1D Euler Solver

Modular solver for the unsteady 1D Euler equations (ideal gas), written in Python/NumPy. Interactive Streamlit interface to compare 59 finite volume schemes on classical Riemann problems and smooth solutions, with mesh convergence studies and Fourier analysis.

Try it online: https://euler-1d-solver.streamlit.app/

Prerequisites

• Python >= 3.10
• NumPy, Pandas, Matplotlib, Plotly, Streamlit

pip install -r requirements.txt

Getting started

streamlit run Solveur_Euler_1D.py

Code architecture

Solveur_Euler_1D.py    Streamlit entry point (main page)
ui_common.py           Plotly styling, constants, shared scheme selectors
pages/
└── 2_Fourier_Analysis.py   Streamlit page: numerical dissipation and dispersion

euler1d/
├── config.py          Configuration dataclasses (gas, mesh, time, problem)
├── physics.py         EOS, primitive/conservative conversions, physical flux
├── riemann.py         Exact Riemann solver (vectorized NumPy)
├── boundary.py        Boundary conditions (ghost cells)
├── solver.py          Time-stepping loop (RK1 to RK5)
├── results.py         Exact solution, error norms, convergence
├── fourier_analysis.py  Numerical dissipation/dispersion analysis
├── test_cases.py      Predefined test cases
└── schemes/
    ├── base.py              Abstract base class NumericalScheme
    ├── flux/
    │   ├── rusanov.py         Rusanov (local Lax-Friedrichs)
    │   ├── hll.py             HLL (2-wave)
    │   ├── hllc.py            HLLC (3-wave)
    │   ├── roe.py             Roe with entropy fix
    │   ├── godunov.py         Godunov (exact solver)
    │   ├── ausm_plus.py       AUSM+ and AUSM+-up
    │   ├── lax_friedrichs.py  Lax-Friedrichs (centered)
    │   ├── lax_wendroff.py    Lax-Wendroff (centered)
    │   └── jst.py             JST (centered)
    └── reconstruction/
        ├── limiters.py    Slope limiters (minmod, van-leer, superbee, mc, van-albada)
        ├── muscl.py       MUSCL reconstruction (order 2)
        ├── eno.py         ENO2 reconstruction (order 2)
        ├── weno3.py       WENO3 JS and Z (order 3)
        └── weno5.py       WENO5 JS and Z (order 5)

Available schemes

Numerical fluxes (order 1)

Scheme	Key	Description
Lax-Friedrichs	lax-friedrichs	Global dissipation, centered scheme
Rusanov	rusanov	Local dissipation (local Lax-Friedrichs)
HLL	hll	2-wave, no contact resolution
HLLC	hllc	3-wave, captures the contact
Roe	roe	Linearization + entropy fix
Roe (no correction)	roe-nc	Roe without entropy fix
Godunov	godunov	Exact Riemann solver
AUSM+	ausm+	Mach/pressure splitting
AUSM+-up	ausm+-up	Improved AUSM+ (low Mach)
Lax-Wendroff	lax-wendroff	2nd order in time and space, centered scheme
JST	jst	Jameson-Schmidt-Turkel, centered scheme

High-order reconstructions

Reconstruction	Key	Order	Composable with
MUSCL	muscl-{flux}	2	Rusanov, HLL, HLLC, Roe, Roe-NC, Godunov, AUSM+, AUSM+-up
ENO2	eno2-{flux}	2	same
WENO3-JS	weno3-{flux}	3	same
WENO3-Z	wenoz3-{flux}	3	same
WENO5-JS	weno5-{flux}	5	same
WENO5-Z	wenoz5-{flux}	5	same

8 composable fluxes × 6 reconstructions + 11 standalone schemes = 59 schemes total.

MUSCL limiters: minmod, van-leer (default), superbee, mc, van-albada.

Streamlit interface

Main page: 1D Euler Solver

Tab	Description
Convergence order	Profiles (ρ, u, p) + exact solution, L1/L2/L∞ errors, mesh convergence
Time evolution	Time animation with slider, exact solution at arbitrary t

Page: Fourier Analysis

Dissipation |G(θ)| and dispersion φ/φ_exact curves for each scheme, obtained by linearization around a uniform state.

Available test cases

Case	Function	Type	Description
Sod	sod_shock_tube()	Riemann	Shock + contact + rarefaction
Lax	lax_test()	Riemann	More severe than Sod
Double rarefaction	double_rarefaction()	Riemann	Two rarefactions, low pressure
Stationary contact	stationary_contact()	Riemann	Stationary contact, diffusion measurement
Near vacuum	near_vacuum()	Riemann	Extreme double rarefaction (93% of vacuum threshold)
Two shocks	two_shocks()	Riemann	Strong shock collision (Toro test 4)
Shu-Osher	shu_osher()	Mixed	Mach 3 shock in sinusoidal density
Entropy wave	entropy_wave()	Smooth	Sinusoidal advection, order verification
Acoustic wave	acoustic_wave()	Smooth	Isentropic perturbation, order verification

Usage as a library

from euler1d.schemes import get_scheme
from euler1d.solver import run_simulation
from euler1d.test_cases import sod_shock_tube
from euler1d.results import compute_errors, compute_exact_solution, result_to_dataframe

config = sod_shock_tube(n_cells=200)
scheme = get_scheme("muscl-hllc", limiter="van-leer")
result = run_simulation(config, scheme)

df = result_to_dataframe(result)
exact = compute_exact_solution(config)
errors = compute_errors(result, exact)
print(errors["rho"])  # {'L1': ..., 'L2': ..., 'Linf': ...}

Theoretical documentation

Detailed mathematical documentation is in the docs/ folder:

1. 1D Euler equations
2. Riemann problem
3. Finite volume method
4. Flux schemes
5. High-order reconstructions
6. Time integration
7. Fourier analysis
8. Test cases
9. Bibliography

References

1. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd Edition, Springer, 2009.
2. B. van Leer, "Towards the ultimate conservative difference scheme. V.", J. Comput. Phys., 32, 1979.
3. P. L. Roe, "Approximate Riemann solvers", J. Comput. Phys., 43, 1981.
4. G.-S. Jiang, C.-W. Shu, "Efficient implementation of weighted ENO schemes", J. Comput. Phys., 126, 1996.