# Non-dispersive conservative regularisation of nonlinear shallow water (and isothermal Euler) equations

## TL;DR

This paper introduces a novel non-dispersive, conservative regularisation for shallow water and Euler equations that preserves key physical quantities and produces smooth, non-oscillatory shocks, demonstrated through dam-break simulations.

## Contribution

It presents a new variational, non-dissipative regularisation method that maintains shock speeds and conserves mass, momentum, and energy in hyperbolic PDEs.

## Key findings

- Regularised shocks propagate at the same speed as original shocks.
- The method produces smooth, non-oscillatory hydraulic jumps.
- Numerical tests confirm the effectiveness on dam-break problems.

## Abstract

A new regularisation of the shallow water (and isentropic Euler) equations is proposed. The regularised equations are non-dissipative, non-dispersive and possess a variational structure. Thus, the mass, the momentum and the energy are conserved. Hence, for instance, regularised hydraulic jumps are smooth and non-oscillatory. Another particularly interesting feature of this regularisation is that smoothed `shocks' propagates at exactly the same speed as the original discontinuous ones. The performance of the new model is illustrated numerically on some dam-break test cases, which are classical in the hyperbolic realm.