6620-2024-05

(20 pts)

1. (10 pts) Compute coefficients \(A\), \(B\), \(C\) in the 3-point one-sided approximation \[ u''(x) \approx Au(x) + Bu(x+h) + Cu(x+2h) \]

   that has optimal \(h\)-order of accuracy. What is this order of accuracy?

2. (10 pts) Compute weights for the following quadrature rule that makes it exact for all polynomials of degree 2. \[ \int _{-1}^1 f(x)\,dx = w_0 f(0) + w_1 f'(0) + w_2 f''(0), \]

(20 pts)

1. (10 pts) Compute the stability/amplification factor for forward Euler to solve \(u' = f(t,u)\), and plot the region of the stability.
2. (10 pts) Compute the stability/amplification factor for the trapezoidal rule/Crank- Nicolson scheme and use this to show that the scheme is A-stable.

(20 pts) Consider the multi-step method, \[ u_{n+1}+\alpha _1 u_n = k\beta _0 f_{n+1}+k\beta _1 f_n, \] where \(u_n \approx u(t_n)\), \(f_n = f(t_n,u_n)\), \(k = t_{n+1}-t_n\), and \(u' = f(t,u)\).

1. (10 pts) Identify the coefficients \(\alpha ,\beta \) that yield a scheme of optimal \(k\)-order of accuracy, and identify this order of accuracy.
2. (10 pts) Determine whether or not this scheme is 0-stable and/or A-stable.

(20 pts)

1. (10 pts) Use von Neumann stability analysis to determine a stability condition for, \[ D^+u_j^n = D_0u_j^n, \] where \(D^+u_j^n = (u_j^{n+1}-u_j^n)/k\), \(D_0u_j^n = (u_{j+1}^n-u_{j-1}^n)/(2h)\), and \(D_\pm u_j^n = \pm (u_{j\pm 1}^n-u_j^n)/h\).
2. (10 pts) Use von Neumann stability analysis to determine a stability condition for \[ D^+u_j^n = D_+D_-u_j^n. \]

(20 pts) For the ODE \(u' = f(t,u)\) with initial condition \(u(0)\) and \(f\) globally Lipschitz continuous in \(u \in \mathbb {R}^M\) uniformly in \(t\), show that forward Euler with initial state \(u_0 = u(0)\) is convergent to first order. A possibly helpful definition: a scheme is 0-stable if, \[ \max _{n\in [N]}\|e_n\| \le C \max _{n\in [N]}\|R_n(u(t_n))\|, \] where \(e_n\) is the time-\(t_n\) error, and \(R_n(u(t_n))\) is the scheme residual using the exact solution.