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：本文考虑使用先前论文在常微分方程框架下发展的高阶线性隐式方法对半线性演化偏微分方程进行数值积分。这些方法以配点龙格-库塔方法为基础，附加变量通过显式更新，使配点龙格-库塔方法的隐式部分仅保持线性隐式。本文针对基础龙格-库塔方法及为适应演化偏微分方程场景所需的附加变量显式步骤，引入了若干稳定性概念。基于先前提出的稳定性假设，我们证明了这些线性隐式方法在偏微分方程框架下具有高阶收敛性的主要定理。虽以非线性薛定谔方程和热方程作为主要算例，但我们的结论可推广至这两类演化偏微分方程之外。我们在1维和2维空间中通过数值实验验证了主要结论，并将线性隐式方法与文献中其他方法进行了效率比较。此外，我们通过数值算例说明了主要结论中稳定性条件的必要性。