A second-order semi-Lagrangian backward differentiation formula with Picard iterations (PICSL) for the spherical shallow-water equations

• Novel PICSL scheme combines semi-Lagrangian advection with second-order backward differentiation formula and Picard iterations, eliminating the need for off-centering while maintaining unconditional stability. • Theoretical analysis proves the absence of spurious computational modes and resonance issues that plague traditional semi-Lagrangian methods. • Multiple test cases confirm the scheme’s correct convergence behavior and conservation properties. • Provides computational performance comparable to existing implicit semi-Lagrangian schemes while improving accuracy. Semi-Lagrangian methods are widely employed in numerical weather prediction models due to their efficiency and enhanced stability compared to explicit Eulerian methods. However, achieving second-order accuracy in time while maintaining sufficient damping of spurious oscillations has remained challenging, often requiring off-centering techniques that degrade temporal accuracy. This paper introduces PICSL (Picard semi-Lagrangian), a new numerical scheme that combines a second-order backward differentiation formula (BDF2) with Picard iterations. It is designed to solve efficiently the nonlinear discrete system that results from temporal discretization. Theoretical analysis demonstrates that PICSL achieves second-order accuracy without the need for off-centering or filtering. The scheme avoids spurious computational modes and resonance problems that plague traditional schemes. Since BDF2 is A-stable, PICSL achieves unconditional stability for all Courant numbers without off-centering. The only restriction on time-step size arises from trajectory calculation convergence, governed by a Lipschitz condition that limits it in regions with large velocity gradients. Dispersion analysis confirms favorable wave propagation characteristics for both Rossby and gravity waves. The method ensures numerical consistency between trajectory calculations and governing equations through an iterative process with convergence-based stopping criteria. Numerical validations with shallow-water benchmarks (steady-state geostrophic flow, Rossby-Haurwitz waves, flow over topography, and barotropic instability) show that PICSL produces solutions comparable to centered semi-Lagrangian schemes while avoiding the excessive damping introduced by off-centered methods. Conservation properties are also thoroughly examined, and demonstrate improved preservation of total energy and potential enstrophy compared to off-centered methods. These results establish PICSL as a promising advancement for semi-Lagrangian time integration in geophysical fluid dynamics.