=====================================================

          MATLAB 1D Time-Dependent PDE Solver

=====================================================

This project is written in MATLAB and provides a solver for 1D time-dependent linear equations. It supports two types of shape functions: P2 and P3. The solver includes validation scripts and the ability to perform both time and space convergence analysis.

---

FILES

• main.m : Contains the main validation script.
• main.txt : Contains a validation test using both time and space refinement. Errors are saved into Excel files.
• rapport.pdf : Describes the equation being solved and provides details about the numerical method.
• commands.txt : Lists the commands and parameters used to run the solver.
• [P2/P3 Scripts]: Located in the project folder, each script corresponds to a specific shape function.

---

HOW TO USE THE SOLVER

Boundary Conditions:

• cl1: Left-side boundary condition (vector of two values)

  • First value: 0 for Dirichlet, 1 for Neumann
  • Second value: Value of u(0) or u'(0)

• cl2: Right-side boundary condition (vector of two values)

  • First value: 0 for Dirichlet, 1 for Neumann
  • Second value: Value of u(L) or u'(L)

Solver Function:

Solver1D_P2_Ref_T_EDP_a(a, b, h, t0, tf, ht, u, mrf, gordre, alpha)

Parameters: - a : Left boundary node - b : Right boundary node - h : Space step size - t0 : Initial time - tf : Final time - ht : Time step size - u : Exact solution (function handle) - mrf : Use reference element technique (1) or not (0) - gordre : Gauss quadrature order - alpha : Coefficient of the PDE (α * du/dt)

Example:

sll = Solver1D_P2_Ref_T_EDP_a(0, 4, 0.01, 0, 1, 0.01, u, 1, 5, 1);

Object Methods:

sll = sll.CLS(cl1, cl2)
    → Store boundary conditions for later use.

sll = sll.initialize(u0)
    → Initialize the first time step with the initial condition.

sll = sll.LoopTime()
    → Loop over time steps:
        - Assemble local and global matrices
        - Apply boundary conditions
        - Solve the system

sll = sll.erreur()
    → Compute the error in space and time.
    → Result stored in: sll.er

---

NOTES

• The project includes comments in the MATLAB code to explain each section.
• Error data is saved in Excel format for both time and space validations.
• Make sure to define the exact solution function u(t, x) and initial condition u0 before running the solver.

---