# Euler Multi-quadrants - Riemann problems (compressible, inviscid fluid)

**NOTE:** this dataset is distributed in two separate datasets: `euler_multi_quadrants_openBC` with open boundary conditions and `euler_multi_quadrants_periodicBC` with periodic boundary conditions.

**One line description of the data:**  Evolution of different gases starting with piecewise constant initial data in quadrants.

**Longer description of the data:**  The evolution can give rise to shocks, rarefaction waves, contact discontinuities, interaction with each other and domain walls.

**Associated paper**: [Paper](https://epubs.siam.org/doi/pdf/10.1137/S1064827595291819?casa_token=vkASCwD4WngAAAAA:N0jy0Z6tshitF10_YRTlZzU-P7mAiPFr3v58sw7pmRsZOarAi824-b1CWhOQts1rvaG3YpJisw).

**Domain experts**: [Marsha Berger](https://cs.nyu.edu/~berger/)(Flatiron Institute & NYU), [Ruben Ohana](https://rubenohana.github.io/) (CCM, Flatiron Institute & Polymathic AI), [Michael McCabe](https://mikemccabe210.github.io/) (Polymathic AI).

**Code or software used to generate the data**: [Clawpack (AMRClaw)](http://www.clawpack.org/).

**Equation**: Euler equations for a compressible gas:

$$
\begin{align*}
U_t + F(U)_x + G(U)_y &= 0 \\
\textrm{where} \quad U = \begin{bmatrix} \rho \\ \rho u \\ \rho v \\ e \end{bmatrix}, \quad F(U) = \begin{bmatrix} \rho u \\ \rho u^2 + p \\ \rho u v \\ u(e + p) \end{bmatrix},& \quad G(U) = \begin{bmatrix} \rho v \\ \rho u v \\ \rho v^2 + p \\ v(e + p) \end{bmatrix}, \quad \\ e = \frac{p}{(\gamma - 1)} + \frac{\rho (u^2 + v^2)}{2}&, \quad p = A\rho^{\gamma}.
\end{align*}
$$

with $\rho$ the density, $u$ and $v$ the $x$ and $y$ velocity components, $e$ the energy, $p$ the pressure, $\gamma$ the gas constant, and $A>0$ is a function of entropy.

![Gif](https://users.flatironinstitute.org/~polymathic/data/the_well/datasets/euler_multi_quadrants_openBC/gif/density_normalized.gif)

| Dataset    | FNO | TFNO  | Unet | CNextU-net
|:-:|:-:|:-:|:-:|:-:|
| `euler_multi-quadrants_periodicBC`  | 0.4081  | 0.4163 |0.1834 |$\mathbf{0.1531}$|

Table: VRMSE metrics on test sets (lower is better). Best results are shown in bold. VRMSE is scaled such that predicting the mean value of the target field results in a score of 1.

## About the data

**Dimension of discretized data:** 100 timesteps of 512x512 images.

**Fields available in the data:** density (scalar field), energy (scalar field), pressure (scalar field), momentum (vector field).

**Number of trajectories:** 500 per set of parameters, 10 000 in total.

**Estimated size of the ensemble of all simulations:** 5.17 TB.

**Grid type:** uniform, cartesian coordinates.

**Initial conditions:** Randomly generated initial quadrants.

**Boundary conditions:** Periodic or open.

**Simulation time-step:** variable.

**Data are stored separated by ($\Delta t$):** 0.015s (1.5s for 100 timesteps).

**Total time range ($t_{min}$ to $t_{max}$):** $t_{min} = 0$, $t_{max}=1.5s$.

**Spatial domain size ($L_x$, $L_y$):** $L_x = 1$ and  $L_y = 1$.

**Set of coefficients or non-dimensional parameters evaluated:** all combinations of $\gamma$ constant of the gas at a certain temperature: $\gamma=${1.13,1.22,1.3,1.33,1.365,1.4,1.404,1.453,1.597,1.76} and boundary conditions: {extrap, periodic}.

**Approximate time to generate the data:** 80 hours on 160 CPU cores for all data (periodic and open BC).

**Hardware used to generate the data and precision used for generating the data:** Icelake nodes, double precision.

## What is interesting and challenging about the data:

**What phenomena of physical interest are catpured in the data:** capture the shock formations and interactions. Multiscale shocks.

**How to evaluate a new simulator operating in this space:** the new simulator should predict the shock at the right location and time, and the right shock strength, as compared to a pressure gauge monitoring the exact solution.

Please cite the associated paper if you use this data in your research:

```
@article{mandli2016clawpack,
  title={Clawpack: building an open source ecosystem for solving hyperbolic PDEs},
  author={Mandli, Kyle T and Ahmadia, Aron J and Berger, Marsha and Calhoun, Donna and George, David L and Hadjimichael, Yiannis and Ketcheson, David I and Lemoine, Grady I and LeVeque, Randall J},
  journal={PeerJ Computer Science},
  volume={2},
  pages={e68},
  year={2016},
  publisher={PeerJ Inc.}
}
```