Notes on Galerkin-finite element methods for the Shallow Water equations with characteristic boundary conditions
Provides theoretical error bounds for a numerical method applied to a specific class of hyperbolic PDEs with transparent boundary conditions, which is an incremental contribution for researchers in numerical analysis and computational fluid dynamics.
The paper proves L^2 error estimates for Galerkin-finite element semidiscretizations of the Shallow Water equations with characteristic boundary conditions, and demonstrates via numerical experiments that fully discrete schemes using explicit Runge-Kutta time stepping achieve excellent absorption of outgoing waves.
We consider the Shallow Water equations in the supercritical and subcritical cases in one space variable,posed in a finite spatial interval with characteristic boundary conditions at the endpoints, which, as is well known, are transparent,i.e. allow outgoing waves to exit without generating spurious reflected waves. Assuming that the resulting initial-boundary-value problems have smooth solutions,we approximate them in space using standard Galerkin-finite element methods and prove L^2 error estimates for the semidiscrete problems on quasiuniform meshes.We discretize the problems in the temporal variable using an explicit,fourth-order accurate Runge-Kutta scheme and check, by means of numerical experiment, that the resulting fully discrete schemes have excellent absorption properties.