Research Projects

This project develops a high-order and robust method for simulating compressible fluid flows based on the Continuous Galerkin Spectral Element Method (CGSEM). CGSEM combines the smoothness of continuous Galerkin methods with the high accuracy of spectral elements. It makes use of Gauss–Lobatto-Legendre (GLL) collocation points and tensor-product polynomial bases, which provide diagonal mass matrices and fulfil the summation-by-parts (SBP) property. Compared to Discontinuous Galerkin (DG) methods, which require about three times more degrees of freedom in 3D for the same polynomial degree, CGSEM achieves similar accuracy at a much lower cost. However, classical CGSEM schemes are not suitable for convection-dominated flows due to dispersion errors and numerical instabilities (see Figure 1a in the pdf). To address this, we will incorporate SBP-compatible split-form fluxes, entropy and kinetic energy preserving discretizations, and a gradient-jump penalty (GJP) (see Figure 2a in the pdf). These features can enable CGSEM to deliver stable, physically consistent simulations of compressible turbulence in both LES and DNS regimes. We will use two-point flux functions that satisfy Tadmor's entropy identities. Such fluxes ensure discrete conservation of kinetic energy or entropy under periodic or appropriately posed boundary conditions. When combined with SBP structure from GLL quadrature, these formulations support energy-stable and high-order accurate schemes for nonlinear hyperbolic systems. To suppress high-frequency numerical oscillations and enhance robustness in under-resolved regimes, we add a gradient-jump penalty (GJP) term, which is a function of local wave speed, a penalty coefficient, a characteristic length scale, jump in the gradient of variable and jump in gradient of basis function. The GJP term vanishes for smooth fields and activates only across interfaces with large normal derivative jumps, providing localized, high-wavenumber dissipation. These ingredients yield a discrete entropy inequality ensuring numerical stability and thermodynamic consistency. The resulting method can function as an implicit LES scheme, capturing inertial-range dynamics and anomalous dissipation (in the sense of Onsager) without relying on explicit subgrid-scale models. The project aims to establish CGSEM—with modern, physics-compatible stabilization—as a competitive alternative to DG for high-fidelity compressible flow simulation (also see Figure 2 in the pdf).

CGSEM, SBP, Split‐Form fluxes, GJP Stabilization, Energy/Entropy Conservation, Euler/Navier-Stokes Equations