This paper considers the numerical integration of semilinear evolution PDEs using the high order linearly implicit methods developped in a previous paper in the ODE setting. These methods use a collocation Runge--Kutta method as a basis, and additional variables that are updated explicitly and make the implicit part of the collocation Runge--Kutta method only linearly implicit. In this paper, we introduce several notions of stability for the underlying Runge--Kutta methods as well as for the explicit step on the additional variables necessary to fit the context of evolution PDE. We prove a main theorem about the high order of convergence of these linearly implicit methods in this PDE setting, using the stability hypotheses introduced before. We use nonlinear Schr\''odinger equations and heat equations as main examples but our results extend beyond these two classes of evolution PDEs. We illustrate our main result numerically in dimensions 1 and 2, and we compare the efficiency of the linearly implicit methods with other methods from the litterature. We also illustrate numerically the necessity of the stability conditions of our main result.

翻译：本文考虑采用先前在常微分方程框架中发展的线性隐式方法对半线性发展型偏微分方程进行数值积分。这些方法以配置龙格-库塔方法为基础，通过显式更新的附加变量使配置龙格-库塔方法的隐式部分仅保持线性隐式性质。针对发展型偏微分方程的应用场景，我们引入了基础龙格-库塔方法及其附加变量显式步的多种稳定性概念。基于上述稳定性假设，我们证明了此类线性隐式方法在偏微分方程框架下具有高阶收敛性的主定理。本文以非线性薛定谔方程和热方程为主要算例，但研究结果可推广至更广泛的发展型偏微分方程类别。我们通过一维和二维数值实验验证了主要结论，并比较了线性隐式方法与文献中其他方法的计算效率，同时数值展示了主定理稳定性条件的必要性。