A numerical stabilization scheme for the shallow shelf approximation

Abstract: We present the Thickness Stabilization Scheme (TSS), a numerical stabilization scheme suitable for the Shallow Shelf Approximation (SSA), one of the most widely-used models for large-scale Antarctic and Greenland ice sheet simulations. The TSS is constructed by inserting an adapted, explicit Euler thickness evolution equation into the driving stress term, thereby treating the term implicitly. We investigate the applicability of TSS across low- and high-shear idealized scenarios, by altering the inflow velocity and initial ice thickness. TSS demonstrates an increase in the numerical stability of SSA, allowing large time-step sizes of dt = 50-100 years to remain numerically stable and accurate, while time-step sizes of dt > 5 in high-shear simulations show significant error without TSS. Remarkably, a time step size as large as dt = 10 000 years is numerically stable with TSS, albeit with a reduction in accuracy. TSS offers greater flexibility for ice-sheet modeling by allowing the re-allocation of computational resources. This method is applicable not only to ice-sheet modeling, including in coupled frameworks, but also to other vertically-integrated computational fluid dynamics problems that couple momentum and geometry evolution equations.