A compact h⁴ scheme of central type for the advection-diffusion equation is developed, based on the central difference scheme and via the perturbation technique, and applied to flow model equations with promising results.

The perturbational fourth-order finite difference scheme of exponential type proposed by the authors is improved, by replacing the exponential coefficients in the source term perturbation with rational expressions, and is applied, aimed at the basic difficulties in computational fluid mechanics, to one-to three-dimensional model equations and a problem of natural convection with large Rayleigh-numbers, turning out the outstanding advantages of the perturbational scheme in such aspects as adaptability to flows with large Reynolds-numbers, resolution to ‘shock wave’-and ‘viscous boundary layer’-like effects, and saving computing CPU time.

A kind of exponential finite difference schemes with h⁴ consistency are developed in this study. The h⁴ scheme is obtained from a second-order modification of the convective coefficients and the source term in an h² scheme, and the modification could be determined once and for all from computational information of the h² scheme, which bring great convenience to the h⁴ scheme. The proposed exponential scheme are unconditionally stable, and show a excellent accuracy and adaptability to great gradient variation when applicated in illustrative computations of 1D to 3D fluid flow model problems.