Application of the parabolic spline method (PSM) to a multi-dimensional conservative semi-Lagrangian transport scheme (SLICE)

The recently devised one-dimensional parabolic spline method (PSM) for efficient, conservative, and monotonic remapping is introduced into the semi-Lagrangian inherently-conserving and efficient (SLICE) scheme for transport problems in multi-dimensions. To ensure mass conservation, an integral form of the transport equation is used rather than the differential form of classical semi-Lagrangian schemes. Integrals within the SLICE scheme are computed using multiple sweeps of PSM along flow-dependent cascade directions to avoid the large timestep-dependent splitting errors associated with traditional fixed-direction splitting. Accuracy of the overall scheme, including at large timestep, is demonstrated using two-dimensional test problems in both Cartesian and spherical geometries and compared with that of the piecewise parabolic method (PPM) applied within the same SLICE framework.