GWsim: a code to simulate gravitational waves propagating in a potential well

We present a code to simulate the propagation of GWs in a potential well in the time domain. Our code uses the finite element method (FEM) based on the publicly available code {\it deal.ii}. We test our code using a point source monochromatic spherical wave. We examine not only the waveform observed by a local observer but also the global energy conservation of the waves. We find that our numerical results agree with the analytical predictions very well. Based on our code, we study the propagation of the leading wavefront of GWs in a potential well. We find that our numerical results agree with the results obtained from tracing null geodesics very well. Based on our simulations, we also test the accuracy of the thin-lens model in predicting the positions of the wavefront. We find that the analytical formula of the Shapiro-time delay is only accurate in regimes that are far away from the center of the potential well. However, near the optic axis, the analytical formula shows significant differences from the simulated ones. Besides these results, we find that unlike the conventional images in geometric optics, GWs can not be sheltered by the scatterer due to wave effects. The signals of GWs can circle around the scatterer and travel along the optic axis until arrive at a distant observer, which is an important observational consequence in such a system.