Numerical Methods for Unsteady Compressible Flow Problems

Preface. 1. Introduction. 1.1. The method of lines. 1.2. Hardware. 1.3. Notation. 1.4. Outline. 2. The Governing Equation. 2.1. The Navier-Stokes Equations. 2.2. Nondimensionalization. 2.3. Source terms. 2.4. Simplifications of the Navier-Stokes equations. 2.5. The Euler Equations. 2.6. Solution theory. 2.7. Boundary layers. 2.8. Boundary layers. 2.9. Laminar and turbulent flows. 3. The Space discretization. 3.1. Structured and unstructured Grids. 3.2. Finite Volume Methods. 3.3. The Line Integrals and Numerical Flux Functions. 3.4 Convergence theory for finite volume methods. 3.5. Source Terms. 3.6. Finite volume methods of higher order. 3.7. Discontinuous Galerkin methods. 3.8. Convergence theory for DG methods. 3.9. Boundary Conditions. 3.10. Spatial Adaptation. 4. Time Integration Schemes. 4.1. Order of convergence and order of consistency. 4.2 Stability. 4.3. Stiff problems. 4.4. Backward Differentiation formulas. 4.5. Runge-Kutta methods. 4.6. Rosenbrock-type methods. 4.7. Adaptive time step size selection. 4.8. Operator Splittings. 4.9. Alternatives to the method of lines. 4.10. Parallelization in time. 5. Solving equation systems. 5.1. The nonlinear systems. 5.2. The linear systems. 5.3. Rate of convergence and error. 5.4. Termination criteria. 5.5. Fixed Point methods. 5.6. Multigrid methods. 5.7. Newton's method. 5.8. Krylov subspace methods. 5.9. Jacobian Free Newton-Krylov methods. 5.10. Comparison of GMRES and BiCGSTAB. 5.11. Comparison of variants of Newton's method. 6. Preconditioning linear systems. 6.1. Preconditioning for JFNK schemes. 6.2. Specific preconditioners. 6.3. Preconditioning in parallel. 6.4. Sequences of linear systems. 6.5. Discretization for the preconditioner. 7. The final schemes. 7.1. DIRK scheme. 7.2. Rosenbrock scheme. 7.3. Parallelization. 7.4. Efficiency of Finite Volume schemes. 7.5. Efficiency of Discontinuous Galerkin schemes. 8. Thermal Fluid Structure Interaction. 8.1. Gas Quenching. 8.2. The mathematical model. 8.3. Space discretization. 8.4. Coupled time integration. 8.5. Dirichlet-Neumann iteration. 8.6. Alternative solvers. 8.7. Numerical Results. A. Test problems. A.1. Shu-Vortex. A.2. Supersonic Flow around a cylinder. A.3. Wind Turbine. A.4. Vortex shedding behind a sphere. B. Coefficients of time integration methods. Bibliography. Index.