Liner shipping is an important branch of maritime transportation. As bunker fuel consumption causes high operating cost and harmful gas emissions, bunker fuel management is a great challenge and a hot research topic in liner shipping. Bunker charging can be achieved at ports with diverse prices, and it is recognized that appropriately managing bunker fuel and sailing speed can improve liner shipping performance and reduce environmental pollution. Most existing works assume that the container demand is deterministic. However, in practice, it is usually difficult to exactly estimate the volume of containers to be shipped due to various factors. This paper studies a liner ship bunkering and speed optimization problem under uncertain container demand. For the problem, a two-stage stochastic and non-linear programming formulation is proposed. To split the complexity of the problem, the complicated bunker consumption function is approximated by piecewise linear ones. To solve the problem, a classic sample average approximation (SAA) method, and the SAA based on scenario reduction, and an L-shaped method are developed and compared. Numerical results show that the L-shaped method outperforms the two SAA methods, in terms of solution quality and computational time.