Assess advanced understanding of trade-offs among accuracy, numerical stability (A- and L-stability, stability regions), and computational cost when selecting and tuning numerical methods for stiff ordinary differential equations. Tasks include comparing implicit and explicit strategies (e.g., BDF, Rosenbrock, implicit Runge–Kutta), analyzing how stiffness ratio and stability region constrain step size and error control, estimating cost contributions from Jacobian evaluation, linear solves, and adaptive time-stepping, and making justified solver and parameter choices for given stiff test problems with quantified error-versus-runtime comparisons and proposed preconditioning or solver strategies.