The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge–Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of \(\left| s\right| \) (up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map.

分析了由波动方程建模的声音软散射问题的半离散化。通过积分方程法进行空间处理。比较了两种基于Runge-Kutta卷积正交的时间离散化：一种依赖于入射波作为输入数据，另一种依赖于其时间导数。后者的收敛速度显示出高于先前文献中确定的收敛速度。数值结果表明分析的清晰度。该证明取决于对某些亥姆霍兹问题的从狄利克雷到阻抗图的新颖估计。也就是说，与Dirichlet-to-Neumann映射相比，频率依赖性可以降低\（\ left | s \ right | \）的幂（最多为多边形域的对数项）。