Application of High‐Order Direct Flux Reconstruction and Stiffness‐Resilient Time Integration to Simulations of Idealized Atmospheric Flows

ABSTRACT High‐order accurate discretizations in space and time are applied to the compressible Euler equations on the rotated cubed‐sphere grid. The proposed methodology combines the Direct Flux Reconstruction (DFR) method for spatial discretization and stiffness‐resilient exponential time integration for temporal evolution. The DFR method is quadrature‐free and offers high‐order accuracy, good conservation properties, and geometric flexibility. Stiffness‐resilient exponential integrators allow for larger time steps than explicit methods while maintaining accuracy and stability. The governing equations are formulated using a space‐time tensor formalism, allowing for a general representation in any curvilinear coordinate system. Discretization of these equations presents a number of challenges, including the numerical evaluation of geometric terms, pressure gradients, gravitational forcing, and aliasing errors. Stabilization techniques—such as filtering, numerically consistent calculation of Christoffel symbols, and vertical logarithmic reconstruction—are proposed to address these issues and compute physically plausible solutions. The numerical schemes are evaluated using a series of standard numerical tests.