In this paper, we study nonsymmetric and highly nondiagonally dominant linear systems that arise from discretizations of constant-coefficient first-order partial differential equations (PDEs). We apply the generalized minimal residual method [Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston] for solving the system with a preconditioner based on the fast sine transform. An analytic formula for the eigenvalues of the preconditioned matrix is derived and it is shown that the eigenvalues are clustered around 1 except some outliers. The outlier eigenvalues are bounded and well separated from the origin when the size of system increases. In numerical experiments, we compare our preconditioner with the semi-Toeplitz preconditioner proposed in [SIAM J. Sci. Comput. 17 (1996) 47]. We refer to [J. Numer. Linear Algebra Appl. 1 (1992) 77, Numer. Math. J. Chinese Univ. 2 (1993) 116, BIT 32 (1992) 650, Linear Algebra Appl. 293 (1999) 85] for the early works on preconditioning techniques for PDEs.