Question

For the following initial-value (IVB) problem, du/dt = f (t,u) where f (t,u) = -10u and u(0) = 1 (A) Compute the values of u up to t=0.2 with Δt=0.1 using the following methods: 1. Using the backward Euler method; 2. Using the Crank-Nicolson method; (B) For the current IVB, what is the limit on the time increment Δt for the Forward Euler method to be stable? (C) Compute the local discretization error for the Forward Euler method.

Answer 1

The question deals with an initial-value problem du/dt = -10u, with initial condition u(0) = 1, where the student must compute values using numerical methods and analyze the Forward Euler method's stability and discretization error.

Step-by-step explanation:

The problem presented is an initial-value problem (IVP) with a linear ordinary differential equation (ODE): du/dt = -10u with initial condition u(0) = 1. The student is asked to compute values of u using different numerical methods and analyze the Forward Euler method's stability.

Backward Euler Method

For the backward Euler method, this implicit method requires solving an equation at each step.

Crank-Nicolson Method

The Crank-Nicolson method is a combination of the forward and backward Euler methods and will give an averaged solution.

Stability of Forward Euler Method

The Forward Euler method's stability is given by the stability condition which is ∆t < 2/|λ|, where λ is the eigenvalue of the system, in this case, -10.

Local Discretization Error

The local discretization error for the Forward Euler method is the difference between the exact solution's derivative and the approximation given by the method at each step.