  In this paper, we study a numerical artifact of solving the nonlinear shallow
water equations with a discontinuous bottom topography. For various first-order
schemes, the numerical solution of the momentum will form a spurious spike at
the discontinuous points of the bottom, which should not exist in the exact
solution. The height of the spike cannot be reduced even after the mesh is
refined. For subsonic problems, this numerical artifact may cause the wrong
convergence to a function far away from the exact solution. To explain the
formation of the spurious spike, we perform a convergence analysis by proving a
Lax--Wendroff type theorem. It is shown that the spurious spike is caused by
the numerical viscosity in the computation of the water height at the
discontinuous bottom. The height of the spike is proportional to the magnitude
of the viscosity constant in the Lax--Friedrichs flux. Motivated by this
conclusion, we propose a modified scheme by adopting the central flux at the
bottom discontinuity in the equation of mass conservation, and show that this
numerical artifact can be removed in many cases. For various numerical tests
with nontransonic Riemann solutions, we observe that the modified scheme is
able to retrieve the correct convergence.
