="post-metadata">

**Author:** ![WillT](https://avatars.discourse-cdn.com/v4/letter/w/bc79bd/32.png) [@WillT](https://pyfr.discourse.group/u/WillT)\
**Post date:** [24 October 2022 09:37 UTC](https://pyfr.discourse.group/t/euler-time-integrator-for-unsteady-flows/712/2 "2022-10-24T09:37:06Z")

</div>

Euler methods aren’t very accurate and this can impact there convergence. A good paper that considers this is by Lai Wang et al. [https://arxiv.org/pdf/1904.04825.pdf](https://arxiv.org/pdf/1904.04825.pdf)

Ultimately, if you perform a step forwards you what to get the most for your money and for incompressible Navier–Stokes using dual time stepping, that often means using a DIRK scheme. Euler is good to have as it very dissipative which can be very useful in testing or, as you say, initialisation. Or I know in more complex physics, say multi-phase with mateirals etc, they like it due to it’s dissipation.

When using dual time stepping dt is the physical time step and pseudo-dt is the pseudo time step. When usinf the standard time integrator there is only dt and this is the physical time step.