6620-2024-08

(20 pts) On an equidistant mesh \(x_j = jh\), define the operator \(\widetilde {D}_0\) as, \[ \widetilde {D}_0u(x_j) = \frac {u(x_j+h/2)-u(x_j-h/2)}{h}, \] For a given smoooth \(\kappa \) and \(f\), consider the scheme, \[ -\widetilde {D}_0\left (\kappa (x_j)\widetilde {D}_0u_j\right ) = f(x_j), \] for the ODE \[ -\frac {d}{dx}\left (\kappa (x)\frac {d}{dx}u(x)\right ) = f(x). \] What order is the local truncation error?

(20 pts) Consider the multi-step method, \[ u_{n+1}+\alpha _1u_n+\alpha _2u_{n-1}=k\beta _1 f_n+k\beta _2 f_{n-1}, \] 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. Is the scheme convergent?

(20 pts) Consider the PDE, \[ u_t + au_x = 0,\qquad u(x,0)=u_0(x),\qquad a > 0, \] with periodic boundary conditions over the spatial domain \(x \in [0, 1)\). Use the standard notation, \(u_j^n \approx u(x_j,t^n)\) with \(x_j = jh\) and \(t^n = nk\) with \(h = \frac {1}{M} > 0\) for some \(M \in \mathbb {N}\) and \(k > 0\). Consider the following semi-Lagrangian scheme: \[ u_j^{n+1}=p_j(y),\qquad y := X(t^n;x_j,t^{n+1}), \] where \(X(t; x, t_0)\) is the time-\(t\) spatial location of the PDE’s characteristic curve passing through spatial location \(x\) at time \(t_0\), and \(p_j(\cdot )\) is the linear interpolant formed by the data \((x_{j-1},u_{j-1}^n)\) and \((x_j,u_j^n)\).

1. (10 pts) Write this scheme as an explicit function of the \(u_j^n\) values.
2. (10 pts) Use von Neumann stability analysis to determine a stability condition for this scheme.

(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.