I have some notes on CFD.    I thought that you might find them useful or interesting.    Questions, suggestions, and comments are welcome [ E-mail ].    Feel free to use the notes with appropriate referencing. ENO Schemes (Math671 report, 1998)

List of Notes

Dr. FUN3D

1. On Dissipation in CESE Method: [ pdf ]
I got interested in CESE method, and looked at dissipative schemes. I think various well-known numerical fluxes can be incorporated into CESE method.
2. The Roe-Averaged Density: [ html ]
Is your Roe flux implemented correctly? If you use the density, there is a unique averaging for the density:

$\hat{\rho} = \sqrt{\rho_L\rho_R}.$

No other averaging is allowed. Here, I explain why it is so.
Web-published Notes (March 3, 2011).
3. Diffusion Scheme for Discontinuous Data: [ html ]
"Given the left and right states at an interface, we compute the numerical flux". It sounds familiar for advection schemes, but it is not typical for diffusion schemes. There are, however, diffusion schemes that are implemented in the same way. Here I give you an example.
Web-published Notes (February 25, 2011).
Reference: [ AIAA2010-5093 ]
4. Table of Diffusion Schemes: [ html ]
Confused with diffusion schemes? Here is a table of diffusion schemes categorized by the approach to the construction.
Web-published Notes (December 16, 2010).
5. Numerical Solutions on Tangled Mesh: [ html ]
Numerical scheme can converge on a tangled mesh having negative volumes!
Unpublished results (1997).
The technique is described in [ PhD Thesis ]: Section 5.1, page 73.
6. Checkerboard Error Gone with Mesh Adaptation: [ html ]
Checkerboard error is a notorious error mode that arises typically on quadrilateral grids. It is usually eliminated by a deliberately designed dissipation or a staggered mesh. But look at this. It is eliminated by a little mesh movement; the mesh has now checkerboard perturbation!
More details can be found in [ PhD Thesis ]: Section 5.2.5, page 115.
7. On Positivity of Galerkin Scheme for Diffusion: [ pdf ]
Galerkin discretization for diffusion is shown to lose positivity given any nonzero nodal perturbation to a regular triangular and tetrahedral stencil.
Unpublished notes (2010).
Note: Positivity enforcement makes the scheme inconsistent [ AIAA 2010-5093 ] (See Appendix E).
8. Finite-Volume Integration Formulas: [ pdf ]
Finite-volume integration formulas are derived that are exact for linear fluxes. A remarkably simple derivation of interesting weights for the boundary integration are given.
Unpublished notes (2009).
An extended version is available in [ AIAA 2010-5093 ] (See Appendix), which includes formulas for all types of elements (triangle, quad, tetra, hexa, prism, pyramid).
9. Multidimensional FV/RD/FE Schemes: [ pdf ]
Inspired by Peter Gnoffo's multidimensional reconstruction scheme, I wrote down my understanding of his scheme, and attempted to devise another multidimensional scheme by combining the residual-distribution and finite-volume methods.
Unpublished notes (2008).
Reference: P. A. Gnoffo, AIAA Paper 2009-599 [ pdf available at FUN3D Web site ].
10. Galerkin Scheme and Finite-Difference Scheme: [ pdf ]
The Galerkin discretization for diffusion reduces to a standard finite-difference formula on regular triangular grids. Did you know that it is independent of the direction of diagonals?
Unpublished notes (2008).
A short version was included in [ AIAA Journal, 2010 v.48, no.7 ] (See Appendix).
11. Future Navier-Stokes Codes (NIA seminar 2007): [ pdf ]
A seminar given at NIA in 2007, describing various topics in CFD and finally giving a form of future Navier-Stokes codes. It was the first time that I talked about the future Navier-Stokes codes.
Future Navier-Stokes codes are described also in [ AIAA 2009-3648 ].
12. Entropy Generation and Dissipation: [ pdf ]
Entropy generation is described from a viscous equation. An entropy-consistent flux is then constructed for Burgers' equation and its extension to systems is discussed.
Unpublished notes (2005).
See [ Farzad and Roe, JCP 2009 ] for an extensive study on entropy-preserving/consistent fluxes.
13. High-Order Diffusion Schemes: [ pdf ]
I constructed a lot of high-order schemes for diffusion. A particularly interesting one is the P2 Galerkin scheme, which is shown to be Richardson's extrapolation of the P1 Galerkin scheme.
It was significantly expanded and became [ VKI Lecture Notes 2005 ].
14. Norm-Reducing Residual-Distribution Schemes: [ pdf ]
Residual-distribution schemes are discussed in terms of a norm-reducing (or residual-minimization) property: the residuals are minimized in a certain norm. It was found that the LDA, PSI, Lax-Wendroff, and SUPG schemes minimize the residuals while N-scheme does not.
Unpublished notes (2005).
15. Hypersonic Flows Computed by CFL3D with the Real-Gas Roe Solver: [ pdf ]
Numerical results are shown for hypersonic flow computations by CFL3D with the real-gas Roe solver. Unfortunately, the real-gas version of CLF3D is lost. I left the code in my computer account in Michigan when I moved to Virginia (I was not allowed to take it away), and it was wiped out by the university several months later.
Unpublished notes (2005).
16. A Real-Gas Roe Solver for CFL3D: [ pdf ]
A real-gas Roe solver is described. I wrote this when I implemented a real-gas Roe solver into NASA's CFL3D.
Unpublished notes (2004).
17. Decomposition of the 2D Euler Equations: [ pdf ]
Elliptic/Hyperbolic decomposition of the 2D Euler equations is discussed in relation to fluctuation-splitting (residual-distribution) schemes.
Unpublished notes (2004).
References: Ph.D. theses by L. Mesaros (1996) and M. Rad (2001), University of Michigan.
18. Conservative Linearization of Euler Equations: [ pdf ]
Conservative linearization of the Euler equations is discussed for fluctuation-splitting (residual-distribution) schemes. A general approach to conservative linearization on general elements is described.
Unpublished notes (2004).
19. Accurate Change of Variables in DG: [ pdf ]
Accuracy of change of variables in Discontinuous Galerkin methods is discussed. It is straightforward for P1, but not really for P2. An error term is derived; its impact on accuracy remains to be demonstrated.
Unpublished notes (2004).
20. On Simple Wave Solutions: [ pdf ]
A general derivation of exact simple wave solutions for conservation laws is described. Examples are given for the Euler and the ideal MHD systems. General exact solutions are derived for entropy wave, acoustic waves, and Alfven waves are derived, which can be used for code verification.
Originally unpublished notes (2003), now included in " I do like CFD, VOL.1 ", pp. 140-146, 2009.
21. Truncation Errors of RD Schemes: [ pdf ]
Truncation errors are shown for residual-distribution (RD) (fluctuation-splitting) schemes for advection and diffusion. It is shown that advection and diffusion schemes lose accuracy when simply added for the advection-diffusion equation.
Unpublished notes (2003).
See [ ICCFD 2004 ] for the accuracy problem and a possible solution.
22. DG Formulation for the 3D Ideal MHD System: [ pdf ]
Detailed P2/P3 Discontinuous-Galerkin formulation of the ideal MHD equations is given for a Cartesian mesh.
Unpublished notes (2003).
23. Modified Roe Matrix for Preconditioned System: [ pdf ]
The Roe flux is constructed analytically for a general preconditioned differential system. The dissipation matrix for the Euler equations is derived as an example.
Unpublished notes (2003).
It was applied to the 2D ideal MHD equations in [ AIAA2003 paper ].
24. Analytical Solutions of MHD Nozzle Flows: [ pdf ]
Analytical solutions are derived for MHD flows through a nozzle: aligned flows (B parallel to the flow) and transverse flows (B perpendicular to the flow ). Gasdynamics solution is also derived.
Unpublished notes (2003).
The aligned flow solution was used in [ AIAA2003 paper ].
25. A Fast and Robust Quartic Equation Solver: [ pdf ]
A quartic equation can be numerically solved very efficiently by Newton's method if a good initial guess is available. The point is to factor the equation as soon as a root is found, before going to the next root.
Unpublished notes (2003).
The method was devised to compute the eigenvalues of the 2D MHD system in [ AIAA2003 paper ].
26. Consistent LSQ Norm of the 2D Euler Equations: [ pdf ]
A dimensionally-consistent discrete least-squares norm for the 2D Euler equations is derived. It can be used to solve the Euler equations by the discrete least-squares method.
Unpublished notes (2001).
It is included in Appendix of [ PhD Thesis ] (See Appendix E).
27. Forms of the 2D Euler Equations: [ pdf ]
Various forms of the 2D Euler Equations are shown. I wrote this to seek dimensionally-consistent variables for the Euler equations for constructing an accurate least-squares scheme.
Unpublished notes (2001).
A dimensionally-consistent least-squares norm is described in [ PhD Thesis ](See Appendix E).
28. Area Change Formula: [ pdf ]
A formula is derived for computing the change of the area of a triangle whose vertices move. It can be used to detect nonlinear waves (shock/expansion) for the Euler equations by taking the characteristic speed as the vertex speed.
Unpublished notes (2000).
Applications of the formula can be found in [ ICCFD 2000, IJNMF 2008 ].
29. LSQ Residual-Minimization and FEM: [ pdf ]
It shows that solving the Cauchy-Riemann system by the residual-minimization scheme is equivalent to solving a pair of Laplace equations by FEM.
This is included in [ PhD Thesis ]: Section 5.1.4., page 79.
30. L2 Error Estimate for Triangular Elements: [ pdf ]
An L2 error estimate for triangular elements is derived. It is shown to be second-order accurate for exact or 4th order accurate nodal solution values.
This is included in [ PhD Thesis ] (See Appendix C).
31. Shooting Method for Node Generation Algorithm: [ pdf ]
A shooting method is described for the adaptive node-generation algorithm to solve a boundary-value problem for stiff ordinary differential equations. It generates nodes from one end to the other simultaneously computing the solution.
This is included in [ PhD Thesis ] (See Appendix B).
32. Adaptive Node Generation Algorithm: [ pdf ]
A very fast algorithm is proposed for adaptively generating nodes over a curve that ensures a specified L2 error. Successful applications to discretization of a curve, numerical integration, numerical solutions of stiff ordinary differential equations for both IVP and BVP are demonstrated.
This is included in [ PhD Thesis ] (See Appendix A).
33. On Mesh Movement Schemes: [ pdf ]
Minimization, vertex-spring, segment-spring mesh movement schemes are described. A formula based on equidistribution is also given and its relation with the spring-analogy schemes is described.
Unpublished notes (2000).
34. Difficulty in Computing a Potential Vortex: [ pdf ]
It shows how difficult it is for a least-squares finite-element (residual-minimization) scheme to compute a potential vortex. The problem is that the scheme does not accurately preserve the circulation.
Unpublished notes (2000).
More details can be found in [ PhD Thesis ]: Section 5.2.1., page 96.
35. Three-Dimensional Streamfunctions: [ pdf ]
Governing equations for three-dimensional streamfunctions are described. Also shown is that the volume flow is given by the product of the streamfunction differences.
Unpublished notes (1999).
A part of the notes is included in [ I Do Like CFD, VOL.1 ]: Section 4.16, page 117.
36. Polynomial Integration Formulas on Triangle: [ pdf ]
Formulas for the integration of a family of polynomials over a triangle is given, which is expressed in terms of the vertex coordinates. I think I did this for deriving truncation errors of some schemes or gradient reconstruction scheme.
Unpublished notes (1999).
It was used to compute the mass matrix for DG schemes in [ AIAA 2010-5093 ]: Section 5.6, page 32.
37. Elliptic Grid Generation by Cauchy-Riemann: [ pdf ]
This shows that elliptic grid generation is possible for triangular grids by solving the Cauchy-Riemann equations instead of Laplace equations. It can be a practical tool if source terms are implemented to control the grid quality.
Unpublished notes (1999).
38. Comparison of Numerical Fluxes for the Euler Equations: [ pdf ]
I wrote this for the final assignment in Math671 ( Numerical Methods for Hyperbolic Conservation Laws, taught by Professor S. Karni) at the University of Michigan in 1998. The actual assignment was to write a second-order Euler with the Roe flux, but I was curious about other fluxes and decided to try out 7 more fluxes: Lax-Friedrichs, Richtmyer, MacCormack, Steger-Warming, Van Leer, Osher, and Godunov fluxes.
Unpublished notes (1998).
Subroutines are available at www.cfdbooks.com/cfdcodes.html
39. ENO Schemes: [ pdf ]
This is a report prepared for Math671 ( Numerical Methods for Hyperbolic Conservation Laws, taught by Professor S. Karni) at the University of Michigan in 1998. Each student picks up an advanced topic in CFD algorithm and gives a lecture in the calss. I chosen ENO scheme. I remember I found the reconstruction by the primitive so interesting.
40. Derivation of Osher's Riemann Solver: [ pdf ]
A full detail of the derivation of Osher's Riemann solver is given.
Unpublished notes (1998).
A f90 subroutine is available at www.cfdbooks.com/cfdcodes.html
41. A Formula for Lift due to False Theory of Flight: [ pdf ]
I derived a formula for the lift based on the popular theory of flight: air flows faster on the upper surface of an airfoil, causing a lower pressure... The formula is useful in proving that the theory is impractical; an example is given to show that the theory greatly underestimates the lift.
Unpublished notes (1998).
42. Geometric Interpretation of Numerical Solution: [ pdf ]
The notes show an interesting geometrical interpretation of numerical solutions for differential equations, using the exterior calculus: the residual measures an error in approximating a solution surface by elements.
Unpublished notes (1997).
I explored its application to mesh adaptation in [ PhD Thesis ] (See Chapter 1 and Appendix D).
43. Introduction to Waveriders: [ pdf ]
I was going to design a waverider using CFD for my PhD ( I changed the subject a year later to mesh adaptation). I wrote this as a preparation for the waverider design project in 1995.
Unpublished notes (1995).
44. Aerodynamic Heating with Turbulent Flows: [ pdf ]
I wrote this for a research project in AE525 (Introduction to Turbulence, taught by Professor L. Bernal) at the University of Michigan in 1994.
Unpublished notes (1994).