The shear shallow water model provides a higher order approximation for shallow water flows by including the effect of vertical shear in the model. This model can be derived from the depth averaging process by including the second order velocity fluctuations which are neglected in the classical shallow water approximation. The resulting model has a non-conservative structure which resembles the 10-moment equations from gas dynamics. This structure facilitates the development of path conservative schemes and we construct HLL, 3-wave and 5-wave HLLC-type solvers. An explicit and semi-implicit MUSCL-Hancock type second order scheme is proposed for the time integration. Several test cases including roll waves show the performance of the proposed modeling and numerical strategy. Figure 1: Shallow water approximation: The free surface is given by x 3 = ξ(x 1 , x 2 , t) and the bottom surface is given by(acoustic) and b-waves (shear), and develops an approximate Riemann solver for each one independently. Each of these sub-systems is also augmented with the energy conservation equation (3) which is used to derive some jump conditions required to develop the Riemann solvers. The approach in [4] also uses the same acoustic and shear sub-systems and develops fluctuation splitting schemes for each sub-system on unstructured grids, but does not make use of the total energy equation (3).In the present work, we cast the SSW equations in a particular non-conservative form which is similar to the 10-moment equations [15, 2] from gas dynamics. In this model, instead of equations for the stress P, we have equations for an energy tensor E, while the mass and momentum equations remain unchanged. This form of the equations naturally arises when we perform the depth averaging of the 3-D Euler equations and the derivation is given in Appendix A. In fact, the equation for the energy tensor appears in [19] but it has not been used by any of the researchers to develop a numerical approximation. We suggest that the form of the equations is important and hence we retain the equation structure arising from depth averaging to build a numerical approximation. The non-conservative terms in this form contain only derivatives of the water depth h unlike model (2) which has derivatives of v, P in the non-conservative terms. The presence of only the derivatives of h in the non-conservative terms facilitates the construction of path conservative schemes [8]. By using the generalized Rankine-Hugoniot (RH) jump conditions arising from taking a linear path in the state space, we build HLL-type Riemann solvers for the new system. We construct the HLL, a 3-wave HLLC and a 5-wave HLLC solver, with the last one including all the waves in the Riemann problem. Unlike previous works, we do not split the model in several sub-systems but instead we construct a unified Riemann solver for the full system. A higher order version of the scheme is constructed following the MUSCL-Hancock approach [21] where we make the source terms implicit. The resulting semi-implici...

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We develop a positivity-preserving finite difference WENO scheme for the Ten-Moment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability preserving Runge-Kutta scheme. The positivity of the forward Euler scheme is obtained under a CFL condition by using a scaling type limiter, while the solution of the source operator is performed exactly and is positive without any restriction on the time step. The proposed method can be used with any WENO reconstruction scheme and we demonstrate it with fifth order accurate WENO-JS, WENO-Z and WENO-AO schemes. An adaptive CFL strategy is developed which can be more efficient than the use of reduced CFL for positivity preservation. Numerical results show that high order accuracy and positivity preservation are achieved on a range of test problems.

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