Primitive variable regularization to derive novel Hyperbolic Shallow Water Moment Equations

Abstract: Shallow Water Moment Equations are reduced-order models for free-surface flows that employ a vertical velocity expansion and derive additional so-called moment equations for the expansion coefficients. Among desirable analytical properties for such systems of equations are hyperbolicity, accuracy, correct momentum equation, and interpretable steady states. In this paper, we show analytically that existing models fail at different of these properties and we derive new models overcoming the disadvantages. This is made possible by performing a hyperbolic regularization not in the convective variables (as done in the existing models) but in the primitive variables. Via analytical transformations between the convective and primitive system, we can prove hyperbolicity and compute analytical steady states of the new models. Simulating a dam-break test case, we demonstrate the accuracy of the new models and show that it is essential for accuracy to preserve the momentum equation.