Examples of Resolution (global spectral model)
Ocean circulation models and boundary conditions
Classes of models:Stochastic (different time & length scales)
Finite differences: Diffusion-advection
2D Staggered grids: Arakawa
Fourier–Galerkin: Spectral methods in AGCMs and their resolution
Finite differences and finite element methods
Example: Ocean model in finite elements
Numerics: Euler integrator
Coarse graining -> Stochastic equations; Climate Application
Play with the parameter h which is the step size in time units !
Can you find a crtical h when the numerical scheme does not work anymore?
Which value of h shall be taken in order to have an accurate solution (close to the analytical one) ?
#demonstration of Euler method in 1st order ODE: dy/dt=A*y
#the function dy/dt<-f(y,A,t)
f<-function(y,A,t)
{
return(A*y)
}
#constants
A<- -0.02 #growth / decay rate
T<- 500 #integration time in time units
h<- 10 #step size in time units
Y0<- 8 #inital value
n<-T/h #number of time steps (time / timestep)
t<-(0:(n-1))*h #create a vector of discrete timesteps
y<-vector() #define an empty vector for the state variable y(t)
y[1]<-Y0 #assign initial value
#integration loop
for (i in 1:(n-1))
{
y[i+1]<-y[i]+h*f(y[i],A,t[i]) #Euler forward: y[t+h]<-y[t]+h*A*y[t]
}
plot(t,y,type="p") #plot the result against time
#additionaly plot the analytical solution in red
lines(t,Y0*exp(A*t),col="red",type="l")
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