Spline Shallow Water Moment Equations

Reduced models for free-surface flows are required due to the high dimensionality of the underlying incompressible Navier-Stokes equations, which need to fully resolve the flow in vertical direction to compute the surface height. On the other hand, standard reduced models, such as the classical Shallow Water Equations (SWE), which assume a small depth-to-length ratio and use depth-averaging, do not provide information about the vertical velocity profile variations. As a compromise, a recently proposed moment approach for shallow flow using Legendre polynomials as ansatz functions for vertical velocity variations showed the derivation of so-called Shallow Water Moment Equations (SWME) that combine low dimensionality with velocity profile modeling. However, only global polynomials are considered so far. This paper introduces Spline Shallow Water Moment Equations (SSWME) where piecewise defined spline ansatz functions allow for a flexible representation of velocity profiles with lower regularity. The local support of the spline basis functions opens up the possibility of adaptability and greater flexibility regarding some typical profile shapes. We systematically derive and analyze hierarchies of SSWME models with different number of basis functions and different degrees, before deriving a regularized hyperbolic version by performing a hyperbolic regularization with analytical proof of hyperbolicity for a hierarchy of high-order SSWME models. Numerical simulations show high accuracy and robustness of the new models.