论文标题
半线性indegro-differention方程的显式指数runge-kutta方法
Explicit exponential Runge-Kutta methods for semilinear integro-differential equations
论文作者
论文摘要
本文的目的是构建和分析线性和半线性间差方程的时间离散化的明确指数runge-kutta方法。通过根据解决方案扩展数值方法的误差，我们得出了构成误差界限的基础的顺序条件。订单条件进一步用于构建数值方法。收敛分析是在Hilbert空间设置中进行的，在Hilbert空间设置中，分解家族的平滑效果大量使用。对于线性案例，我们将在满足这些条件时得出通用订单$ p $的订单条件，并证明订单$ p $的融合。在半线性的情况下，我们考虑了通过光谱盖金方法的空间离散化，我们需要局部Lipschitz连续非线性。我们得出了一个和两个订单的顺序条件，构建满足这些条件并证明其收敛性的方法。最后，给出了一些说明我们理论结果的数值实验。
The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order $p$ and prove convergence of order $p$, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given.
