A two-level linearized scheme for two-dimensional Riesz space fractional nonlinear advection-dispersion equations

Abstract

Nonlinear Riesz space fractional advection-dispersion equations (SFADEs) find widespread applications in many fields, and numerical solutions are the primary means of solving such equations due to the complexity of nonlinear problems. A majority of the existing numerical techniques for solving nonlinear equations are based on three-level schemes, thereby leading to increased computational overheads and theoretical challenges. A novel approach is developed to solve twodimensional nonlinear Riesz SFADEs, which employs a two-level linearized finite difference method and incorporates state-of-the-art linearization techniques. The proposed approach involves using the Crank-Nicolson approach to process the time component and approximating the Riesz space fractional derivative using the weighted and shifted Grünwald difference (WSGD) formula. The resulting difference scheme is observed to exhibit second-order convergence in both temporal and spatial coordinates. The efficiency and accuracy of the proposed method are confirmed by conducting numerical experiments.