Implicit Runge-Kutta with spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian

An efficient numerical method with high accuracy both in time and in space is proposed for solving d$$ d $$-dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an s$$ s $$-stage implicit Runge-Kutta method and approximating the space by a spectral Galerkin method with Fourier-like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high-dimensional problems. The proposed method is showed to be stable and convergent with at least order s+1$$ s+1 $$ in time, when the implicit Runge-Kutta method with classical order p$$ p $$ (p >= s+1$$ p\ge s+1 $$) is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter alpha$$ \alpha $$. This improves the previous result which depends on the fractional parameter alpha$$ \alpha $$. Numerical experiments verify and complement our theoretical results.