Math 6620 • Analysis of Numerical Methods II

(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.