For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $\frac{3}/{2}$ and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

中文翻译：

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一种指数随机 Runge-Kutta 类型阶数方法，最高为 1.5，适用于 Nemytskii 型 SPDE


对于随机偏微分方程的解的近似，获得高收敛阶并同时涉及合理计算成本的数值方法特别值得关注。因此，我们提出了一种新的指数随机 Runge-Kutta 类型的数值方法，该方法允许以高达 $\frac{3}/{2}$ 的时间顺序收敛，并且可以与多个空间离散化相结合。开发的无导数方案系列是为 Nemytskii 类型的随机偏微分方程量身定制的，即具有逐点乘法噪声运算符。我们在均方根意义上证明了这些方案的强收敛性，并提供了一些揭示理论结果的数值示例。