Embedded domain Reduced Basis Models for the shallow water hyperbolic equations with the Shifted Boundary Method

TL;DR

This paper develops a reduced basis modeling approach for shallow water hyperbolic equations using the Shifted Boundary Method on a fixed mesh, enabling efficient simulations across different geometries.

Contribution

It introduces a novel combination of the Shifted Boundary Method with reduced basis models and an interpolation pre-processing step for untested geometries.

Findings

Effective reduced-order models for parametrized geometries

Improved solution approximation for untested shapes

Validated approach on geometrically varying test cases

Abstract

We consider fully discrete embedded finite element approximations for a shallow water hyperbolic problem and its reduced-order model. Our approach is based on a fixed background mesh and an embedded reduced basis. The Shifted Boundary Method for spatial discretization is combined with an explicit predictor/multi-corrector time integration to integrate in time the numerical solutions to the shallow water equations, both for the full and reduced-order model. In order to improve the approximation of the solution manifold also for geometries that are untested during the offline stage, the snapshots have been pre-processed by means of an interpolation procedure that precedes the reduced basis computation. The methodology is tested on geometrically parametrized shapes with varying size and position.