Efficient space and time solution techniques for high-order discontinuous galerkin discretizations of the 3d compressible navier-stokes equations

• Autores: Andrés Mauricio Rueda Ramírez
• Directores de la Tesis: Eusebio Valero Sánchez (dir. tes.) , Esteban Ferrer Vaccarezza (codir. tes.)
• Lectura: En la Universidad Politécnica de Madrid ( España ) en 2019
• Idioma: español
• Tribunal Calificador de la Tesis: Francisco Javier de Vicente Buendía (presid.) , Gonzalo Rubio Calzado (secret.) , Oriol Lehmkuhl Barba (voc.) , David Kopriva (voc.) , Sonia Fernández Méndez (voc.)
• Enlaces
  • Tesis en acceso abierto en: Archivo Digital UPM
• Resumen

  • In this thesis, we develop space and time solution techniques for high-order Discontinuous Galerkin (DG) methods. Even though the main focus is the accurate and computationally efficient solution of the 3D compressible Navier-Stokes equations using the Discontinuous Galerkin Spectral Element Method (DGSEM), the methods developed here can be applied to other systems of nonlinear conservation laws and other DG methods. The solution techniques can be classified into two groups: (i) improved solution techniques for temporal discretizations, and (ii) local p-adaptation techniques that enhance the spatial discretization.

    The first group of techniques decreases the computational time for a given spatial discretization with a fixed number of degrees of freedom (NDOF) by improving the linear and nonlinear solver algorithms. In that regard, special attention is paid to the development of efficient implicit time-integration solvers and to multigrid methods (both explicit and implicit in time). We derive a methodology to obtain the analytical Jacobian of DG discretizations of nonlinear advection-diffusion equations. Furthermore, we show that time-implicit DGSEM discretizations on Gauss-Lobatto points can be formulated as Schur complement problems and solved using the static-condensation method. This enables one to solve smaller, better-conditioned problems, which yields shorter computation times for moderate polynomial orders.

    The second group of techniques modifies the spatial discretization by reducing the NDOF while maintaining similar accuracy. To do that, we develop a novel truncation error estimator that uses the tau-estimation method for the p-anisotropic DGSEM. The new method is more accurate and computationally cheaper than previous approaches. The error estimator is then successfully applied to perform local anisotropic p-adaptation for steady and unsteady flows. We show that the truncation error estimator can be readily coupled with multigrid techniques to obtain further speed-ups. Moreover, we develop two p-adaptation strategies for unsteady flows. One of them is a dynamic tau-based anisotropic p-adaptation method. The other one is a static tau-based anisotropic p-adaptation method, which may be computationally cheaper in statistically steady flows.