半线性波动方程指数积分器的全离散化误差分析

Benjamin Dörich, Jan Leibold
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摘要

本文证明了半线性二阶演化方程的完全离散化误差界。考虑将时间指数积分法应用于空间抽象非协调半离散问题。由于全离散格式涉及到空间离散半群,因此误差分析的关键在于消除精确解表示中的连续半群。因此,我们推导了一个由空间离散半群驱动的修正常数变分公式,该公式具有离散误差。我们的主要结果提供了指数Adams方法和显式指数Runge - Kutta方法的完全离散误差的边界。我们证明了相应指数积分器的刚性阶在时间上的收敛性,以及由空间离散引起的误差。作为抽象理论的应用,我们考虑了一个具有动力学边界条件的声波方程,并给出了一些数值实验来说明我们的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Full discretization error analysis of exponential integrators for semilinear wave equations
In this article we prove full discretization error bounds for semilinear second-order evolution equations. We consider exponential integrators in time applied to an abstract nonconforming semi discretization in space. Since the fully discrete schemes involve the spatially discretized semigroup, a crucial point in the error analysis is to eliminate the continuous semigroup in the representation of the exact solution. Hence, we derive a modified variation-ofconstants formula driven by the spatially discretized semigroup which holds up to a discretization error. Our main results provide bounds for the full discretization errors for exponential Adams and explicit exponential Runge– Kutta methods. We show convergence with the stiff order of the corresponding exponential integrator in time, and errors stemming from the spatial discretization. As an application of the abstract theory, we consider an acoustic wave equation with kinetic boundary conditions, for which we also present some numerical experiments to illustrate our results.
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