28.7.4. Shallow water#

28.7.4.1. Integral equations#

  • Intgral equation for a control volume at rest \(V\)

  • Intgral equation for an arbitrary domain \(v_t\), with Reynolds’ transport theorem

28.7.4.1.1. Jump conditions#

  • From the integral equation for an arbitrary domain \(v_t\), collapsed on a surface

28.7.4.2. Differential equations#

  • In regions where the fields are smooth, from integral to differential equations with “divergence theorem”

28.7.4.2.1. Spectral decomposition#

  • Eigenvalues, right and left eigenvectors

  • Characteristic lines/surfaces, and compatibility conditions

28.7.4.3. Riemann problem#

  • Useful in numerical schemes in finite volume methods, using Godunov flux

28.7.4.3.1. Linearization - Roe intermediate state#

  • Local linearization of the problem, to reduce the computational cost of solving non-linear Riemann problems at all the interfaces in a grid in FVM

28.7.4.4. Boundary conditions#

  • characteristic-based

  • wall

  • …