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.