One-Dimensional Time-Dependent Advection Equations Simulator

Description

This project is an interactive web application built with Python and Streamlit that simulates the one-dimensional time-dependent advection equation. It serves as a pedagogical tool to visualize and compare various finite-difference numerical schemes against the exact analytical solution.

The application allows users to modify simulation parameters in real-time—such as the Courant number, spatial resolution, and wave velocity—to observe how different discretization methods handle stability, dissipation, and dispersion.

Mathematical Background

The simulator solves the linear advection equation, a partial differential equation (PDE) that describes the transport of a quantity (scalar field $u$) by a flow with velocity $v$:

$$ \frac{\partial u}{\partial t} + v \frac{\partial u}{\partial x} = 0 $$

Where:

• $u(x,t)$ is the scalar field being advected.
• $v$ is the constant velocity.

Features

• Interactive Simulation Controls: Adjust Domain Length ($L$), Advection Velocity ($v$), Simulation Time ($T$), and Spatial Resolution ($N_x$).
• Courant Number Analysis: Dynamically adjust the Courant-Friedrichs-Lewy (CFL) condition ($C = v \frac{dt}{dx}$) to observe stability limits.
• Multiple Initial Conditions:
  • Gaussian Bell
  • Square Pulse
  • Triangle Wave
  • Cosine Hat
• Numerical Schemes: The application implements and compares the following schemes:
  • Exact Solution (Analytical)
  • Forward Euler Centered Space (FECS)
  • Upwind Scheme
  • Leapfrog Scheme
  • Lax-Wendroff Scheme
  • Crank-Nicolson (Implicit)
  • Backward Euler (Implicit)
• Visualizations:
  • Animated wave propagation (Plotly).
  • Space-Time (Hovmöller) heatmaps for every solver.
  • Mass conservation integrals to analyze numerical dissipation and instability.

Installation and Usage

Prerequisites

Ensure you have Python 3.8+ installed.

Setup

1. Clone the repository:

   git clone https://github.com/hsannav/advection-simulator.git
   cd advection-simulator

2. Create a virtual environment (recommended):

   python -m venv venv
   source venv/bin/activate  # On Windows use `venv\Scripts\activate`

3. Install the required dependencies:

   pip install -r requirements.txt

4. Run the Streamlit application:

   streamlit run app.py

Project Structure

• app.py: The main application script containing the PDE solvers and Streamlit interface.
• requirements.txt: List of Python dependencies.
• README.md: Project documentation.

Numerical Methods Overview

This project demonstrates the behavior of different discretization strategies:

• FECS: Unconditionally unstable.
• Upwind: First-order accurate, stable if $C \leq 1$, but diffusive.
• Leapfrog: Second-order accurate, non-dissipative, but dispersive.
• Lax-Wendroff: Second-order accurate, minimizes dissipation compared to Upwind.
• Implicit Schemes (CN, BE): Unconditionally stable, requiring matrix inversion at each time step.

Authors

• Fernando Blanco
• Hugo Sánchez