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Modern Numerical Methods for Fluid Flow

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Phillip Colella & Elbridge Gerry Puckett Class Notes,
University of California, Berkeley

TOC

1. Finite difference methods for linear scalar problems
1.1 Consistency, stability, and convergence
1.2 The Von Neumann stability analysis
1.3 Some simple finite difference schemes
1.3.1 Upwind Differencing
1.3.2 Downwind Differencing
1.3.3 Centered Differencing
1.3.4 The Two-Step Lax-Wendroff Scheme
1.4 Upwind schemes and the geometric interpretation of finite differences
1.5 Fourier analysis and accuracy
1.5.1 Amplitude error
1.5.2 Phase errors
1.6 The modified equation
1.7 Discontinuities
1.7.1 Max-norm boundedness and Godunov’s Theorem
1.8 Weak solutions, conservative finite difference methods and the Lax-Wendroff
1.9 Limiters
1.9.1 Flux-corrected transport
1.9.2 Geometric limiters
1.9.3 Design criteria for schemes with limiters

2. Nonlinear scalar problems
2.1 Weak solutions of nonlinear hyperbolic problems
2.1.1 Nonuniqueness of weak solutions
2.1.2 The entropy condition
2.2 Strategies to enforce the entropy condition
2.2.1 Artificial viscosity
2.2.2 The first-order Godunov method
2.2.3 The second-order Godunov method
2.2.3.1 Outline of the method
2.2.3.2 Analysis of the method
2.2.4 The convexification of the Riemann problem
2.2.5 The Engquist-Osher flux

3. Systems of conservation laws
3.1 The linearized perturbation equations
3.1.1 Perturbations of the Riemann problem
3.1.2 The first order Godunov method
3.1.3 The first-order Godunov method (continued)
3.1.4 The second-order Godunov method
3.1.4.1 Stability of the method
3.1.4.2 Local truncation error
3.2 The effect of a nonlinear change of variables
3.2.1 The general case
3.2.2 Gasdynamics

4. Nonlinear systems of conservation laws
4.1 The Riemann problem
4.2 The entropy condition
4.3 Solution procedure for the approximate Riemann problem
4.3.1 The solution in phase space
4.3.2 The solution in physical space
4.3.3 Miscellaneous tricks
4.4 Temporal evolution
4.4.1 First-order Godunov
4.4.2 Second-order Godunov

II. Incompressible flow

5. Introduction
6. The Poisson equation
6.1 Direct solvers
6.2 Iterative solvers
6.2.1 Convergence and stability
6.2.2 Procedure for implementation of multigrid
6.2.2.1 Point Jacobi iteration
6.2.2.2 Gauss-Seidel relaxation with red/black ordering
6.2.2.3 Solvability conditions — the Fredholm alternative
6.2.2.4 Boundary conditions
6.2.3 Performance of multigrid
6.2.4 Time step considerations
6.2.4.1 Forward Euler
6.2.4.2 Backward Euler
6.2.4.3 Crank-Nicholson

7. The prototype advection equation
8. The Navier-Stokes equations
8.1 The treatment of the nonlinear advection terms
8.2 The incompressible Navier-Stokes equations
8.2.1 The divergence, gradient and inner product
8.2.2 The Hodge decomposition
8.2.3 The projection operator
8.3 The projection method
8.3.1 The discrete divergence operator
8.3.2 The discrete gradient operator
8.3.3 The discrete projection operator
8.3.4 Chorin’s projection method
8.3.5 A second-order projection method
8.3.6 Second-order cell-centered projection methods

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