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 method has noticeable displacement errors and larger diffusion than the other methods. The second consists of two-dimensional interpolants with formal definitions of a two-dimensional monotonic surface and application of a two-dimensional monotonicity constraint. This approach is examined for the Hermite cubic interpolant with cubic derivative estimates and produces very good results. The additional complications expected in extending to it 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. Although a description of the properties of the implied interpolating surface is difficult to obtain, the results show this to be a viable approach. Of the schemes considered, the Hermic cubic coupled with the Akima derivative estimate modified to satisfy a C⁰monotonicity condition produces the best solution to our test cases. The C₁monotonic forms of the Hermite cubic have serious differential phase errors that distort the test patterns. The C₁ forms of the rational cubic do not show this distortion and produce virtually the same solutions as the corresponding C₀forms. The second best scheme (or best C₁ continuity is desired) is the rational cubic with Hyman derivative approximations modified to satisfy C₁ monotonicity condition. 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. A procedure is also introduced to calculate the departure point in spherical geometry. The scheme uses local geodesic coordinate systems based on each arrival point. It is shown to be comparable in accuracy to the one proposed Ritchie, which uses a Cartesian system in place of the local geodesic system.