Exponential High-order Compact Difference Schemes For Solving Steady Convection Diffusion Equation

In this paper, two types of exponential high-order compact (EHOC) finite difference schemes for solving steady convection diffusion equations are proposed. Firstly, based on the method of undetermined coefficients, an exponential compact difference scheme on uniform grid is proposed for solving the one, two and three-dimensional steady convection diffusion equations. Where, the scheme for one dimensional problems has ninth order accuracy. The Fourier analysis is utilized to show that the present method has a spectral like resolution. The scheme for solving the two and three dimensional problems have only fourth order accuracy because of the influence on the mixed derivatives. The computed results show that the new method is better than some other high order difference schemes given in the literature. And then, novel EHOC method on the nonuniform orthogonal grid is proposed for solving the one, two and three dimensional steady convection diffusion equations with constant or variable convection coefficients. The present method has at least third order accuracy. When the grids reduce to the uniform grids, the present method has fourth order accuracy in theory. The advantages of the present schemes show in the two aspects. On one hand, the present scheme can introduce an artificial diffusion that preserves the upwind effect to improve the computational accuracy. On the other hand, the use of the nonuniform grid strategy which can cluster the grid points in the regions of larger gradients and distribute a relatively coarse grid in the smooth regions. So the present method can improve the resolution without refining the mesh. The present method is well suitable for solving the problems with the high gradients or boundary layers. Some numerical experiments are preformed to demonstrate the superiority of the present method. Numerical results show that the present method has a significant accuracy and higher resolution than previous any other published schemes.