Efficient Runge–Kutta Scheme Combined with Richardson Extrapolation for Nonlinear Fractional Partial Differential Equations Involving the Fractional Laplacian

In this paper, an efficient numerical framework combined with RK4 method and Richardson extrapolation is proposed to solve nonlinear time-dependent partial differential equations involving the Riesz fractional Laplacian operator ${(-\Delta )}^s$. The RK4 method guarantees fourth-order temporal accuracy and L-stability, whereas the spatial fractional operator is discretized using a second-order central finite difference scheme. Based on the consistency conditions of the underlying spatial discretization, and by constructing a Vandermonde matrix to determine the extrapolation coefficients, novel high-order Richardson extrapolation formulas are derived, achieving a maximum convergence order of $\mathcal{O}\left(h^{2n}\right)$. Numerical experiments, covering 1D variable-coefficient cases, 2D cases with equal/unequal spatial steps, and 3D equidistant differencing cases, demonstrate that the proposed method stably upgrades the convergence order from second-order to fourth-order and further to sixth-order under oscillatory and nonlinear variable-coefficient conditions, with the extrapolated numerical errors reduced to the magnitude of ${10}^{-13}$. Asynchronous convergence observed in 2D unequal-step cases validates Theorem 3, while fourth-order convergence is achieved via extrapolation in 3D complex domains. This method possesses prominent advantages of high accuracy, strong robustness, and high efficiency, breaking through the dimensionality and convergence order limitations of traditional high-precision numerical algorithms.