On the accuracy of high-order finite elements in curvilinear coordinates

The governing equations for shallow water flow on the sphere are formulated in generalized curvilinear coordinates. The various analytic expressions for the differential operators are all mathematically equivalent. However, numerical approximations are not equally effective. The accuracy of high-order finite element discretizations are evaluated using the standard test problems proposed by Williamson et al (1992). The so-called strong conservation formulation is far more accurate and results in standard error metrics that are at least two orders of magnitude smaller than the weak conservation form, Jorgensen (2003), Prusa and Smolarkeiwicz (2003). Moreover, steady state solutions can be integrated much longer without filtering when time-stepping the physical velocities.