1. Differences Among Difference Schemes
NUMERICAL METHODS FOR PDES
William L. Ruys

2. Objectives
• Understand the classification of Second Order Linear PDEs
• Understand the Finite Difference Methods
• How to discretize an equation
• How to analyze convergence
• Implement and Analyze Schemes for
• A Parabolic PDE: 1D Heat-Diffusion Equation
• An Elliptic PDE: Laplace’s Equation
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2

3. Second Order Linear PDEs
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2
𝐴
𝜕2 𝑢
𝜕𝑥2 + 𝐵
𝜕2 𝑢
𝜕𝑥𝜕𝑦
+ 𝐶
𝜕2 𝑢
𝜕𝑦2 + 𝐷
𝜕𝑢
𝜕𝑥
+E
𝜕𝑢
𝜕𝑦
+ 𝐺 = 0
• 𝐵 − 4𝐴𝐶 = 0
• Evolution with DiffusionParabolic
• 𝐵 − 4𝐴𝐶 < 0
• Steady StateElliptic
• 𝐵 − 4𝐴𝐶 > 0
• Evolution with ConservationHyperbolic

4. Finite Difference Approximations
FORWARD DIFFERENCEBACKWARD DIFFERENCE
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2
SECOND ORDER CENTERED DIFFERENCE
𝜕𝑢𝑖,𝑗
𝜕𝑥
=
𝑢𝑖+1,𝑗 − 𝑢𝑖,𝑗
Δ𝑥
𝜕𝑢𝑖,𝑗
𝜕𝑥
=
𝑢𝑖,𝑗 − 𝑢𝑖−1,𝑗
Δ𝑥
𝜕2
𝑢𝑖,𝑗
𝜕𝑥2
=
𝑢𝑖+1,𝑗 − 2𝑢𝑖,𝑗 + 𝑢𝑖−1,𝑗
Δ𝑥 2

5. Von-Neumann Analysis
• A Finite Difference Equation converges to its PDE when it is stable
• Lax-EquivalenceTheorem
• Break the solution into its Fourier modes and examine one of them.
• Let 𝑢𝑖
𝑛
= 𝑄 𝑚
𝑒 𝑖𝛼𝑥
in the difference equation (Where 𝑖 is the imaginary unit)
• Solve for the amplification factor, 𝑄
• Stable when 𝑄 ≤ 1
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2

6. 1D Heat-Diffusion Equation
A PARABOLIC PDE – TEMPERATURE DISTRIBUTION IN A ROD
𝜕𝑢
𝜕𝑡
= 𝑘
𝜕2
𝑢
𝜕𝑦2
𝑢𝑖
𝑛+1
= 𝑢𝑖
𝑛
+
𝑘Δ𝑡
Δ𝑥2 [𝑢𝑖+1
𝑛
− 2 𝑢𝑖,𝑗+1 + 𝑢𝑖−1
𝑛
]
Centered Space Difference !
𝑢𝑖
𝑛
=
−𝑘Δ𝑡
Δ𝑥2 𝑢𝑖+1
𝑛+1
+ 1 +
𝑘Δ𝑡
Δ𝑥2 𝑢𝑖
𝑛+1
+
−𝑘Δ𝑡
Δ𝑥2 𝑢𝑖−1
𝑛+1
BTCS FTCS

7. Comparison of FTCS and BTCS
BTCS
• Implicit
• Slow
• Unconditionally Stable
FTCS
• Explicit
• Fast
• Conditionally Stable
• Only when
𝑘Δ𝑡
Δ𝑥2 ≤
1
2
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2

8. Error Analysis
• Both schemes have an error O(Δ𝑥2) and O(Δt)
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2

9. Different Boundary Conditions - Example
DIRICHLET
• Heat escapes out of the sides
• Temperature at boundaries is fixed
NEUMANN
• Heat is trapped
• Heat Flux at boundaries is fixed
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2

10. Laplace’s Equation
AN ELLIPTIC PDE – STEADY STATE 2D HEAT
𝜕2
𝑢
𝜕𝑥2
+
𝜕2
𝑢
𝜕𝑦2
= 0
𝑢𝑖,𝑗 =
𝑢𝑖+1,𝑗 + 𝑢𝑖−1,𝑗 + 𝑢𝑖,𝑗+1 + 𝑢𝑖,𝑗−1
4

11. Jacobi Iteration
Iterations through Grid: 999
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2
𝜕2
𝑢
𝜕𝑥2 +
𝜕2
𝑢
𝜕𝑦2 = 0
𝑢 𝑥, 0 = 50
𝑢 𝑥, 𝐻 = 0
𝑢 0, 𝑥 = 0
𝑢 𝐿, 𝑥 = 0
𝑢𝑖,𝑗
𝑛+1
=
𝑢𝑖+1,𝑗
𝑛
+ 𝑢𝑖−1,𝑗
𝑛
+ 𝑢𝑖,𝑗+1
𝑛
+ 𝑢𝑖,𝑗−1
𝑛
4

12. Gauss-Seidel
Iterations through Grid: 692
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2
𝜕2
𝑢
𝜕𝑥2 +
𝜕2
𝑢
𝜕𝑦2 = 0
𝑢 𝑥, 0 = 50
𝑢 𝑥, 𝐻 = 0
𝑢 0, 𝑥 = 0
𝑢 𝐿, 𝑥 = 0
𝑢𝑖,𝑗
𝑛+1
=
𝑢𝑖+1,𝑗
𝑛
+ 𝑢𝑖−1,𝑗
𝑛+1
+ 𝑢𝑖,𝑗+1
𝑛
+ 𝑢𝑖,𝑗−1
𝑛+1
4

13. Successive
Over Relaxation
Iterations through Grid: 395
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2
𝜕2
𝑢
𝜕𝑥2 +
𝜕2
𝑢
𝜕𝑦2 = 0
𝑢 𝑥, 0 = 50
𝑢 𝑥, 𝐻 = 0
𝑢 0, 𝑥 = 0
𝑢 𝐿, 𝑥 = 0
𝑢𝑖,𝑗
𝑛+1
= 1 − 𝜔 𝑢𝑖,𝑗
𝑛
+ 𝜔
𝑢𝑖+1,𝑗
𝑛
+ 𝑢𝑖−1,𝑗
𝑛+1
+ 𝑢𝑖,𝑗+1
𝑛
+ 𝑢𝑖,𝑗−1
𝑛+1
4

14. Iterations Needed to Increase Accuracy
0
2000
4000
6000
8000
10000
12000
0 1 2 3 4 5 6 7 8 9 10
NumberofIterations
Number of Times Error Threshold is Halved from 0.10
SOR Jacobi Gauss Linear (SOR) Poly. (Jacobi) Poly. (Gauss)
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2

15. High Frequency Error Dampening
𝑒 𝑖𝛼𝑥
𝐹∘
𝛻2Jacobi Iteration Gauss-Seidel