This paper proposes a high-order fast two-grid compact difference method for solving a class of two-dimensional (2D) nonlocal nonlinear wave equations (NNWE). In the temporal direction, a fast L2- \(1_\sigma \) ( \(\mathcal {F}\) L2- \(1_\sigma \) ) scheme, developed via order-reduction techniques and sum-of-exponentials (SOE) approximation, is employed to discretize the Caputo derivative. Spatial discretization utilizes a high-order compact difference operator to enhance accuracy. To improve computational efficiency, a two-grid strategy is adopted: a nonlinear problem is first solved on a coarse grid, followed by a linearized correction on a fine grid. A high-order mapping operator ensures consistency and accuracy between grids. Rigorous theoretical analysis establishes unconditional stability and optimal \(H^1\) -norm error estimates. Numerical experiments validate the theoretical findings and demonstrate that the proposed method achieves the expected accuracy with significantly reduced computational cost compared to standard nonlinear schemes.
