This is part of the process mod­el­ing and sim­u­la­tion col­lec­tion.

Introduction

These are a com­pi­la­tion of search­able com­po­nents which are typ­i­cally asked. (top level head­ings are cur­rently lower [FIX LATER])

Ordinary Differential Equations

Common: f(x_0,y_0)=y'

  • h is the step size
  • x_0 and y_0 are start­ing val­ues

Runge Kutta Methods

The gen­eral form is

y_1=y_0+k

Runge Kutta I

k=hf(x_0,y_0)

Runge Kutta II

k_1=hf(x_0,y_0)

k_2=hf(x_0,y_0+k_1)

k=\frac{1}{2}(k_1+k_2)

Runge Kutta III

k_1=hf(x_0,y_0)

k_2=hf(x_0+\frac{h}{2},y_0+\frac{k_1}{2})

k'=hf(x_0+h,y_0+k_1)

k_3=hf(x_0+h,y_0+k')

k=\frac{1}{6}(k_1+4k_2+k_3)

Runge Kutta IV

k_1=hf(x_0,y_0)

k_2=hf(x_0+\frac{h}{2},y_0+\frac{k_1}{2})

k_3=hf(x_0+\frac{h}{2},y_0+\frac{k_2}{2})

k_4=hf(x_0+h,y_0+k_3)

k=\frac{1}{6}(k_1+2k_2+2k_3+k_4)

Milne’s Predictor Corrector

y_4^P=y_0+4\frac{h}{3}(2f_1-f_2+2f_3)

y_4^C=y_2+\frac{h}{3}(f_2+4f_3+f_4)

Root Finding Methods

Newton Raphson

x_{i+1}=x_i-\frac{f(x_i)}{f'(x_i)}

Regula Falsi

With a and b as the in­ter­val points.

Evaluate f(a) and f(b).

x_r=\frac{af(b)-bf(a)}{f(b)-f(a)}

Check

f(a)f(x_r)<0\implies{b}=x_r f(a)f(x_r)>0\implies{a}=x_r

Secant

x_{i+1}=\frac{x_{i-1}f_i-x_i{f_{i-1}}}{f_i-f_{i-1}}

Bisection

With a and b as the in­ter­val points such that f(a)f(b)<0.

x_r=\frac{a+b}{2}

Check

f(a)f(x_r)<0\implies{b}=x_r f(a)f(x_r)>0\implies{a}=x_r

SOR

x_i^k=x_i^{k-1}+w[x_{gs}^k-x_i^{k-1}]

Where:

  • x_{gs}^k is the k^{th} it­er­a­tion Gauss-Seidel vec­tor
  • w\in[1,2] for speed and is usu­ally taken as 1.2

x_{gs}^{k+1}=C+{B}X^k