1. Convection-Dominated Problems
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 b
t = t1
t = t0
t = t2
shock a

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)
t = t0
t = t1
t = t2

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)
Chain rule

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

20. Burgers’ Equation

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)

29. Burgers’ Equation

30. Burgers’ Equation Propagating Shock Solution
Rcell = 1.0, C = 0.25

31. Burgers’ Equation: Propagating Shock
Rcell = 3.33, C = 1.0
Rcell = 100, C = 1.0

32. Velocity distribution at t = 2.0; Rcell = 100

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
Continuity equation
X-momentum equation
Energy equation

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

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
Conservative form
Non-conservative form

42. One-dimensional Burgers’ equation
Conventional and group FEMs

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

46. Exact solution for 2D Burgers’ equation

47. 2D Burgers’ Equation – Exact u
a1= a2= 1.3*1013, a3= a4= 0, a5 = 1,  = 25, x0 =1,  = 0.04

48. 2D Burgers’ Equation – Exact v
a1= a2= 1.3*1013, a3= a4= 0, a5 = 1,  = 25, x0 =1,  = 0.04

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

57. Computer Program - TWBURG

64. Approximate Factorization

65. Error Distributions at y/ymax = 0.4