A study of the highorder accuracy and positivedefinite conformal advection scheme in the GRAPES model Ⅰ: Scientific design and idealized tests

Accuracy of transport scheme has an important influence on the performance of a numerical weather prediction (NWP) model. How to develop a highaccuracy scalar transport scheme is of significance in improving the accuracy of the semiLagrangian model. This study applies a new highorder accuracy and positivedefine conformal advection scheme into a semiLagrangian model by using the cell-integrated scheme. It not only remains the large time step and high computational efficiency of the semiLagrangian time integration scheme, but also gives scope to the new advection scheme’s advantages of highorder accuracy and positive define conformal computation. The new scalar advection scheme, named the Piecewise Rational Method （PRM）, is a deformation of the existing highorder Godunov scheme based on the piecewise rational function. This scheme is simple, practical and easy to program. It also can keep the scalar variables conservative in the advection process. The one and two-dimensional ideal experiments were designed to test the PRM scheme. It is found that the PRM has great ability in treating the large spatial variations compared with the PPM （Piecewise Parabolic Method） and the cubicLagrangian interpolation method. The dissipation error of the PRM is smaller than that of the cubicLagrangian interpolation scheme. The PRM is much similar to the PPM with almost the same advection effect. But the PRM makes use of the convexity preserving nature of the rational function, and avoids the adjustments of the cellinterface values to enforce the monotonicity in the PPM. In addition, another ideal experiments were designed where the PRM is applied in place of the original water substances advection scheme utilized in the GRAPES model in the spherical coordinate system. The result further shows the advantages of the PRM and its feasibility in the GRAPES.