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lecture/21 boundary value problem.ipynb
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%% Cell type:markdown id: tags:
# boundary value problem (poisson equation in 1d)
also known as a 2-point boundary value problem
Find a function
$$u:[0,1] \quad \rightarrow \quad \mathbb{R},$$
such that
$$-u''(x) = e^x, \quad x \in (0,1)$$
$$u(0) = u(1) =0$$
There is no general closed solution because of the boundary conditions <br />
$\Rightarrow$ numerical approximation of the solution.
## finite difference
- discretice: $0=x_0 < \dots < x_{n}=1$ with $x_i=\frac{i}{n}$
- difference quotient:
$$u''(x_i) \sim \frac{u(x_{i-1}) - 2 u(x_i) + u(x_{i+1})}{
h^2}, \quad h:=\frac{1}{n}$$
- plug in $-u''(x)=e^x$ yields
$$-u(x_{i-1}) + 2 u(x_i) - u(x_{i+1}) = h^2 e^{x_i}, \quad
i=1,\dots ,n-1$$
boundary condition $\Rightarrow$ $u(x_0)=u(x_n)=0$. These would lead to zero entries in the matrix and right hand side.
- $\Rightarrow$ linear system for $u(x_1), \dots ,u(x_{n-1})$.
Define $z=(z_1,\dots ,z_{n-1})^t=(u(x_1), \dots ,u(x_{n-1}))^t$.
solve the linear system $Az=F$ with
$$A:=
\left( \begin{array} {ccccccc}
2 & -1 & & & 0 \\
-1 & 2 & -1 & & \\
& \ddots & \ddots & \ddots &\\
& & -1 & 2 & -1 \\
0 & & & -1 & 2 \\
\end{array} \right), \ F:=
h^2 \left( \begin{array}{c} e^\frac{1}{n}\\ \vdots \\ e^\frac{n-1}{n}
\end{array} \right) .$$
### Implementation - OLD
%% Cell type:code id: tags:
``` haskell
import Numeric.Container
import Graphics.Rendering.Chart.Easy hiding (Matrix)
import Graphics.Rendering.Chart.Backend.Cairo
-- indexfunction for creating the matrix
idxf (i,j)
| i == j = 2
| i == j-1 || i == j+1 = -1
| otherwise = 0
twopoint = (x,a <\> (asColumn f)) -- solve the linear system
where
n = 100
-- divide the interval
x = linspace n (0,1::Double)
-- cut of boundary points (they are zero anyway)
xi = subVector 1 (n-2) x
-- build the matrix
a = buildMatrix (n-2) (n-2) (idxf) :: Matrix Double
-- right hand side f=exp(x)
f = mapVector (\xi -> (1/(fromIntegral n))^2*exp(xi)) xi
(x,sol) = twopoint
-- fill up boundary values
solfull = [0] ++ toList (flatten sol) ++ [0]
toRenderable $ do
layout_title .= "2-point boundary value problem"
plot (line "sol" [zip (toList x) solfull])
```
%% Cell type:markdown id: tags:
### HELPER
%% Cell type:code id: tags:
``` haskell
:ext FlexibleContexts
import Data.Array.Repa
import Numeric.LinearAlgebra.Helpers
diag :: Num a => a -> Int -> Array D DIM2 a
diag x n = fromFunction (ix2 n n) f
where
f (Z :. i :. j) | i == j = x
| otherwise = 0
:t diag
ident = diag 1
:t ident
-- TODO check this
computeS $ ident 3 :: Array U DIM2 Int
```
%% Output
%% Cell type:code id: tags:
``` haskell
:ext FlexibleContexts
import Data.Array.Repa
band :: Num a => [a] -> Int -> Array D DIM2 a
band xs n = if even (length xs)
then error "band: only odd vector lengths allowed!"
else fromFunction (ix2 n n) f
where
m = length xs `div` 2
f (Z :. i :. j) | i == j = xs !! m
| i < j && j-i <= m = xs !! (m+j-i)
| j < i && i-j <= m = xs !! (m-i+j)
| otherwise = 0
%% Cell type:code id: tags:
``` haskell
computeS $ band [1,2,3] 4 :: Array U DIM2 Int
```
%% Output
%% Cell type:code id: tags:
``` haskell
linspace :: (Fractional e) => Int -> (e, e) -> Array D DIM1 e
linspace 0 _ = fromFunction (ix1 0) (const 0)
linspace 1 (a,b) = fromFunction (ix1 1) (const (a+b/2))
linspace n (a,b) = fromFunction (ix1 n) f
where f (Z :. x) = a + (b-a)*(fromIntegral x) / fromIntegral (n-1)
(computeS $ linspace 5 (1,3)) :: Array U DIM1 Double
```
%% Output
%% Cell type:markdown id: tags:
### Implementation
%% Cell type:code id: tags:
``` haskell
import Graphics.Rendering.Chart.Easy hiding (Matrix)
import Graphics.Rendering.Chart.Backend.Cairo
import Numeric.LinearAlgebra.Repa
import Data.Array.Repa as R hiding ((++))
import Data.Array.Repa.Repr.ForeignPtr
twopoint = (x,solveS a f) -- solve the linear system
where
n = 100
-- divide the interval
x = linspace n (0,1::Double)
-- cut of boundary points (they are zero anyway)
xi = extract (ix1 1) (ix1 (n-2)) x
-- build the matrix
a = band [-1,2,-1] (n-2)
-- right hand side f=exp(x)
f = R.map (\xi -> (1/(fromIntegral n))^2*exp(xi)) xi
(x,sol) = twopoint
-- fill up boundary values
solfull = [0] ++ toList sol ++ [0]
toRenderable $ do
layout_title .= "2-point boundary value problem"
plot (line "sol" [zip (toList x) solfull])
```
%% Output
%% Cell type:code id: tags:
``` haskell
```
......
lecture/22 ODE.ipynb
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lecture/23 runge-kutta methods.ipynb
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%% Cell type:markdown id: tags:
 
# Runge-Kutta methods
 
Runge-Kutta class methods are a implicit and explicit methods for temporal discretization for ODE's.
 
Let it be a initial value problem
$$y^\prime = f(t,y)$$
with $y(t_0)=y_0$. find $y(t)$.
 
then runge-kutta methods have the form of
$$y_{n + 1} = y_n + h \sum_{i = 1}^s b_i k_ i $$
 
$$k_i = f ( t_n + c_i h , y_n + h \sum_{j = 1}^s a_{ij} k_j )$$
 
with some initial value $y_0$ and parameters $a_{ij}, b_i$ and $c_i$
 
## implicit Euler
 
the simplest *implicit* runge-kutta method is the *implicit Euler method* (explicit is an exercise) with $c_i = 1$, $a_{ij} = 1$ and $b_1 = 0$ which then yields the update formula
$$y_{n + 1} = y_n + h f ( t_{n+1} , y_{n+1} )$$
 
Now there is on both sides the unkown function and we have to solve an algebraic equation for this. This can be done with newton's method or with solving a linear equation depending on the problem. Because our example with the harmonic oscillator is linear but multidimensional we can solve a linear system of equations for $y_{n+1}$
 
%% Cell type:code id: tags:
 
``` haskell
import Numeric.Container
import Data.Array.Repa