Research
Nonlinear Stable Schemes
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.
Extended the methods to ensure entropy dissipation for viscous flows.
Developed a fully-discrete entropy stable scheme with the use of the relaxation Runge-Kutta time-stepping scheme.
Introduced positivity-preserving limiters to capture shock waves.
PDE-Constrained Optimization
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. Current research topics in this area:
Approximate Hessian for Accelerated Convergence of Aerodynamic Shape Optimization.
Linear Solvers for Adjoint Systems.
Inexactly Constrained Discrete Adjoint Approach.
Adaptive High-Order Methods
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.
High-Performance computing
New massively parallel hardware for scientific computation promise new levels of performance for computational fluid dynamics codes. However, for leveraging such hardware for implicit solvers, fine-grain parallel algorithms are needed. Current research topics in this area:
Scaled-Additive Multigrid Methods.
Asynchronous Fine-grain Parallel Implicit Smoothers and Preconditioners.
Dynamically Deflated Krylov Solvers for Adjoint Systems.
