Lecture 16: Convection and Diffusion (Cont’d)

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

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

CDS Model Equations

Pure convection equation:

Apply CDS:

Expand in Taylor series

Do same type of expansion in y direction

Model Equation (Cont’d)

Subtract to obtain:

Do same in y direction:

Substitute into discrete equation

Dispersion Term

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

First-Order Schemes Based on Exact Solutions

1D Convection-diffusion equation

x

Pe

-Pe

Pe=0

What are the limits of this equation for different Pe?

Exponential Scheme

Use 1-D exact solution as profile assumption in doing

discretization

Consider convection-diffusion equation:

Integrate over control volume:

Exponential Scheme (Cont’d)

Area vectors

Flux*Area:

Use exact solution to write convection and diffusion terms

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

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

Hybrid Difference Scheme

Consider the aE coefficient in exponential scheme

Limits with respect to Pe:

Hybrid Difference Scheme (Cont’d)

Instead of using the exact curve for aE/De, use three

tangents

Similar manipulation for other coefficients

Hybrid Difference Scheme (Cont’d)

Guaranteed bounded

solutions

Satisfies Scarborough

criterion

O(x) accurate

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

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

exp( / )A UX BY C X

Y

U

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

(i,j)(i-1,j) (i+1,j)

(i,j+1)

(i, j-1)

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