Question

1 Runge-Kutta Method The discretization of the spatial derivatives of a PDE often results in a system of ODEs of the fornm du Runge-Kutta methods are the most commonly used schemes for numerically integrating in time the ODE system. We will numerically implement the "standard" third-order Runge-Kutta method. To advance the solution u from time t to t + Δ1, three sub-steps, are taken. If the solution at time t is un the following three steps are taken to advance the solution to un+1 at t + Δt Un+1 1. Implement the Runge-Kutta method of (2 4) 2. Verify the implementation by meically integrating dt Take u(1-2) 1 as the initial condition, choose a Δι-1/N, where N is the number of time steps, and integrate up to t 3. Using a p-norm of the error show that the rate of convergence is third order with respect to At. The exact solution is ult) 1/-)
Both parts please!

Answer 1

du/dt = u^2 Because f (u) = u^2 let N = 2 Delta t = 1/N = 1/2 u_0 = u (t = 2) = -1 u_0 + alpha = u_0 + Delta t 1/2 f (u_0) (using 2) = -L + (1/2) times 1/2 times (- 1)^2 = -1 + 1/4 = -4 + 1/4 = -3/4 u_0 + alpha_1 = u_0 + Delta t (- f/u) + 2 f (u_0 + alpha)) = (- 1) + 1/2 (- (- 1)^2 + 2 (- 3/4)^2) = -1 + 1/2 (- 1 + 2 times 9/16) = -1 + 1/2 (- 1 + 9/8) = -1 + 1/2 times 1/8 = -1 + 1/16 = -15/16 u_1 u_0 + 1 = u_0 + Delta t (1/6 f u_0) + 2/3 f (u_0 + alpha) + 1/6 f (u_0 + alpha_1)) \r\n Therefore u_1 = (- 1) + 1/2 (1/6 times 1 + 2/3 times times (- 3/4)^2 + 1/6 (- 15/16)^2) = (- 1) + 1/2 (1/6 + 3/8 + 1/6 times 225/256) = (- 1) + 0.3440755 = -0.6559245 2^nd