In this paper, we propose a $\mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a d-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show that the action of the exponentials, i.e. the corresponding batched matrix-vector products, can be implemented efficiently on modern computer systems. We further explain how $\mu$-mode products can be used to compute spectral transformations efficiently even if no fast transform is available. We illustrate the performance of the new integrator by solving three-dimensional linear and nonlinear Schr\"odinger equations, and we show that the $\mu$-mode integrator can significantly outperform numerical methods well established in the field. We also discuss how to efficiently implement this integrator on both multi-core CPUs and GPUs. Finally, the numerical experiments show that using GPUs results in performance improvements between a factor of 10 and 20, depending on the problem.
翻译:在本文中, 我们提出一个 $\ mu$- mode 集成器, 用于计算僵硬进化方程式的解决方案 。 集成器以二维分解法为基础, 使用精确( 通常预先计算) 的一维矩阵指数。 我们显示, 指数的动作, 即相应的分批矩阵- 矢量器产品, 可以在现代计算机系统中高效实施 。 我们进一步解释 $\ mu$- mode 产品如何用于高效计算光谱转换 。 我们通过解决三维线性线性和非线性Schr\ " 指数方程式, 我们通过解答三维线性线性线性和非线性Schr\ " 方程式, 演示新的集成器的性能, 我们显示, $\ mu$- modrod 集成器可以显著地超过实地确立的数值方法 。 我们还讨论如何在多核心 CPU 和 GPU 上高效地实施这一集成器 。 最后, 数字实验显示, 使用 GPUP 在10 和 20 之间的性工作表现改进了 。