3D Euler Equation

Mathematical description

The PDEs is given by

$\begin{aligned} \rho_t + \nabla \cdot (\rho\bold{v})&=0, \\ \rho(\partial_t\bold{v}+\bold{v} \cdot \nabla\bold{v})+\nabla p&=0, \\ \partial_t[\epsilon+\frac{\rho v^2}{2}]+\nabla \cdot [(\epsilon+p+\frac{\rho v^2}{2})\bold{v}]&=0, \end{aligned}$

where $\epsilon=3p/2$ and $v=||\bold{v}||$ .

The domain is defined as

$(\bold{x},t) \in (0,1)^3 \times [0,1]$

The initial condition is

$\bold{v}(\bold{x},0)=\sum_{i=1}^nA_i\sin(k_i\bold{x}+\phi_i),$

where $n=4,A_i=c_s/|k_i|^2$ and $c_S$ is the speed of sound.

All PDEs are subject to periodic boundary conditions due to the domain being defined as a periodic cubic domain $(0,1)^3$ . Periodic boundary conditions imply that the values of the variables wrap around the boundaries of the domain, maintaining periodicity in the solution.

Dataset file information

File name: 3D_CFD_rand_M1.0_periodic_Train.hdf5

File hierarchy:

/
|-- Vx
|-- Vy
|-- Vz
|-- density
|-- pressure
|-- x-coordinate
|-- y-coordinate
|-- z-coordinate
|-- t-coordinate

Datasets information:

• Vx
• Shape: (100, 21, 128, 128, 128)
• Data Type: float
• Description: velocity component in the x-direction
• Vy
• Shape: (100, 21, 128, 128, 128)
• Data Type: float
• Description: velocity component in the y-direction
• Vz
• Shape: (100, 21, 128, 128, 128)
• Data Type: float
• Description: velocity component in the z-direction
• density
• Shape: (100, 21, 128, 128, 128)
• Data Type: float
• Description: density
• pressure
• Shape: (100, 21, 128, 128, 128)
• Data Type: float
• Description: gas pressure
• x-coordinate
• Shape: (128,)
• Data Type: float
• Description: x-axis spatial coordinates
• y-coordinate
• Shape: (128,)
• Data Type: float
• Description: y-axis spatial coordinates
• z-coordinate
• Shape: (128,)
• Data Type: float
• Description: z-axis spatial coordinates
• t-coordinate
• Shape: (22,)
• Data Type: float
• Description: temporal coordinates

Data generation method

We use the data generation code from PDEBench to generate equation data. The numerical solution is calculated with the temporally and spatially 2nd-order HLLC scheme with the MUSCL method for the inviscid part, and the central difference scheme for the viscous part. Specifically, we run the shell script: bash data_gen/data_gen_NLE/CompressibleFluid/run_trainset_3D.sh, which will in turn run the python script data_gen/data_gen_NLE/CompressibleFluid/CFD_multi_Hydra.py and update nn and key args in config file data_gen/data_gen_NLE/CompressibleFluid/config/args/3D_Multi_Rand.yaml . Finally, we run the data merge script python data_gen/data_gen_NLE/Data_Merge.py to merge data.