论文标题
高效的指数runge - 高级kutta方法：施工和实施
Efficient exponential Runge--Kutta methods of high order: construction and implementation
论文作者
论文摘要
指数runge - kutta方法已证明对刚性半线性抛物线PDE的时间整合具有竞争力。但是，当前僵硬准确的指数runge的构造 - 但是，kutta方法依赖于收敛的结果，该结果需要削弱许多顺序条件，从而导致了必须以序列方式实现阶段的方案。在这项工作中，在显示出更强的融合结果之后，我们能够得出两个新的和五阶指数式runge的新家庭 - kutta方法，与现有方法相反，它们具有多个阶段，它们彼此独立并共享相同的格式，从而使它们在与之相处或同时相同的方法中实现，并且可以与他人相同，并且可以用来努力。此外，它们的所有阶段仅涉及$φ$ functions（使用相同参数）的乘积的一种线性组合。总体而言，与现有订单的现有方法相比，这些新方法可以更有效地实施。给出了一维半线性抛物线问题，非线性schrödinger方程和二维灰色 - 斯科特模型的数值实验，以确认两种新构建方法的准确性和效率。
Exponential Runge--Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge--Kutta methods, however, relies on a convergence result that requires weakening many of the order conditions, resulting in schemes whose stages must be implemented in a sequential way. In this work, after showing a stronger convergence result, we are able to derive two new families of fourth- and fifth-order exponential Runge--Kutta methods, which, in contrast to the existing methods, have multiple stages that are independent of one another and share the same format, thereby allowing them to be implemented in parallel or simultaneously, and making the methods to behave like using with much less stages. Moreover, all of their stages involve only one linear combination of the product of $φ$-functions (using the same argument) with vectors. Overall, these features make these new methods to be much more efficient to implement when compared to the existing methods of the same orders. Numerical experiments on a one-dimensional semilinear parabolic problem, a nonlinear Schrödinger equation, and a two-dimensional Gray--Scott model are given to confirm the accuracy and efficiency of the two newly constructed methods.
