The streamline subgrid integration method: I. Quasi-monotonic second-order transport schemes

A new finite-volume method has been developed for conservative and monotonic transport in multiple dimensions without any sort of dimension splitting to emphasize the space–time integrity of the fluid system. The streamline subgrid integration (SSI) method is a combination of the semi-Lagrangian and the finite-volume methods with generalized multidimensional subgrid distributions. Second-order transport schemes are constructed in two dimensions, and their extensions to three dimensions are also discussed. Spurious divergence is rigorously controlled with Lagrangian control volumes, monotonicity is well preserved for both compressible and incompressible flows, and positive-definiteness is always guaranteed. The accuracy and the numerical properties of these schemes are evaluated with both continuous and discontinuous solutions in flows of rigid motion and incompressible deformation where analytic solutions are available. Icosahedral geodesic grids are selected to demonstrate the general nature of the SSI method in spherical geometry that the numerical solutions are virtually not affected by the irregularity of grid structure. We have found that the numerical solutions converge quickly to the analytic solutions, and the SSI second-order schemes are relatively more beneficial for high-resolution applications.