Question

Consider the nonlinear problem $$ -\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)-e^{u}=0 $$ defined in the unit square \(0

Short answer

To summarize, the steps to approximate the two solutions to the given nonlinear PDE are: 1. Discretize the problem using finite differences on a uniform grid. 2. Implement Newton's method with appropriate initial guesses. 3. Solve the resulting linear system in MATLAB using the backslash operator. 4. Plot the two solutions, display their scaled norms, and calculate the infinity norms of their exponentials. 5. Determine the number of iterations taken by Newton's method to converge. By following these steps, you can find approximations for the two solutions and the number of iterations required for convergence.

Step-by-step solution

Discretize the problem using finite differences on a uniform grid

We can approximate the Laplacian term (using central differences) as follows: $$ \frac{\partial^2 u}{\partial x^2} \approx \frac{u_{i+1, j} -2u_{i, j} + u_{i-1, j}}{h^2} $$ and $$ \frac{\partial^2 u}{\partial y^2} \approx \frac{u_{i, j+1} -2u_{i, j} + u_{i, j-1}}{h^2} $$ Thus, the discretized PDE becomes: $$ -\left(\frac{u_{i+1, j} -2u_{i, j} + u_{i-1, j}}{h^2} + \frac{u_{i, j+1} -2u_{i, j} + u_{i, j-1}}{h^2}\right) - e^{u_{i,j}} = 0 $$

Implement Newton's method with appropriate initial guesses

To apply Newton's method, we need to linearize the nonlinear equation (by taking the first order Taylor series expansion) around a guess for the solution: $$ F(u) = -\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)-e^{u} \approx F(u_0) + \frac{dF}{du}\Bigg|_{u_0} (u-u_0) $$ We need to solve the linearized system iteratively for \(u\) until convergence. For the initial guesses, one can start with random values or zeros in the interior grid points.

Solve the resulting linear system directly using MATLAB's backslash

After obtaining the linearized system, we can solve for the updated solution \(u_{k+1}\) using MATLAB's backslash operator as follows: ``` matlab A = ... % Construct the matrix of the linearized system F = ... % Calculate the residual using the current guess deltaU = A \ (-F); % Solve for the correction in the solution guess u = u + deltaU; % Update the solution guess ```

Plot the two solutions and display their scaled norms

After converging to the solutions, we can plot them using MATLAB's `surf` or `contour` functions, and calculate the scaled norms using `norm` function, as follows: ``` matlab figure; surf(X, Y, u1); % Plot the first solution title('First solution'); figure; surf(X, Y, u2); % Plot the second solution title('Second solution'); scaled_norm_u1 = norm(u1, 2) / sqrt(n); scaled_norm_u2 = norm(u2, 2) / sqrt(n); infty_norm_exp_u1 = norm(exp(u1(:)), inf); infty_norm_exp_u2 = norm(exp(u2(:)), inf); disp(['Scaled norm of first solution: ', num2str(scaled_norm_u1)]); disp(['Scaled norm of second solution: ', num2str(scaled_norm_u2)]); disp(['Infinity norm of exp(u1): ', num2str(infty_norm_exp_u1)]); disp(['Infinity norm of exp(u2): ', num2str(infty_norm_exp_u2)]); ```

Determine the number of iterations taken by Newton's method to converge

To find the number of iterations for convergence, simply count the iterations inside the loop of Newton's method: ``` matlab iter = 0; tol = 1e-6; residual = norm(F, inf); while residual > tol % Perform one iteration of Newton's method iter = iter + 1; % Calculate the residual and check for convergence residual = norm(F, inf); end disp(['Number of iterations for convergence: ', num2str(iter)]); ``` Performing these steps, one can find approximations for the two solution functions and determine the number of iterations taken by Newton's method to converge.

Key concepts

Finite Difference Method

The Finite Difference Method (FDM) is an essential numerical technique to approximate the solution of partial differential equations (PDEs). As in our exercise, where we aim to solve a nonlinear PDE, FDM allows us to replace continuous derivatives with discrete approximations. By using a uniform grid with step size h, we estimate the second derivatives of a function u with respect to x and y by considering the values of u at corresponding points on the grid.

When applying FDM to our problem, we create a system of algebraic equations that can be handled by computers, transforming a complex calculus problem into a manageable linear algebra one. This approach divides the domain into a grid and, utilizing central differences, yields an approximation that can be incredibly accurate depending on the grid resolution, which is controlled by the value of h.

Newton's Method

Newton's method, also known as the Newton-Raphson method, is a powerful tool in numerical analysis for finding successively better approximations to the roots (or zeroes) of a real-valued function. In the context of our nonlinear PDE, we're using this iterative method to tackle a challenging problem.

Initially, a guess for the solution is made, and then the method proceeds to improve this guess iteratively. The key here is the process of linearization, which simplifies the nonlinear equation at each step, making it more straightforward to solve. In our case, this leads to creating a sequence of linear systems, each bringing us closer to the PDE's solution. The choice of initial guess greatly influences the convergence speed, which is why having a good first approximation can be beneficial.

Linearization

Linearization is a strategy where we approximate a nonlinear function with a linear one, near a specific point. By using the first-order Taylor series expansion, we linearize our nonlinear PDE around our initial guess for the solution. This approximation simplifies calculations immensely.

The linearization process converts the complex nonlinear problem into a series of linear steps that are far easier to manage computationally. Each iteration of this procedure brings us a step closer to the actual solution, relying on the principle that the linear approximations will converge towards the nonlinear problem's solution. The accuracy of linearization depends on the nonlinearity's nature and the proximity of the initial guess to the actual solution.

MATLAB Numerical Computation

MATLAB is a widely-used software for numerical computation, providing a robust environment for matrix operations, algorithm implementation, and data visualization. In the context of our PDE problem, MATLAB's functionality is crucial as it handles matrix constructions, operations, and sophisticated functions for dealing with linear systems.

The use of MATLAB's backslash operator for solving linear systems is an example of its power. This operator performs a matrix division, essentially solving the equation Ax = b for x. It chooses the best method (like LU decomposition or Cholesky factorization) based on the matrix A's properties. After each iteration of Newton's method, we employ the backslash operator to update our solution, demonstrating MATLAB’s efficiency in numerical computations, and moving forward towards the PDE's solutions.