Numerical methods. Note 9


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Lectures

Integration of systems of differential equations

We shall discuss here the generalization of the Runge-Kutta methods for a system of differential equations y'(x) = f(x,y), where y = {y₁,y₂,...,y_n} is a column of unknown functions and f = {f₁,f₂,...,f_n} is a column of the right-hand-sides. The initial condition is now also a column y(x₀) = y₀. The user-supplied subroutine to calculate the right-hand-sides should be modified like, for example

 
subroutine dydx(n,x,y,f); dimension y(n),f(n)
f(1)= ... ; f(2)=...; ...; f(n)=...; return; end

where the column of the right-hand-sides is returned in an array f.

Midpoint Runge-Kutta method for system of differential equations. For simplicity we shall only implement the midpoint method for systems of equations:

k₀ = f(x,y₀) , k_1/2 = f(x+^h/₂,y₀ + ^h/₂k₀) , y₁ = y₀ + hk_1/2 , err = h|k₀-k_1/2|/2

Finite difference methods.

Problems

Generalize your Runge-Kutta routine to system of differential equations (use midpoint method for simpicity). Your step-size controlling (driver) subroutine may have the following prototype
```
subroutine rk(a,b,f,n,nmax,y0,x,y,k,kmax,h,eps,acc,gerr,ak0,ak12)
dimension x(kmax),y(nmax,kmax),y0(nmax),ak0(nmax),ak12(nmax); external f
```
where y(i,j)=y_i(x_j) is the solution, ak0 and ak12 is the storage for k₀ and k_1/2, gerr is the estimated global error and h is the initial step.
The driver calls the stepper -- the subroutine that performs the very Runge-Kutta step:
```
subroutine rkstep(n,x,y0,y1,f,h,err,ak0,ak12)
dimension y0(n),y1(n),ak0(n),ak12(n)
```
Integrate the equation y''(x)=-y , y(0)=0 , y'(0)=1 from a=0 to b=4π.