Transport Equation Solver

This project implements numerical schemes for solving the one-dimensional transport equation $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0$ with a periodic initial condition using finite difference methods. It compares different discretization schemes and evaluates their accuracy and stability.

Dependencies

The code requires the following Python libraries:

• numpy for numerical operations
• plotly for visualization
• copy for deep copying arrays

Implementation

Initialization

• Defines the problem parameters (time interval, wave speed, and CFL condition).
• Specifies the initial condition $u(x,0) = \sin(2 \pi x)$.
• Provides an analytical solution for validation.

Spatial and Temporal Discretization

The function espaceDiscretise(J, lam) computes:

• dx: spatial step size
• dt: temporal step size
• X: array of discretized spatial points
• M: number of time steps

Finite Difference Schemes

Several numerical schemes are implemented in a dictionary Schema:

• Centered Difference (C)
• Upwind (DAG/DAD)
• Lax-Friedrichs (LF)
• Lax-Wendroff (LW)

The function schema(dx, dt, X, M, c_1, c0, c1) computes the numerical solution for a given scheme.

Numerical Solution and Visualization

• numerical_solution(Schema, J): Computes and visualizes numerical solutions against the analytical solution.
• plot_norm_Qn(Schema, J): Evaluates stability by plotting norms of matrix powers.
• convergenceDeltaX(Schema, J): Analyzes error convergence with respect to spatial discretization.

Usage

Run the script to visualize results for different schemes:

python transport_solver.py

Results

• The numerical solutions are plotted and compared against the analytical solution.
• Stability analysis evaluates growth in matrix norms.
• Convergence studies confirm expected error behavior for different schemes.