A key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also determine how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29-58]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065], which introduces the hydrostatic reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro D´ıaz, and C. Par´es, Appl. Math. Comput., 219 (2013), pp. 9012-9032], which analyzes whether a wave has enough energy to overcome a step. The third is our new scheme, which borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane with no-slip condition on the velocity field, perfectly conducting wall condition on the magnetic field and Dirichlet boundary condition on the temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer which is described by a Prandtl-type system. Under the non-degeneracy condition on the tangential magnetic field instead of monotonicity of velocity, by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work [27], we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.

In this paper, we prove the global existence of solutions with analytic regularity to the 2D magnetohydrodynamic (MHD) boundary layer equations in the mixed Prandtl and Hartmann regime derived by formal multiscale expansion in [D. Gerard-Varet and M. Prestipino, <i>Z. Angew. Math. Phys.</i>, 68 (2017), 76]. The analysis shows that the combined effect of the magnetic diffusivity and transverse magnetic field on the boundary leads to a linear damping on the tangential velocity field near the boundary. And this damping effect yields the global-in-time analytic norm estimate in the tangential space variable on the perturbation of the classical steady Hartmann profile.