1. Objective
Numerical methods

2. Convection term dxw P dxe W E Dx – Central difference scheme:
- Upwind-scheme:
If Vx>0
and
If Vx<0
and

3. Diffusion term W
dxw P
dxe E Dx w e

4. Summary: Steady–state 1DI)
X direction
If Vx > 0,
If Vx < 0,
Convection term - Upwind-scheme: W P
dxw
dxe E
and a)
and Dx w e
Diffusion term: b)
When mesh is uniform:
DX = dxe = dxw
Assumption:
Source is constant over the control volume c)
Source term:

5. 1D example - uniform mesh
After substitution a), b) and c) into I):
We started with partial differential equation:
same
and developed algebraic equation:
We can write this equation in general format:
Unknowns
Equation coefficients

6. 1D example multiple (N) volumes
N unknowns 1 2 3 i
N-1 N
Equation for volume 1 N
equations
Equation for volume 2
……………………………
Equation matrix:
For 1D problem
3-diagonal matrix

7. 3D problem Equation in the general format: H N W P E S L
Wright this equation for each discretization volume
of your discretization domain A F
60,000 elements
60,000 cells (nodes)
N=60,000 x =
60,000 elements
7-diagonal matrix
This is the system for only
one variable ( )
When we need to solve p, u, v, w, T, k, e, C
system of equation is larger

8. Higher order differencing scheme
for advection W Dx P Dx E
Upwind scheme:
Vx<0 Dx
Vx>0
Central differencing scheme:
Higher order differencing scheme:
Quadratic upwind differencing
Scheme (QUICK)
N-2
N-1 N
N+1
N+2 WW P W E EE
We need to find coefficients
aP, aW, aE, aWW, aEE,

9. Quadratic upwind differencing
Scheme (QUICK)
Coefficients:
Advection coefficient:
Source:
Diffusion coefficients :
For advection only:

10. Matrix for QUICK method
Use Spars Matrix Solver

11. Iteration method
Alternative to use matrix solver tool is to use iterations
You can use excel if you are not familiar with matrix solver tools
General Iteration Procedure:
1) Express equation in explicit form
2) Guess initial values
3) Substitute initial values and calculate new values
4) Substitute new values and calculate newer values
6) Repeat step 4) until convergence is achieved
example
Iterations -residual
Value: T1
Residual
initial guess 22
iteration 1 23
iteration 2
23.25
iteration 3
iteration 4
iteration 5 ……
---
Iteration 98 2 2 2
Difference of value
between two iteration 2 2

12. Numerical instability divergency
divergence
variable
solution
convergence
iteration

13. General Transport Equation unsteady-stateH N
Equation in the algebraic format: W P E S L
We have to solve the system matrix for each time step !
Transient term:
Are these values for step  or + ?
Unsteady-state 1-D
If: -  explicit method
- + - implicit method

14. General Transport Equation unsteady-state
Fully explicit method:
Or different notation:
Implicit method
For Vx>0
For Vx<0

15. Steady state vs. Unsteady state
We use iterative solver to get solution
Unsteady state
We use iterative solver to get solution and
We iterate for each time step
Make the difference between
- Calculation for different time step
- Calculation in iteration step