Well-balanced finite difference WENO schemes for the Ripa model

• A high-order well-balanced weighted essentially non-oscillatory (WENO) scheme is designed for the Ripa model.
• The scheme preserves the well-balanced property.
• The scheme achieves high order accuracy for smooth solutions.
• The scheme possesses sharp shock transition.

The Ripa model is a non-homogeneous hyperbolic system with a non-zero source term, which is deduced from the shallow water equations by incorporating the horizontal temperature gradients. The Ripa model can maintain steady state solutions in which the flux gradients are non-zero but exactly balanced by the source term. It is a challenging task to design genuinely high order accurate numerical schemes which preserve exactly these steady state solutions. In this paper we design high order well-balanced finite difference weighted essentially non-oscillatory (WENO) schemes to this model. Rigorous theoretical analysis as well as extensive 1D and 2D numerical results all demonstrates that the present schemes share high order accuracy, maintain the well-balanced property, and keep good resolution for both smooth and discontinuous solutions.