Problem 1
1+hλ=−3, whose magnitude exceeds 1, so the numerical solution grows/oscillates unstably.
Faculty use
Absolute Stability and Stiff Equations
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Widely separated time scales force explicit methods to take very small steps for stability rather than accuracy.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind using the test equation to understand why a stable exact solution can produce unstable numerics. Use the formulas y′=λy; Euler amplification 1+hλ.
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to simulation, control, orbital motion, chemical kinetics, heat transfer, and computational engineering, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.