The Finite Difference WENO Scheme For Shallow Water Flow Models Under The Influence Of Temperature Field

The Ripa model is deduced from the shallow water equations by incorporating the horizontal temperature gradients.It belongs to non-homogeneous hyperbolic systems with a non-flat bottom topography and a geometric source.It is worth noting that the Ripa model can maintain steady state solutions in which the flux gradients are non-zero but exactly balanced by the source term.The standard numerical processing for the source term can not maintain the constant solution,which can lead to false oscillation.Even if the use of grid refinement strategy,we can not eliminate the oscillation.A numerical scheme which can maintain the constant solution at discrete state is named well-balanced property.In this paper,we design high order well-balanced finite difference WENO schemes to this model by a special splitting of the source term and designing high order finite difference operator to approximate flux gradient and derivative in the source term.Rigorous theoretical analysis as well as extensive numerical results all demonstrates that the present scheme maintains the well-balanced property.One-and two-dimensional numerical results also show this scheme have high order accuracy,and keep good resolution for discontinuous solutions.