Matrix- and tensor-oriented numerical schemes for the evolutionary space-fractional complex Ginzburg--Landau equation

In this manuscript, we propose matrix- and tensor-oriented methods for the numerical solution of the multidimensional evolutionary space-fractional complex Ginzburg--Landau equation. After a suitable spatial semidiscretization, the resulting system of ordinary differential equations is time integrated with stiff-resistant schemes. The needed actions of special matrix functions (e.g., inverse, exponential, and the so-called $\varphi$-functions) are efficiently computed in a direct way by exploiting the underlying tensor structure of the task and taking advantage of high performance BLAS and parallelizable pointwise operations. Several numerical experiments in 2D and 3D, where we apply the proposed technique in the context of linearly-implicit and exponential-type schemes, show the reliability and superiority of the approach against the state-of-the-art, allowing to obtain speedups which range from one to two orders of magnitude. Finally, we demonstrate that in our context a single GPU can be effectively exploited to boost the computations both on consumer- and professional-level hardware.