‌The research in the Advance Algorithms and Numerical Simulation (AANS laboratory focuses on the stability and efficiency of spatial and temporal discretizations and mimetic & structure-preserving techniques which allow to transfer the results from the continuous level to the discrete one.

We design and analyze of numerical schemes that approximate the solution to hyperbolic PDEs, e.g., shallow water, compressible Euler, or ideal magnetohydrodynamic (MHD) as well as mixed hyperbolic-parabolic PDEs, e.g, compressible Navier-Stokes. In particular, in our laboratory we develop hp-adaptive high-order nodal discontinuous Galerkin (DG) numerical methods for linear and non-linear PDEs. With a special design, a particular flavour of collocation DG scheme is created with discrete differentiation operators that satisfy the summation-by-parts (SBP) property. This is important because the DG approximation can then be constructed to conserve primary quantities, like the density, as well as incorporate auxiliary physical principles, like the second law of thermodynamics. Generally, this builds a DG framework that can discretize split forms of the governing PDEs at high-order. 

Numerical Analysis

• Numerical spatial discretizations for hyperbolic and mixed hyperbolic-parabolic conservation laws: Discontinuous Galerkin methods, finite volume methods, finite difference schemes
• Entropy and energy stability: Summation-by-parts operators, mimetic properties
• Implicit and explicit Runge-Kutta methods: stability and efficiency

Scientific Computing

• Compressible Euler and Navier-Stokes equations, shallow water equations, magnetohydrodynamics, dense gas flow
• High performance computing with CPUs and GPUs
• Modeling and analysis of physical processes

Applications

• Computational aerodynamics
• Study of highly separated turbulent flows
• Computational aeroacoustics
• Combustion
• Aerodynamic shape optimization
• Study of hemodynamic flows
• Pollution transport