Tutorial 10

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Introduction to PDEs and Numerical Methods
Tutorial 10.
Hyperbolic equations– numerical solution
Dr. Noemi Friedman, 23.01.2015.
Overview of this tutorial
 Introduction (classification of PDEs, examples of important PDEs)
 1D – Poisson equation −𝑢′′ 𝑥 = 𝑓(𝑥)
 Analytical solution (existence, uniqueness of solution, stability)
 Finite difference approximation of the Poisson eqaution
 Eigenvalue problem, Fourier transform
 Elliptic equations (−𝑎𝑎′′ 𝑥 + 𝑏 𝑥 𝑢′ 𝑥 + 𝑐 𝑥 𝑢 𝑥 = 𝑓 𝑥 )
 Analytical solution: existence, and uniqueness of solution, and its bounds
 FD approximation
 Consistency, stability, convergence
 FD approximation of PDEs on non-equidistant grids
 Higher order FD schemes
 With Taylor expansion
 With Lagrange interpolation
 Compact schemes
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 2
Overview of this tutorial
 Schemes for the numerical approximation of initial value problems




Explicit Euler
Implicit (backward) Euler
Theta-methods
Runge-Kutta methods
 Heat eqaution with 1d spatial dimension
 analytical solution (seperation of variables)
 Spatial and time discretisation, consistency and stability analysis
 Transport and wave equations with 1d spatial dimension
 analytical solution
 numerical soluion
 Multidimensional problems
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 3
Numerical solution of transport equation – downwind method
𝜕𝑢(𝑥, 𝑡)
𝜕𝜕
+𝑐
=0
𝜕𝑡
𝜕𝜕
Numerical approximation
𝑡
1
𝑐
𝑢
− 𝑢𝑛,𝑗 + 𝑢𝑛,𝑗+1 − 𝑢𝑛,𝑗 = 0
∆𝑡 𝑛+1,𝑗
ℎ
𝑐∆𝑡 𝑐∆𝑡 𝑖𝑘𝑘
−
𝑒
ℎ
ℎ
for 𝑐 > 0 unconditionally instable
𝐺 𝑘 =1+
transport direction
𝑐>0
𝑥
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 4
downwind
upwind
𝑡
Numerical solution of transport equation – upwind method
𝜕𝑢(𝑥, 𝑡)
𝜕𝜕
+𝑐
=0
𝜕𝑡
𝜕𝜕
Numerical approximation
𝑡
backward differences
1
𝑐
𝑢
− 𝑢𝑛,𝑗 + 𝑢𝑛,𝑗 − 𝑢𝑛,𝑗−1 = 0
∆𝑡 𝑛+1,𝑗
ℎ
𝐺 𝑘 =1−
for 𝑐 > 0 stable when
𝑐∆𝑡 𝑐∆𝑡 𝑖𝑘𝑘
+
𝑒
ℎ
ℎ
(CFL)
transport direction
𝑐>0
𝑥
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 5
downwind
upwind
𝑡
Numerical solution of transport equation – NAIVE scheme
𝜕𝑢(𝑥, 𝑡)
𝜕𝜕
+𝑐
=0
𝜕𝑡
𝜕𝜕
𝑡
1
𝑐
𝑢
− 𝑢𝑛,𝑗 +
𝑢
− 𝑢𝑛,𝑗−1 = 0
∆𝑡 𝑛+1,𝑗
2ℎ 𝑛,𝑗+1
𝐺 𝑘 =1−
𝑐∆𝑡
𝑖 sin(𝑘𝑘)
ℎ
𝑥
unconditionally instable
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 6
𝑡
Numerical solution of transport equation –Lax Friedrich’s
method
𝜕𝑢(𝑥, 𝑡)
𝜕𝜕
+𝑐
=0
𝜕𝑡
𝜕𝜕
from central differences:
1
𝑢
+ 𝑢𝑛,𝑗+1
2 𝑛,𝑗−1
1
𝑐
𝑢𝑛+1,𝑗 − 𝑢𝑛,𝑗 +
𝑢𝑛,𝑗+1 − 𝑢𝑛,𝑗−1 = 0
∆𝑡
2ℎ
1
1
𝑢
− 𝑢
+ 𝑢𝑛,𝑗+1
∆𝑡 𝑛+1,𝑗 2 𝑛,𝑗−1
𝑐∆𝑡
𝑖 sin(𝑘𝑘)
ℎ
2
𝑐∆𝑡
= cos2 (𝑘𝑘) −
sin2 (𝑘𝑘)
ℎ
𝐺 𝑘 = cos(𝑘𝑘) −
𝐺 𝑘
2
𝑥
𝑐
+
𝑢
− 𝑢𝑛,𝑗−1 = 0
2ℎ 𝑛,𝑗+1
𝐺 𝑘
2
≤1
𝑐∆𝑡
≤1
ℎ
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 7
𝑡
Numerical solution of transport equation – Lax-Wendroff method
𝜕𝑢(𝑥, 𝑡)
𝜕𝜕
+𝑐
=0
𝜕𝑡
𝜕𝜕
Approximate second derivate with the three point stencil (spatial discretisation of the heat eq.)
from Fridrich’s scheme
𝑥
from central differences
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 8
𝑡
Hints for next assignment
≤1
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 9
Hints for next assignment
1) Get scheme from plugging in the first equation to the second one
2) If you can bring it to the form:
then the scheme is conservative
3) If
the scheme is consistent
23. 01. 2015. | Dr. Noemi Friedman | PDE tutorial | Seite 10