  In this paper, a high-order exponential scheme is developed to solve the 1D
unsteady convection-diffusion equation with Neumann boundary conditions. The
present method applies fourth-order compact exponential difference scheme in
spatial discretization at all interior and boundary points. The Pad\'e
approximation is used for the discretization. The resulting scheme obtains
fourth-order accuracy in both spatial and temporal discretization. In each
iterative loop, the scheme corresponds to a strictly diagonally dominant
tridiagonal matrix equation, which can be inverted by simple tridiagonal
Gaussian decomposition. The developed scheme is proved numerically
unconditionally stable for convection dominated problems. Four typical PDEs
with Neumann boundary conditions are provided to verify the accuracy of the
proposed scheme. The results are compared with analytical solutions and
numerical results calculated by different numerical methods. It shows that the
new scheme produces high accuracy solutions for all the test problems and it is
more suitable for dealing with convection dominated problems.
