From Well-Posed Initial-Boundary-Value Problems for the Shallow Water Equations to the Hyperbolicity of the Grass Equation

In this presentation, we begin with the linearised shallow water equations, where we derive well-posed boundary conditions using the energy method by determining the required number, location, and form of the boundary conditions for the initial-boundary-value problem. Based on this analysis, we develop a stable finite-volume method within the summation-by-parts framework, with the boundary conditions imposed weakly through penalty terms, and establish discrete stability through an energy estimate that mimics the continuous analysis. We then extend this framework to the one-dimensional Grass sediment transport model by incorporating the well-posed boundary conditions derived for the shallow water equations. At the continuous level, stability is rigorously established using the energy method, while at the semi-discrete level, dual pairing summation-by-parts (DP-SBP) operators constructed from finite-element operators are employed together with simultaneous approximation term (SAT) penalties to obtain an energy-stable numerical scheme. Finally, we extend the study to the two-dimensional Grass sediment transport model coupled with the linearised shallow water equations. We establish strong hyperbolicity by analysing the characteristic polynomial of the locally frozen system and employ perturbation analysis to determine the signs of the eigenvalues. Although the number of real eigenvalues remains unchanged as the flow regime transitions from subcritical to supercritical, one characteristic speed changes sign. We further show that each flow regime admits a complete set of eigenvectors, thereby guaranteeing strong hyperbolicity. Numerical experiments are presented throughout to verify the theoretical analysis and demonstrate the stability and effectiveness of the proposed methods.