We propose a novel algorithm for the temporal integration of the magnetohydrodynamics (MHD) equations. The approach is based on exponential Rosenbrock schemes in combination with Leja interpolation. It naturally preserves Gauss's law for magnetism and is unencumbered by the stability constraints observed for explicit methods. Remarkable progress has been achieved in designing exponential integrators and computing the required matrix functions efficiently. However, employing them in realistic MHD scenarios require matrix-free implementations that are competent on modern computer hardware. We show how an efficient algorithm based on Leja interpolation that only uses the right-hand side of the differential equation (i.e. matrix-free), can be constructed. We further demonstrate that it outperforms, in the context of magnetic reconnection and the Kelvin--Helmholtz instability, earlier work on Krylov-based exponential integrators as well as explicit methods. Furthermore, an adaptive step size strategy is employed that gives an excellent and predictable performance, particularly in the lenient to intermediate tolerance regime that is often of importance in practical applications.

翻译：我们建议采用新的算法,将磁力动力学(MHD)等式暂时整合在一起。这个方法以指数性罗森布罗克办法为基础,与Leja内插法相结合。它自然保留高斯的磁法,不受明确方法所观察到的稳定限制。在设计指数性集成器和高效计算所需的矩阵功能方面已经取得了显著进展。然而,在现实的MHD假设情景中运用它们需要能够操作现代计算机硬件的无基体执行。我们展示了如何在只使用差异方(即无矩阵)右侧的Leja内插法的基础上构建高效的算法。我们进一步表明,在磁再连接和凯尔文-赫尔默尔茨不稳定的背景下,它优于先前对基于Krylov的指数化器的工作以及明确的方法。此外,我们采用了适应性步骤规模战略,它提供了优异和可预测的性表现,特别是在实际应用中往往十分重要的宽度至中间容忍制度中。