Two-dimensional semi-Lagrangian transport with shape-preserving interpolation

Abstract

The more attractive one-dimensional, shape-preserving interpolation schemes as determined from a companion study are applied to two-dimensional semi-Lagrangian advection in plane and spherical geometry. Hermite cubic and a rational cubic are considered for the interpolation form. Both require estimates of derivatives at data points. A cubic derivative form and the derivative estimates of Hyman and Akima are considered. The derivative estimates are also modified to ensure that the interpolant is monotonic. The modification depends on the interpolation form. Three methods are used to apply the interpolators to two-dimensional semi-Lagrangian advection. The first consists of fractional time steps or time splitting. The second consists of two-dimensional interpolants with formal definitions of a two-dimensional monotonic surface and application of a two-dimensional monotonicity constraint. The additional complications expected in extending it to three dimensions and the lack of corresponding two-dimensional forms for the rational cubic led to the consideration of the third approach - a tensor product form of monotonic one-dimensional interpolants. The two-dimensional interpolants are easily applied to spherical geometry using the natural polar boundary conditions. No problems are evident in advecting test shapes over the poles. -from Authors