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 $$\varphi $$ 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.

中文翻译：

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高阶高效指数 Runge-Kutta 方法：构造和实现

指数 Runge-Kutta 方法已证明在刚性半线性抛物线偏微分方程的时间积分方面具有竞争力。然而，目前严格精确的指数 Runge-Kutta 方法的构建依赖于需要弱化许多阶条件的收敛结果，导致其阶段必须以顺序方式实现的方案。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 Runge-Kutta 方法系列，与现有方法相比，它们具有多个相互独立的阶段并共享相同的格式，从而允许它们并行或同时实现，并使方法表现得像使用更少的阶段一样。而且，它们的所有阶段都只涉及 $$\varphi $$ φ 函数（使用相同参数）与向量的乘积的一种线性组合。总体而言，与相同订单的现有方法相比，这些功能使这些新方法的实现效率更高。给出了一维半线性抛物线问题、非线性薛定谔方程和二维格雷-斯科特模型的数值实验，以证实这两种新方法的准确性和效率。