Efficient exponential Runge–Kutta methods of high order: construction and implementation
Efficient exponential Runge–Kutta methods of high order: construction and implementation
复制标题
DOI：
10.1007/s10543-020-00834-z
复制
发表时间：
2020-09
影响因子：
1.5
通讯作者：
Vu Thai Luan
中科院分区：
文献类型：
--
作者：
Vu Thai Luan
指数Runge-Kutta方法在求解刚性半线性抛物型偏微分方程的时间积分问题上具有很强的竞争力。然而，目前严格精确的指数龙格-库塔方法的构建依赖于收敛结果，需要削弱许多阶条件，从而导致其阶段必须以顺序的方式实施的方案。在这项工作中，在显示出更强的收敛结果后，我们能够推导出两个新的四阶和五阶指数龙格库塔方法，与现有方法相比，它们具有相互独立的多个阶段，并且共享相同的格式，从而允许它们并行或同时实现，并且使方法表现得像使用更少的阶段。此外，它们的所有阶段只涉及一个线性组合的产品的功能（使用相同的论点）与向量。总的来说，这些特征使得这些新方法与相同阶数的现有方法相比实现起来更加有效。通过一维半线性抛物问题、非线性Schr dinger方程和二维Gray-Scott模型的数值实验，验证了这两种新方法的精度和效率.
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-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.