Lab 7: Solution of Laplace Equation using Gauss-Seidel Iteration

import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D def solve_laplace_gauss_seidel(grid_size=20, max_iter=1000, tolerance=1e-4, w=1.0): “”” Solve Laplace equation using Gauss-Seidel iteration with relaxation Parameters: grid_size : number of grid points along each axis max_iter : maximum number of iterations tolerance : convergence tolerance w : relaxation factor (1.0 for standard Gauss-Seidel) Returns: u : solution array iterations : number of iterations performed “”” # Initialize grid u = np.zeros((grid_size, grid_size)) # Set boundary conditions (example: u=1 on top boundary) u[0, :] = 1.0 # Top boundary # Other boundaries remain at 0 iterations = 0 error = 1.0 while error > tolerance and iterations < max_iter: error = 0.0 for i in range(1, grid_size-1): for j in range(1, grid_size-1): temp = u[i,j] u[i,j] = (1-w)*u[i,j] + w*0.25*(u[i+1,j] + u[i-1,j] + u[i,j+1] + u[i,j-1]) error += abs(u[i,j] - temp) error /= (grid_size-2)**2 # Average error per point iterations += 1 return u, iterations def plot_solution(u, title="Solution of Laplace Equation"): """Plot the solution in 3D surface plot""" x = np.linspace(0, 1, u.shape[0]) y = np.linspace(0, 1, u.shape[1]) X, Y = np.meshgrid(x, y) fig = plt.figure(figsize=(10, 7)) ax = fig.add_subplot(111, projection='3d') surf = ax.plot_surface(X, Y, u, cmap='viridis', rstride=1, cstride=1) ax.set_title(title, fontsize=14) ax.set_xlabel('X', fontsize=12) ax.set_ylabel('Y', fontsize=12) ax.set_zlabel('U', fontsize=12) fig.colorbar(surf, shrink=0.5, aspect=5) plt.show() # Solve and plot solution, iterations = solve_laplace_gauss_seidel(grid_size=20, max_iter=1000, tolerance=1e-5) print(f"Solution converged in {iterations} iterations") plot_solution(solution, "Solution of Laplace Equation using Gauss-Seidel Iteration")