1. Ordinary differential equaltions:
Solutions by finite difference schemes

2. Ordinary differential equations
1st order ode
Taylor expansion of x(t) about time t
we thus have an approximation for x(t+dt) in terms of x(t)
this is the (explicit) Euler discretization scheme

3. we can get a better approximation if we take two steps
to get from t to t+dt
first get an approximation for x(t+dt/2) using the Euler
discretization
then make the full step using the function evaluated at this
midpoint
this is the 2nd order Runge-Kutta discretization

4. Example:
Analytic solution:
Euler method:
So this misses the term as expected
Runge-Kutta method:
So this misses the term, as expected

5. #include <stdio.h>
#include <stdlib.h>
#include <math.h>
/* set up the finite-difference parameters */
const int runLength=1000;
const double delta_t=0.01;
/* set up the function prototypes */
double Euler(double x,double t);
double RungeKutta(double x,double t);
double function(double x,double t);
void printData(double *xEuler,double *xRungeKutta);
main() {
/* define the arrays to store the data */
double xEuler[runLength]={0.0},xRungeKutta[runLength]={0.0},t=0.0;
/* set the initial conditions */
xEuler[0]=1.0;
xRungeKutta[0]=1.0;
/* evolve the solutions using the two integration methods */
for(int n=0;n<runLength-1;++n){
xEuler[n+1]=Euler(xEuler[n],t);
xRungeKutta[n+1]=RungeKutta(xRungeKutta[n],t);
t=t+delta_t; }
printData(xEuler,xRungeKutta);

9. general 2nd order ode:
write as two first order equations
solve using 2nd order Runge-Kutta
half step
full step

10. Example:
Simple harmonic oscillator
Initial conditions:
Analytic solution:
i.e.
Runge-Kutta method:

11. Choosing an algorithm:
accuracy
how accurate is the algorithm?
how accurate does the answer need to be?
efficiency
can I use a different, more efficient algorithm?
can I implement the current algorithm more efficiently?
stability
is the algorithm stable?
Can I avoid instabilities?

12. Numerical stability:
numerical methods approximate continuous differential
equations by discrete difference equations
the difference equations may exhibit instabilities even if the
continuous differential equations don’t
two possible reasons are:
time-step problems
competing solutions

13. Time-step problems: e.g.
has the analytic solution which is a decaying exponential
The Euler approximation has
Which we can analyze by changing dt
good solution
vanishing solution
decaying oscillations
constant amplitude oscillations---
increasing amplitude oscillations-

14. =1.0)

15. Competing solutions: e.g. with boundary conditions:
has the analytic solution which is a decaying exponential:
However, the general solution is
there are small errors in any finite difference scheme
these errors have the effect of introducing some of the
growing mode solution. This leads to a numerical solution
which grows, rather than decays.