A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach, the exponential k-step method would have \(k^{\mathrm{th}}\)-order convergence in approximating a mild solution, possibly nonsmooth at the initial time. In consistency with the theoretical analysis, a numerical example shows that the method can achieve high-order convergence in the maximum norm for semilinear parabolic equations with discontinuous initial data.

中文翻译：

---

具有非光滑初始数据的非线性抛物方程的高阶指数积分器

提出了一种变步长指数多步积分器，其轮廓积分为算子值指数函数的近似值，用于求解初始数据不光滑的半线性抛物方程。通过这种方法，指数k阶方法将具有\（k ^ {\ mathrm {th}} \）阶收敛，以近似一个温和的解，可能在初始时间是不平滑的。与理论分析一致，算例表明，该方法可以在初始数据不连续的半线性抛物方程的最大范数上实现高阶收敛。