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This paper is concerned with modified adaptive difference schemes for solving degenerate nonlinear reaction–diffusion equations with singular source terms. The adaptive mechanisms both in space and time are considered. Modified monitor functions based on arc-lengths of the rate function ut are constructed. Further analyses are given on properties of the numerical schemes derived. We show that, under proper smoothness, consistency and step-size constraints, the numerical solution obtained preserves the monotonicity of the physical solution. Numerical experiments are given to demonstrate the monotonicity and convergence properties of the methods. [Copyright &y& Elsevier]

• Use partial differential equations (PDEs) modeling the fair price of Asian options. • The combination of aPC and a system of PDEs with moving boundary conditions. • The application of probability distributions constructed from real data of historic volatilities. This paper concerns accurate and efficient polynomial chaos expansions (PCEs) for Asian option pricing with uncertain volatilities. While arbitrary distributions of the volatility parameter are applied for estimating real-world option prices, arbitrary polynomial chaos (aPC) are incorporated for approximating raw data of the historical volatility distributions. Rigorous analysis is carried out to ensure the numerical stability of the compact aPC Crank-Nicolson finite difference method accomplished. Numerical results acquired are compared with solutions via standard Monte Carlo schemes (MCSs) and generalized polynomial chaos (gPC) with different random volatilities. Stock data from Asian financial industry are used. It is evident that the novel schemes derived are highly accurate and efficient for evaluating means and variances of uncertain volatility and stochastic Asian option pricing. [ABSTRACT FROM AUTHOR]