document: Numerical studies of 2D free surface waves with fixed bottom The motion of surface waves under the effect of bottom is a very interesting and challenging phenomenon in the nature . we use boundary integral method to compute and analyze this problem . In the linear analysis , the linearized equations have bounded error increase under some compatible conditions . This contributes to the cancellation of instable Kelvin-Helmholtz terms . Under the effect of bottom , the existence of equations is hard to determine , but given some limitations it proves true . These limitations are that the swing of interfaces should be small enough , and the distance between surface and bottom should be large enough . In order to maintain the stability of computation , some compatible relationship must be satisfied . In the numerical examples , the simulation of standing waves and breaking waves are calculated . And in the case of shallow bottom , we found that the behavior of waves are rather singular
doc_bio_tags: B I O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O