Isobaric steady-state preserving adaptive surface reconstruction schemes for the two-dimensional Ripa system on triangles

This work aims to introduce adaptive surface reconstruction schemes to solve the Ripa system on moving triangular meshes. We use surface reconstructions to define approximate Riemann states to preserve stationary steady states, including the still-water steady state and the highly nontrivial isobaric steady state. To prevent spurious pressure oscillations near contact waves, we introduce a provable positivity-preserving parameter. Importantly, the scheme equipped with the parameter is provably positivity-preserving. To enhance numerical accuracy, we reconstruct piecewise linear polynomials that satisfy the local maximum principle, which ensures the positivity-preserving property and eliminates spurious oscillations near solutions with large gradients. In particular, the adaptive surface reconstruction scheme preserves stationary steady states, including the still-water steady state and the isobaric steady state. Finally, we validate these properties by presenting several computed results for the Ripa system.