The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics. This paper analyzes the accuracy of the immersed boundary method. The procedure contains two parts, I.e., the code verification and the accuracy analysis. The code verification provides the confidence that the code used is free of mistakes, and the accuracy analysis gives the order of accuracy of the immersed boundary method. The method of manufactured solutions is taken as a means for both parts. In the first part, the numerical code employs a second-order discretization scheme, I.e., it has second-order accuracy in theory. It matches the calculated order of accuracy obtained in the numerical calculation for all variables. This means that the code contains no mistake, which is a premise of the subsequent work. The second part introduces a jump in the manufactured solution for the pressure and adds the corresponding singular forcing terms in the momentum equations. By analyzing the discretization errors, the accuracy of the immersed boundary method is proven to be first order even though the discretization scheme is second order. It has been found that the coarser mesh may not be sensitive enough to capture the influence of the immersed boundary, and the refinement on the Lagrangian markers barely has any effect on the numerical calculation.