Discontinuous Galerkin methods for compressible flows: higher order accuracy, error estimation and adaptivity

Kurzfassung

1) Introduction<br></br> 2) The Discontinuous Galerkin discretization of hyperbolic conservation equations<br></br> 2.1) Hyperbolic conservation equations<br></br> 2.2) Scalar hyperbolic problem<br></br> 2.3) The compressible Euler equations<br></br> 2.4) The Discontinuous Galerkin discretization<br></br> 2.5) DG for scalar hyperbolic problems<br></br> 2.6) DG for scalar hyperbolic problems: <em>A priori</em> error estimates<br></br> 3) Higher order convergence of the DG method for the 2d compressible Euler equations<br></br> 3.1) The Ringleb flow problem<br></br> 4) A Discontinuous Galerkin discretization of the compressible Navier-Stokes equations<br></br> 4.1) The compressible Navier-Stokes equations<br></br> 4.2) The Interior Penalty Discontinuous Galerkin discretization<br></br> 5) Newton--GMRES algorithm<br></br> 5.1) Jacobians of boundary terms<br></br> 5.2) Jacobians of numerical flux functions<br></br> 5.3) Numerical performance of the Newton iteration<br></br> 6) Higher order convergence of the DG method for the 2d compressible Navier-Stokes equations<br></br> 6.1) Subsonic flow around an airfoil: Convergence of force coefficients<br></br> 6.2) Flow over a flat plate<br></br> 7) Shock-capturing<br></br> 7.1) Introduction<br></br> 7.2) Shock-capturing for the compressible Navier-Stokes equations<br></br> 7.3) The Jacobian of the shock-capturing term<br></br> 7.4) Numerical results<br></br> 7.4.1) Supersonic flow past a wedge<br></br> 7.4.2) Supersonic flow around an airfoil<br></br> 8) Error estimation and adaptivity<br></br> 8.1) Introduction<br></br> 8.2) The dual (adjoint) problem<br></br> 8.3) Adjoint problems for scalar hyperbolic problems<br></br> 8.4) Error estimation<br></br> 8.5) Adaptive mesh refinement<br></br> 8.6) Numerical examples<br></br> 8.6.1) Ringleb flow problem<br></br> 8.6.2) Supersonic flow past a wedge<br></br> 8.6.3) Supersonic flow around an airfoil<br></br> 9) Acknowledgements