Using cubic Hermite polynomials in constructing monotone semi-Lagrangian methods for advection equation

Semi-Lagrangian methods have been widely applied in general circulation models of the atmosphere as they do not suffer from a Courant–Fredericks–Levy (CFL) constraint for computational stability. Ease of application, high accuracy and speed of execution in general circulation models are other reasons for the popularity of semi-Lagrangian methods. Two fundamenta lissues in semi-Lagrangian methods are related to the trajectory computation and interpolation from the regular grid to departure points. If sufficiently accurate schemes are used to solve for trajectories with interpolations, one can expect good performance from the semi-Lagrangian scheme in solving the equations of motion of the atmosphere. Two general methods of solving the trajectory equation are the forward and backward methods. Most semi-Lagrangian methods use backward-trajectory schemes for estimating positions of the air parcels that arrive at the grid points in the future time step. Solving the trajectory equation is carried out by iteration. In the research reported, two iterations are used for trajectory computation. The fundamental difference between the forward and backward trajectory scheme rests in the calculation of advective quantity at the departure and destination points. While in the backward solution procedure, it is necessary to make interpolation from the regular grid to departure points; in the forward scheme, it is necessary to make interpolation from the irregular grid of destination points to the regular grid. The usually used interpolation methods in the semi-Lagrangian method include piecewise cubic Lagrange and Hermite, cascades, and monotone Hermite. Increasing the degree of polynomial interpolation leads to a higher degree of formal accuracy, but it leads to the generation of unwanted oscillation in regions with severe gradients of the transported quantities. Eliminating the unwanted oscillations is done through a variety of methods which generally increase the computational cost and reduce the accuracy of the scheme. To address the issue, in this research, a new selective monotone semi-Lagrangian method is developed and tested along with two standard methods based on the Lagrange and Hermite interpolations. The Lagrange polynomials have been considered by researchers for the high speed of computation in operational models. The fictitious oscillations produced at the edges of sharp gradients of the advected quantities are the main shortcoming of this method. The fictitious oscillations cannot be eliminated by increasing the degree of interpolation polynomials, which can only lead to a reduction in the wavelength of the oscillations. The results presented on increasing the degree of interpolation polynomials clearly show that the removal of the fictitious oscillations requires the use of monotone polynomials for interpolation. It is important to note that the Hermite interpolation polynomials are not inherently monotone. To make them monotone, one needs to manipulate the derivatives at the grid points appropriately. This process, however, may lead to a substantial deteriration of accuracy. For this reason, in this paper, a selective interpolation method is desined to obtain the best accuracy in solution of the advection equation, while preserving monotnonicity and removing the issue with the fictitious oscillations. In the selective method, first the interpolation is done by the non-monotonic cubic Hermite and then a properly designed slope function is calculated at each grid interval. If the slope function takes values outside the range, it indicates that a fictitious oscillation has occurred in the interpolantion. To remove the oscillation, the non-monotone interpolation is abandoned and the monotone interpolation is performed by limiting the derivative to the monotone region. This technique can minimize the error caused by the changes in the derivatives. Results are shown to demonstrate the working and superiority of the seclective montone scheme.