An accurate and efficient numerical framework for adaptive numerical weather prediction

TL;DR

This paper introduces a highly accurate, stable, and computationally efficient adaptive numerical framework for weather prediction that combines semi-Lagrangian methods, high-order spatial discretization, and p-adaptivity to optimize accuracy and cost.

Contribution

It develops a novel adaptive discretization method that integrates semi-Lagrangian, semi-implicit, and high-order finite element techniques without remeshing, suitable for complex weather models.

Findings

Achieves second-order accuracy in time and high spatial polynomial degrees.

Allows large time steps up to 100 times larger than explicit methods.

Reduces computational cost through effective adaptivity.

Abstract

We present an accurate and efficient discretization approach for the adaptive discretization of typical model equations employed in numerical weather prediction. A semi-Lagrangian approach is combined with the TR-BDF2 semi-implicit time discretization method and with a spatial discretization based on adaptive discontinuous finite elements. The resulting method has full second order accuracy in time and can employ polynomial bases of arbitrarily high degree in space, is unconditionally stable and can effectively adapt the number of degrees of freedom employed in each element, in order to balance accuracy and computational cost. The p-adaptivity approach employed does not require remeshing, therefore it is especially suitable for applications, such as numerical weather prediction, in which a large number of physical quantities are associated with a given mesh. Furthermore, although the proposed method can be implemented on arbitrary unstructured and nonconforming meshes, even its application on simple Cartesian meshes in spherical coordinates can cure effectively the pole problem by reducing the polynomial degree used in the polar elements. Numerical simulations of classical benchmarks for the shallow water and for the fully compressible Euler equations validate the method and demonstrate its capability to achieve accurate results also at large Courant numbers, with time steps up to 100 times larger than those of typical explicit discretizations of the same problems, while reducing the computational cost thanks to the adaptivity algorithm.