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.

翻译：本文针对多维演化型空间分数阶复Ginzburg--Landau方程的数值求解，提出了基于矩阵与张量结构的计算方法。通过对空间变量进行适当的半离散化，所得常微分方程组采用刚性稳定格式进行时间积分。通过充分利用问题的内在张量结构，并借助高性能BLAS库与可并行化的逐点运算，我们以直接方式高效计算了所需特殊矩阵函数（如逆矩阵、指数函数及所谓的$\varphi$-函数）的作用量。在二维与三维场景下的多组数值实验中，我们将所提技术应用于线性隐式与指数型格式，结果表明该方法相较于现有最优技术具有可靠性与优越性，可获得一至两个数量级的加速效果。最后，我们证明在所述框架下，单块GPU可有效提升计算性能，该结论在消费级与专业级硬件平台上均成立。