1. Ordinary Differential Equations
Scientific Computing
Numerical Solution Of
Ordinary Differential Equations
- Further Analysis

2. Solving First Order ODE’s
A general form for a first order ODE is
Or alternatively
We desire a solution x(t) which satisfies this ODE and one specified boundary condition.
x(a) = c

3. Numerical Solution
The value of the true solution is approximated at a set of n values of t.
We denote the approximation at these pts by so that

4. Numerical Solution
Let x(t) be the actual solution for dx/dt at the step values.
Then, assuming no roundoff error, the difference in the calculated and true value is the truncation error,

5. Taylor Series
Last time we looked at expressing the solution x(t) about some starting point t0 using a Taylor expansion.
The ODE is given by

6. Error Analysis for Taylor Approximation
Then, if we use the first n terms of the series as an approximation for the solution, we get:

7. Error Analysis for Taylor Approximation
We saw that the local truncation error is bounded as follows:
where

8. Truncation Error vs Roundoff Error

9. Stability of solutions
Solution of ODE is
Stable if solutions resulting from perturbations of initial value remain close to original solution
Asymptotically stable if solutions resulting from perturbations converge back to original solution
Unstable if solutions resulting from perturbations diverge away from original solution without bound

10. Example: Stable Solutions

11. Example: Unstable solution
ODE x’ = x is unstable
(solution is x(t)=cet )
we show solutions with Euler’s method

12. Example: Unstable solution

13. Example: asymptotically stable solutions
ODE x’ = -x is stable
(solution is x(t)= ce-t ) (solution -> 0)
if h too large, numerical solution is unstable
we show solutions with Euler’s method in red

14. x’=-x, stable but slow solution

15. x’=-x, stable, rather inaccurate solution for h large

16. x’=-x, stable but poor solution

17. Stability Analysis
The solution to satisfying some initial condition x(a) = s will be:
unstable if fx ≥ δ for some positive δ and
stable if fx ≤ -δ for some positive δ.
Proof: Let x(t,s) be the solution to the IVP above. Then
Since t and s are independent variables, then dt/ds=0, so ->

18. Stability Analysis
Proof(continued): So, if we assume x is C1
Let
The solution to this diff eq is u(t) = c eQ(t) , where
If q(r) = fx(r,x(r,s)) ≥ δ >0 then Q-> ∞ so, u-> ∞ and the solution is unstable.
If q(r) = fx(r,x(r,s)) ≤ -δ <0 then Q-> -∞ so, u-> 0 and the solution is stable. QED

19. Euler’s Method Review:
The simple Euler method is a linear approximation, and is a perfect solution only if the function is linear (or at least linear in the interval).
This is inherently inaccurate.
Because of this inaccuracy small step sizes are required when using the algorithm.

20. Better Methods
If we took more terms in the Taylor’s Series expansion of f(t,x), we would get a more accurate solution algorithm.
Problem: It is difficult to compute higher-order derivatives numerically.
Idea: Approximate the value of higher derivatives of f(t,x) by evaluating f several times between iterates: ti and ti+1 .

21. Better Methods The ODE +IV x(a)= x0 =s
can be alternatively solved as an integral equation. For example, on the first interval from x0 to x1 :

22. Better Methods In general:
The problem is that we don’t know x(t), so evaluating the integral is not trivial.
We assume that if h is small enough, f(t,x) will not change that much over the interval [ti, ti+1]. Using “left-rectangular integration,”
This is the Euler Method. A family of improved versions are called Runge-Kutta methods

23. Euler Method Euler Solution x xi true solution h ti ti+1 = ti + h
Tuncation error:
true solution

24. Euler vs Actual Solution
Euler Solution
RK methods differ in how they estimate fi
For Euler Method (a type of RK method), fi ≈ f(ti, xi) x xi fi
true solution h ti
ti+1 = ti + h

25. Runge-Kutta Order 2 Methodx xi fi h
Weighted average of two slopes x ti
ti+αh
ti+1