Lecture 16: Convection and Diffusion (Cont’d)

1. Lecture 16: Convection and Diffusion (Cont’d)

2. Last Time … We • Looked at CDS/UDS schemes to unstructured meshes • Look at accuracy of CDS and UDS schemes • Look at false diffusion in UDS using model equation

3. This Time… • We will use model equation to look at behavior of CDS scheme • Look at some first-order schemes based on exact solutions to the convection-diffusion equation • Exponential scheme • Hybrid scheme • Power-law scheme

4. CDS Model Equations • Pure convection equation: • Apply CDS: • Expand in Taylor series Do same type of expansion in y direction

5. Model Equation (Cont’d) • Subtract to obtain: • Do same in y direction: • Substitute into discrete equation Dispersion Term

6. Discussion • Model equation for CDS has extra third-derivative (dispersive) term • This type of odd-derivative term tends to cause spatial wiggles • Note that truncation error for CDS is O( x2 ) • Thus, UDS is dissipative and CDS is dispersive

7. First-Order Schemes Based on Exact Solutions • 1D Convection-diffusion equation -Pe  Pe=0 Pe x What are the limits of this equation for different Pe?

8. Exponential Scheme • Use 1-D exact solution as profile assumption in doing discretization • Consider convection-diffusion equation: • Integrate over control volume:

9. Exponential Scheme (Cont’d) • Area vectors • Flux*Area: • Use exact solution to write convection and diffusion terms

10. Exponential Scheme: Discrete Equations • Both convection and diffusion terms estimated from exact solution • If S=0, we would get the exact solution in 1D problems • But obviously not exact for non-zero S, multi-dimensional problems… • Discretization has boundedness, diagonal dominance • Only first-order accurate

11. Approximations to Exponential Scheme • Exponentials are expensive to compute • Approximations to the exponential profile assumption have been used to offset the cost. • Hybrid difference scheme • Power-law scheme • Both these approximations are also only first-order accurate

12. Hybrid Difference Scheme • Consider the aE coefficient in exponential scheme • Limits with respect to Pe:

13. Hybrid Difference Scheme (Cont’d) Instead of using the exact curve for aE/De, use three tangents Similar manipulation for other coefficients

14. Hybrid Difference Scheme (Cont’d) • Guaranteed bounded solutions • Satisfies Scarborough criterion • O(x) accurate

15. Power-Law Scheme • Employs fifth-order polynomial approximation to • Similar approach to other coefficients • Scheme is bounded and satisfies the Scarborough criterion • Is O(x) accurate

16. X U Y Multi-Dimensional Schemes • Exact solutions have been used as profile assumptions in multi-dimensional situations • Control volume-based finite element method of Baliga and Patankar (1983) • This form is the solution to the 2D convection-diffusion equation

17. (i,j+1) (i-1,j) (i,j) (i+1,j) (i, j-1) Multi-Dimensional Schemes • Finite analytic scheme (Chen and Li, 1979) • Write 2D convection diffusion equation with source term for “element”: • Fix coefficient using (i,j) values • Find analytical solution using separation of variables • Use exact solution for profiles assumptions

18. Closure In this lecture, we • Looked at the model equation for CDS • Shown dispersive nature of model equation • Looked at differencing schemes based on exact solution to 1D convection-diffusion equation • Looked at schemes which are approximations to the exponential scheme • Looked at multidimensional schemes based on exact solutions