A study on the high-order Smolarkiewicz methods

Abstract

A modified Smolarkiewicz positive definite scheme is proposed. The computational cost of the proposed scheme is exactly the same as the high-order (third-order accurate) Smolarkiewicz method. We test numerically the convergence rate, stability, accuracy, peak value maintenance and mass loss for the method. Comparative studies of the modified scheme with other positive definite schemes, the fourth-order finite difference and the semi-Lagrangian method are performed. It is found that the modified scheme provides an error convergence rate which is comparable to both the semi-Lagrangian and fourth-order finite difference schemes. When the advective field is smooth enough, the modified scheme yields improved fourth-order accuracy in the low Courant number situation. When the Courant number is larger than 0.5, the modified procedure yields the same result as the third-order Smolarkiewicz scheme. Other advection calculations with a slotted cylinder yield results which compare favorably with the results of the high-order (third-order accurate) Smolarkiewicz method and the semi-Lagrangian method.