ANNOUNCEMENT:

Dear all,

as we have noticed an increased interest in Jexpresso (this is amazing, thank you for this!), we added a script to automate the installation of Jexpresso.

Moreover, we have added ERA5 reading capabilities and the branch for that will be merged soon into master, as well as an improved handling of boundary conditions from the user's perspective.

Thank you your interest in the Jexpresso!

<img src="./assets/logo-ext2.png" width="500" title="JEXPRESSO logo">

JEXPRESSO

A CPU and GPU research software for the numerical solution of a system of arbitrary conservation laws using continuous spectral elements and finite differences in 1D, 2D, 3D. DISCLAIMER: this will always be WIP! Contact us to join the team of developers!

Suggested Julia version: 1.11.2 or higher.

Installation:

Use the installer as described in INSTALLATION.md

The installer will install all the necessary packages with the correct versions.

Jexpresso uses a few packages whose latest version may be incompatible. Please, enfornce the installation of the following versions:

MPI 0.20.22
MPIPreferences 0.1.11
PackageCompiler 2.2.1
Thermodynamics 0.12.7
PrettyTables 2.4.0
Crayons 4.1.1
UnicodePlots 3.7.2
Gridap v0.18.12
GridapDistributed v0.4.7
GridapGmsh v0.7.2
GridapP4est v0.3.11

If you use Jexpresso please drop us a line to let us know. We'd like to add a link to your paper or work on this page.

Please cite Jexpresso using:

@inproceedings{marrasJexpresso,
  author    = {S. Marras and Y. Tissaoui and H. Wang and S. Stechmann}
  title     = {JEXPRESSO V0. 1: A JULIA-LANGUAGE, USER-FRIENDLY, MULTI-PHYSICS PARALLEL SOLVER FOR THE SOLUTION OF CONSERVATIONS LAWS ON CPUs AND GPUs.},
  booktitle = {Proceedings of the 36th Parallel CFD international conference 2025},
  year      = {2025},
  address   = {Merida, Yucatan, Mexico},
  month     = {November},
  organization = {UNAM},
}

@article{tissaoui2024,
  author = {Y. Tissaoui and J. F. Kelly and S. Marras}
  title = {Efficient Spectral Element Method for the Euler Equations on Unbounded Domains},
  volume ={487},
  pages={129080},
  year = {2024},
  journal = {App. Math. Comput.},
}

Equations:

Jexpresso uses arbitrarily high-order (3rd and above) continuous spectral elements to solve

$$\frac{\partial \bf q}{\partial t} + \sum_{i=1}^{nd}\nabla\cdot{{\bf F}_i({\bf q})} = \mu\nabla^2{\bf q} + {\bf S}({\bf q}) + ~{\rm b.c.}$$

where the vectors ${\bf q}$, ${\bf F}$, and ${\bf S}$ are problem-dependent as shown below, and are taken to be zero vectors of the appropriate size when not explicitly stated otherwise.

The Julia package DifferentialEquations.jl is used for time discretization and stepping.

In order, we provide tests and results for the following equations:

1D wave equation:

$${\bf q}=\begin{bmatrix} u \ v \end{bmatrix}\quad {\bf F}=\begin{bmatrix} v\ u \end{bmatrix}$$

2: 1D shallow water:

$${\bf q}=\begin{bmatrix} h \ u \end{bmatrix}\quad {\bf F}=\begin{bmatrix} Uh + Hu\ gh + Uu \end{bmatrix},$$

where $H$ and $U$ are a reference height and velocity, respectively.

2D Helmholtz:

$${\bf S}=\begin{bmatrix} \alpha^2 u + f(x,z) \end{bmatrix}\quad \mu\nabla^2{\bf q}=\mu\begin{bmatrix} u_{xx} + u_{zz} \end{bmatrix},$$

for a constant value of $\alpha$ and $\mu$, which are case-dependent.

2D scalar advection-diffusion:

$${\bf q}=\begin{bmatrix} q\ \end{bmatrix}\quad {\bf F}=\begin{bmatrix} qu\ \end{bmatrix}\quad {\bf F}=\begin{bmatrix} qv\ \end{bmatrix}\quad \mu\nabla^2{\bf q}=\mu\begin{bmatrix} q_{xx} + q_{zz} \end{bmatrix},$$

2D Euler equations of compressible flows with gravity and N passive chemicals $c_i, \forall i=1,...,N$

$${\bf q}=\begin{bmatrix} \rho \ \rho u\ \rho v\ \rho \theta\ \rho c1\ ...\ \rho cN \end{bmatrix}\quad {\bf F1}=\begin{bmatrix} \rho u\ \rho u^2 + p\ \rho u v\ \rho u \theta\ \rho u c1\ ...\ \rho u cN \end{bmatrix}\quad {\bf F2}=\begin{bmatrix} \rho v\ \rho v u\ \rho v^2 + p\ \rho v \theta\ \rho v c1\ ...\ \rho v cN \end{bmatrix}\quad {\bf S}=\begin{bmatrix} 0\ 0\ -\rho g\ 0\ 0\ ...\ 0 \end{bmatrix}\quad \mu\nabla^2{\bf q}=\mu\begin{bmatrix} 0\ u_{xx} + u_{zz}\ v_{xx} + v_{zz}\ \theta_{xx} + \theta_{zz}\ c1_{xx} + c1_{zz}\ ...\ cN_{xx} + cN_{zz} \end{bmatrix}.$$

3D Euler equations of compressible flows with gravity

$${\bf q}=\begin{bmatrix} \rho \ \rho u\ \rho v\ \rho w\ \rho \theta\ \end{bmatrix}\quad {\bf F}1=\begin{bmatrix} \rho u\ \rho u^2 + p\ \rho u v\ \rho u w\ \rho u \theta\ \end{bmatrix}\quad {\bf F}2=\begin{bmatrix} \rho v\ \rho v u\ \rho v^2 + p\ \rho v w\ \rho v \theta\ \end{bmatrix}\quad {\bf F3}=\begin{bmatrix} \rho w\ \rho w u\ \rho w v\ \rho w^2 + p\ \rho w \theta\ \end{bmatrix}\quad {\bf S}=\begin{bmatrix} 0\ 0\ 0\ -\rho g\ 0\ \end{bmatrix}\quad \mu\nabla^2{\bf q}=\mu\begin{bmatrix} 0\ u_{xx} + u_{yy} + u_{zz}\ v_{xx} + v_{yy} + v_{zz}\ w_{xx} + w_{yy} + w_{zz}\ \theta_{xx} + \theta_{yy} + \theta_{zz}\ \end{bmatrix}.$$

If you are interested in contributing, please get in touch: Simone Marras, Yassine Tissaoui, Hang Wang

Some notes on using JEXPRESSO

To install and run the code assume Julia 1.11.2

Start by cloning Jexpresso and JexpressoMeshes:

git clone https://github.com/smarras79/Jexpresso.git

git clone https://github.com/smarras79/JexpressoMeshes.git

cd Jexpresso

ln -s ../JexpressoMeshes/meshes .

Setup with CPUs

>> cd $JEXPRESSO_HOME
>> julia --project=. -e "using Pkg; Pkg.instantiate(); Pkg.API.precompile()"

followed by the following:

Push problem name to ARGS You need to do this only when you run a new problem

push!(empty!(ARGS), EQUATIONS::String, EQUATIONS_CASE_NAME::String);
include("./src/Jexpresso.jl")

PROBLEM_NAME is the name of your problem directory as $JEXPRESSO/problems/equations/problem_name
PROBLEM_CASE_NAME is the name of the subdirectory containing the specific setup that you want to run:

The path would look like $JEXPRESSO/problems/equations/PROBLEM_NAME/PROBLEM_CASE_NAME

Shallow cumuli:

Example of shallow cumuli simulations (right) for the type of Barbados clouds shown on the left: (picture taken from P. Blossey webpage from U. Washington)

Turbulent ABL

Example of coarse simulation of the turbulent atmospheric boundary layer. Domain size: 10240m X 10240m X 3000m using 64x64x24 spectral elements of order 4. Surface and SGS: Monin-Obukhov Similarity Theory model with Richardson-corrected Smagorinsky. <img src="assets/ABLfullDomain.gif" alt="Markdown icon" style="float: left; margin-right: 5px;" />

Examples available in this branch:

Example 1: to solve the 2D Euler equations with buoyancy and two passive tracers defined in problems/equations/CompEuler/thetaTracers you would do the following:

push!(empty!(ARGS), "CompEuler", "thetaTracers");
include("./src/Jexpresso.jl")

Example 2: to solve the 3D Euler equations with buoyancy defined in problems/equations/CompEuler/3d you would do the following:

push!(empty!(ARGS), "CompEuler", "3d");
include("./src/Jexpresso.jl")

Example 3: to solve the 1D wave equation defined in problems/equations/CompEuler/wave1d you would do the following:

push!(empty!(ARGS), "CompEuler", "wave1d");
include("./src/Jexpresso.jl")

For ready to run tests, there are the currently available equations names:

CompEuler (option with total energy and theta formulation)

The code is designed to create any system of conservsation laws. See CompEuler/case1 to see an example of each file. Details will be given in the documentation (still WIP). Write us if you need help.

More are already implemented but currently only in individual branches. They will be added to master after proper testing.

Laguerre semi-infinite element test suite

This section contains instructions to run all of the test cases presented in

@article{tissaoui2024,
  author = {Y. Tissaoui and J. F. Kelly and S. Marras}
  title = {Efficient Spectral Element Method for the Euler Equations on Unbounded Domains},
  volume ={487},
  pages={129080},
  year = {2024},
  journal = {App. Math. Comput.},
}

Test 1: 1D wave equation with Laguerre semi-infinite element absorbing layers

The problem is defined in [problems/CompEuler/wave1d_lag](https://github.com

Jexpresso

Install / Use

README