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In this paper, a high-order exponential scheme is developed to solve the 1Dunsteady convection-diffusion equation with Neumann boundary conditions. Thepresent method applies fourth-order compact exponential difference scheme inspatial discretization at all interior and boundary points. The Pad\'eapproximation is used for the discretization. The resulting scheme obtainsfourth-order accuracy in both spatial and temporal discretization. In eachiterative loop, the scheme corresponds to a strictly diagonally dominanttridiagonal matrix equation, which can be inverted by simple tridiagonalGaussian decomposition. The developed scheme is proved numericallyunconditionally stable for convection dominated problems. Four typical PDEswith Neumann boundary conditions are provided to verify the accuracy of theproposed scheme. The results are compared with analytical solutions andnumerical results calculated by different numerical methods. It shows that thenew scheme produces high accuracy solutions for all the test problems and it ismore suitable for dealing with convection dominated problems.

A Compact Exponential Scheme for Solving 1D Unsteady Convection-Diffusion Equation with Neumann Boundary Conditions