Research

Our focus is on advancing a new class of methods that discretely satisfy a discrete conservation or dissipation of energy or entropy for nonlinear systems of equations with a specific focus on the flux reconstruction family of schemes. Termed NSFR for Nonlinear Stable Flux Reconstruction, in the past several years, we have

• Demonstrated mathematically and numerically the conservation of energy and entropy for systems of nonlinear equations. (Cicchino et al. 2022, Cicchino et al. 2022, Cicchino & Nadarajah 2024, Cicchino & Nadarajah 2025)

• Demonstrated entropy-stable schemes for large-eddy simulations. (Brillon & Nadarajah 2026)

• Developed a fully-discrete entropy-stable scheme with the use of the relaxation Runge-Kutta time-stepping scheme. (Pethrick & Nadarajah 2026)

• Introduced positivity-preserving limiters to capture shock waves. (Srinivasan & Nadarajah 2026)

Our objective is to create and develop the necessary infrastructure algorithms, methodologies, and toolboxes to enable the transition from current state-of-the-art Aerodynamic Shape Optimization (ASO) methodologies to a fully three-dimensional design environment for complex geometries. Over the past two decades, Adjoint-based PDE-Constrained Optimization has matured to the level of Multi-Disciplinary Design Optimization of complete Aircraft Configurations. Our current research topics in this area:

• Development of the Full-Space Approach for the Concurrent Solve of the Conservation Law, Adjoint System, and the Design Vector. (Shi-Dong & Nadarajah 2018, 2021) 

• Linear Solvers for Adjoint Systems.

• Inexactly Constrained Discrete Adjoint Approach.

Research in high-order numerical methods for the compressible Euler and Navier–Stokes equations has been particularly vigorous over the last decade. Indeed, using asymptotic arguments, it is commonly argued that high-order schemes can provide increased accuracy levels, in a more effective manner than current state-of-the-art second-order flow solvers. At the same time, it is also recognized that the widespread acceptance of high-order methods is hindered by the high cost of such schemes, and truly flexible and robust solvers are still at the development stage. To achieve these goals, advances in efficient high-order discretization, solution procedures and adaptivity are required, and the present study is a contribution in this direction. Current research topics in this area:

• Optimization-based Anisotropic hp-Adaptation for High-Order Methods.

• Stability of Energy Stable Flux Reconstruction for the Diffusion Problem Using Compact Numerical Fluxes.

• Geometry Representation for Discontinuous Galerkin Methods on Domains with Curved Boundaries.