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Nonlinear Convection-Dominated Problems

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**1. **Chapter 10 Nonlinear
Convection-Dominated Problems

**2. **10.1 Burgers’ Equation One-dimensional Burgers’ equation
Conservative form

**3. **Inviscid Burgers’ Equation One-dimensional inviscid Burgers’ equation
Larger values of convect faster and overtake slower
Multi-valued solution may occur
Postulate a shock to allow the development of discontinuous solutions

**4. **Inviscid Burgers’ Equation Formation of multi-valued solution
The nonlinearity allows discontinuous solutions to develop
Shock-fitting

**5. **Viscous Burgers’ Equation Viscous term reduces the amplitude in high gradient regions
Prevents multi-valued solutions from developing (second derivative increases faster than first derivative)

**6. **10.1.2 Explicit Schemes FTCS scheme (non-conservative)
FTCS (conservative form)

**7. **Explicit Schemes Four-point Upwind Scheme
Truncation errors
O(?x2) if q ? 0.5
O(?x3) if q = 0.5

**8. **Lax-Wendroff Scheme Inviscid Burgers’ equation for unsteady one-dimensional shock flows
Replace temporal derivative by equivalent spatial derivative (more complicated for nonlinear case)

**9. **Lax-Wendroff Scheme Central-difference discretization
For Burgers’ equation

**10. **Lax-Wendroff Scheme Temporal derivative
Inviscid Burgers’ equation
Rearrange

**11. **Lax-Wendroff Scheme Linear pure convection equation
Nonlinear - inviscid Burgers’ equation
Equivalent two-stage algorithm (more economical)

**12. **Burgers’ Equation Thommen’s extension of Lax-Wendroff scheme for viscous flow problems
Error in textbook
Stability limit

**13. **10.1.3 Implicit Schemes Burgers’ equation (viscous)
Crank-Nicolson implicit formulation
Thomas algorithm for tridiagonal matrices cannot be used directly due to the appearance of nonlinear implicit term
Use Taylor series expansion of at nth time- level to convert to tridiagonal form

**14. **Crank-Nicolson Scheme Taylor-series expansion (linearlization of F)
Linear tridiagonal system (in terms of ?u or u)

**15. **Crank-Nicolson Scheme Thomas algorithm
The matrix coefficients must be reevaluated at every time step (to recover nonlinearity of the equation)
Truncation error O(?t2, ?x2)
Unconditionally stable in Von Neumann sense (linear)

**16. **Generalized Crank-Nicolson Mass operator and four-point upwind
Truncation error O(?t2, ?x2)

**17. **Generalized Crank-Nicolson Quadridiagonal system of equations – can be solved using generalized Thomas algorithm

**18. **Artificial Dissipation Crank-Nicolson with additional dissipation
For small values of viscosity (high-Re), it is desirable to add some artificial dissipation
Modified Crank-Nicolson
Choose ?a empirically

**19. **10.1.4 BURG: Numerical Comparison Propagation of a shock wave governed by viscous Burgers’ equation
Exact solution

**21. **BURG: Numerical Comparison ME = 1, FTCS scheme
ME = 2, two-stage Lax-Wendroff scheme
ME = 3, Explicit four-point upwind scheme
ME = 4, Crank-Nicolson (CN-FDM): ? = 0, q = 0
ME = 4, Crank-Nicolson (CN-FEM): ? = 1/6, q = 0
ME = 4, Crank-Nicolson, Mass Operator (CN-MO): ? = 1/12, q = 0
ME = 4, Crank-Nicolson, 4-pt. Upwind (CN-4PU): ? = 0, q = 0.5
ME = 5, Crank-Nicolson plus additional dissipation
Note: Optimum ? and q (locally freezing nonlinear coefficients)

**33. **10.2 Systems of Equations Continuity equation
Momentum equations
Energy equation
Equation of state (compressible flows)
Turbulent kinetic energy equation
Rate of turbulent energy dissipation equation
Reynolds stresses equations
Multiphase flows
Chemical reactions

**34. **Systems of Equations 1D unsteady compressible inviscid flow
Continuity equation, x-momentum equation, energy equation

**35. **Two-Stage Lax-Wendroff Single equation
System of equations

**36. **Lax-Wendroff Scheme with Artificial Viscosity

**37. **Crank-Nicolson Scheme System of equations
Linearization
3?3 block tridiagonal system (solved by block Thomas algorithm)

**38. **Crank-Nicolson Scheme Use Von Neumann analysis for the linearized equation
Amplification matrix
Numerical Stability

**39. **10.3 Group Finite Element Method Conventional finite element method introduces a separation approximate solution (trial function, interpolation function) for each dependent variable
Galerkin method produces large numbers of products of nodal values of dependent variables, particularly from the nonlinear convective terms
Inefficient, time-consuming
Group finite element formulation is effective in dealing with convective nonlinearities

**40. **Group Finite Element Method Group finite element formulation
The equations are cast in conservative form
A single approximation solution is used for the group of terms in the differential terms (i.e., approximate F directly instead of the nonlinear convective term u?u/?x)
One-dimensional Group Formulation

**41. **Group Finite Element Method One-dimensional Group Formulation
Conventional finite element

**43. **10.4 2D Burgers’ Equation Two-dimensional Burgers’ equation
Equivalent to 2D momentum equations for incompressible laminar flow with zero pressure gradient

**44. **2D Burgers’ Equation Exact solution
Use Cole-Hopf transformation
Transform the 2D Burgers’ equation into one single equation – 2D diffusion equation

**45. **2D Burgers’ Equation Steady 2D Burgers’ equation
Exact solution

**49. **Multidimensional Group FEM Two-dimensional Burgers’ equation
Approximate solutions for (u,v), and groups (u2, uv, v2) and the components of S
For example (bilinear for rectangular elements)

**50. **Galerkin Finite Element Linear (Chapter 9)
Nonlinear (Group FE formulation)
The equations are treated as linear at the level at which the discretization take place (but indeterminate)
Substitution for the nodal groups in terms of the unknown nodal variables introduces the nonlinearity but also makes the system determinate

**51. **Split Schemes Two-dimensional Burgers’ equations
Similar to those used in Chapters 8 and 9
Additional complication due to nonlinearity
Generalized FEM/FEM with mass operators Mx and My

**52. **Pseudo-Transient Formulation Use pseudo-transient formulation (sect 6.4) for steady-state solution
For steady-state problems, unsteady formulation provides an equivalent underrelaxation parameter for steady iterative schemes
For steady-state solutions, it is desirable to use a simple time discretization (such as two-level fully implicit scheme with ? = 1) to simplify the formulation

**53. **Pseudo-Transient Formulation Two-level fully implicit scheme (? = 1)
Linearize the nonlinear terms F, G, and S in (RHS)n+1

**54. **Pseudo-Transient Formulation Linearization (Jacobian matrices A, B, C)
Approximate Factorization

**55. **Pseudo-Transient Formulation Further simplification to reduce CPU time
Use the same left-hand-side for each scalar component
Perform only one factorization (BANFAC) for different components
Does not affect the steady-state solution since (RHS)n = 0 in the steady state limit

**56. **TWBURG: Numerical Solution Two-dimensional Burgers’ equations
Steady state solution with the following split algorithm
Solution domain
?1? x ? 1 , 0 ? y ? ymax , ymax= ?/6?
Use exact solution for the boundary conditions
Initial conditions obtained from linear interpolation of the boundary condition in the x-direction