A novel fourth-order three-point compact finite difference method for nonlinear fractional diffusion problems on unbounded domains

Nonlinear time-fractional diffusion problems on unbounded domains still suffer from numerical complexity because of the issue of developing efficient non-reflecting boundary conditions (NRBCs) together with preserving the high accuracy of schemes. In this work, we develop a numerical scheme to deal with these issues effectively. The unbounded domain is truncated by applying exact absorbing boundary conditions obtained by taking the Laplace transformation, and a linearized Padé approximation is used to discretize the nonlinearities. For discretizing the space and time directions, a fourth-order compact finite difference and a high-order fractional Adams-type scheme are considered, respectively, which give O ( τ 3 − α ) temporal accuracy under standard regularity assumptions on initial condition and for solutions with weak singularities near ζ = 0 , the accuracy degrades to O ( τ min { 3 − α , α ( 3 − α ) } ) . The stability and convergence of the scheme are proved theoretically for the full discrete system, including the auxiliary boundary variables introduced to reformulate the boundary conditions. Additionally, a perturbation analysis confirms that the Padé linearization error does not affect the stability threshold. Several numerical examples, including a comparative study with the classical L1 scheme, show the superiority of the presented numerical scheme over second-order classical schemes and quantify the individual contributions of the temporal and spatial discretizations.