Khosro Shahbazi
Postdoctoral Research Associate,
Division of Applied Mathematics, Brown University

For more information please contact Dr. George Biros at gbiros@cc.gatech.edu
~~~~~~~~~~~~~~~~~~~~~~

Title:

Efficient High-Order Discontinuous Galerkin Methods for Fluid Flow Simulations

Abstract:

The inadequacy of current production-level computational fluid dynamics codes in delivering sufficient accuracy in numerical flow simulations, as well as in resolving a wide range of turbulence scales for reliable large eddy simulations have been widely realized over the past decade. Since these codes are typically based on low-order finite volume methods, high-order methods such as discontinuous Galerkin (DG) methods have been advocated as alternative discretization techniques. DG methods are weighted residual methods with discontinuous approximate solution spaces typically consisting of polynomials defined on each element of the geometry triangulation. The inter-element connectivities are enforced through a proper definition of numerical fluxes along the shared boundaries between elements. Like more traditional finite element methods (continuous Galerkin methods), DG methods are well-suited for simulations in complex geometries. Moreover, they offer advantages in capturing features of convection-dominated flows, facilitating hp-adaptivity, ease of parallelization, and in the effectiveness of block diagonal iterative solvers. For these advantages to be realized in industrial simulations, efficient solution strategies should be developed for systems arising from high-order DG discretizations. Addressing this issue in the context of solving both incompressible and compressible Navier-Stokes equations is the focus of this talk.

For the unsteady incompressible Navier-Stokes equations, we present an efficient parallel solver based on high-order DG methods using triangular and tetrahedral meshes in two and three space dimensions, respectively. In the context of semi-explicit temporal discretization, we present an algorithm with compact stencil sizes for all discrete equations yielding minimum computation and communication costs.

For the compressible Navier-Stokes equations, we present multigrid algorithms for systems arising from high-order DG discretizations on unstructured meshes. The algorithms are based on coupling both functional and geometric multigrid methods which are used in non-linear or linear forms, and either directly as solvers or as preconditioners to a Newton-Krylov method.

We identify two areas that need further explorations, namely developing new high-order DG algorithms for high-speed flows involving strong shock discontinuities, and methods for efficient simulation of flow-structure interactions. Finally, we describe the motivation behind our current research effort on devising hybrid Fourier continuation-WENO finite difference solvers for simulating high-speed multi-material compressible flows such as Richtmyer-Meshkov instabilities. This is an ongoing collaboration with Prof. Oscar Bruno's research group at the California Institute of Technology.

Bio:

http://www.dam.brown.edu/people/shahbazi/