In collaboration with the Numerical Analysis Group at Texas A&M University, led by Professors Jean-Luc Guermond and Bojan Popov, our research focuses on the development and application of invariant-domain preserving Arbitrary Lagrangian–Eulerian (ALE) methods for the simulation of compressible and multiphase flows. 

We recently proposed an explicit numerical approximation for the Lagrangian hydrodynamics equations. This approach is invariant-domain preserving and locally mass conservative. Its main objective is to serve as a foundation for enhancing higher-order schemes.

📄 Selected publications:

• J.-L. Guermond, B. Popov, L. Saavedra, Y. Yang. Invariant Domains Preserving Arbitrary Lagrangian Eulerian Approximation of Hyperbolic Systems with Continuous Finite Elements. SIAM Journal on Scientific Computing, 2017.

🔗doi.org/10.1137/16M1063034 

•  J.-L. Guermond, B. Popov, L. Saavedra. Second‑order invariant domain preserving ALE approximation of hyperbolic systems. Journal of Computational Physics, 2020.

🔗doi.org/10.1016/j.jcp.2019.108927 

•  J.-L. Guermond, B. Popov, M. Sheridan, L. Saavedra.  Invariant-domain preserving and locally mass-conservative approximation of the Lagrangian hydrodynamics equations. Computer Methods in Applied Mechanics and Engineering, 2025

🔗doi.org/10.1016/j.cma.2025.117927 

Lagrange–Galerkin (LG) methods are semi-Lagrangian schemes that track the flow backward along characteristics to discretize the material derivative. This naturally introduces upwinding, making them suitable for advection-dominated flows such as incompressible and compressible Navier–Stokes equations. 

✅ Advantages:

• Unconditionally stables.

• Do not suffer from mesh deformation.

• Lead to symmetric algebraic systems (solving a linear Stokes problem at each time step).

⚠️ Challenges:

Accurate integration of transported terms can be expensive, especially with low viscosity. To address this, we apply Local Projection Stabilization (LPS) techniques, which improve stability and accuracy without breaking the structure of the method.

🧪 Key Contributions

✅ Stabilized LG schemes for high Reynolds number flows

✅ Use of BDF2 time-stepping for second-order temporal accuracy

✅ Implementation with inf-sup stable finite elements and quasi-local interpolation

📄 Selected publications:

• Bermejo, R., Saavedra, L. A second-order in time local projection stabilized Lagrange–Galerkin method for Navier–Stokes equations at high Reynolds numbers. Computers & Mathematics with Applications,  2016

🔗 doi.org/10.1016/j.camwa.2016.05.012 

• Bermejo, R., Saavedra, L. Lagrange–Galerkin methods for the incompressible Navier–Stokes equations: a review. Communications in Applied and Industrial Mathematics, 2016.

🔗doi.org/10.1515/caim-2016-0021 

• Bermejo, R., Carpio J.,  Saavedra, L. New error estimates of Lagrange–Galerkin methods for the advection equation. Calcolo, 2023.

🔗doi.org/10.1007/s10092-023-00509-5