Jacobian-Free Newton-Krylov Discontinuous Galerkin (JFNK-DG) Method and Its Physics-Based Preconditioning for All-Speed Flows

The Discontinuous Galerkin (DG) method for compressible fluid flows is incorporated into the Jacobian-Free Newton-Krylov (JFNK) framework. Advantages of combining the DG with the JFNK are two-fold: $a)$ enabling \textit{robust and efficient high-order-accurate modeling of all-speed flows on unstructured grids}, opening the possibility for high-fidelity simulation of nuclear-power-industry-relevant flows; and $b)$ ability to \textit{tightly, robustly and high-order-accurately couple with other relevant physics} (neutronics, thermal-structural response of solids, etc.). In the present study, we focus on the physics-based preconditioning (PBP) of the Krylov method (GMRES), used as the linear solver in our implicit higher-order-accurate Runge-Kutta (ESDIRK) time discretization scheme; exploiting the compactness of the spatial discretization of the DG family. In particular, we utilize the \textit{Implicit Continuous-fluid Eulerian (ICE) method} and investigate its efficacy as the PBP within the JFNK-DG method. Using the eigenvalue analysis, it is found that the ICE collapses the complex components of the \underline {all eigenvalues} of the Jacobian matrix (associated with pressure waves) onto the real axis, and thereby enabling at least an order of magnitude faster simulations in nearly-incompressible/weakly-compressible regimes with a significant storage saving.