ExerciseODEsadvanced

Exercise: numerical order estimation (Euler, Heun, RK4)

On the same logistic IVP, Euler's order is estimated with the exact solution, Heun's by comparing successive meshes without an exact solution, and RK4's, confirming orders 1, 2 and 4.

The reference problem

The logistic (Verhulst) IVP $y'(t)=(3-0.1\,y)\,y$ on $[0,2]$ with $y(0)=10$ is used; its exact solution is known and allows measuring true errors:

y(t)=\frac{30}{1+2e^{-3t}}

Each method is run with $N=\{2,4,8,16,32,64\}$ subintervals and the logarithmic ratio from Convergence, consistency and order is applied to the maximum errors.

Euler, with the exact solution

N	Maximum error $E_N$	$\log_2(E_{N/2}/E_N)$
2	2.7167	n/a
4	2.7167	0.0000
8	1.0659	1.3497
16	0.4878	1.1277
32	0.2361	1.0467
64	0.1164	1.0202

Euler errors against the exact solution.

Heun, without the exact solution

To illustrate the technique that needs no exact solution, with Heun each discrete solution is compared with the double-mesh one at the common nodes, $\varepsilon_N=\max_k|y^{(N)}_k-y^{(2N)}_{2k}|$ :

N	$\varepsilon_N$	$\log_2(\varepsilon_{N/2}/\varepsilon_N)$
4	18.2934	n/a
8	1.9319	3.2432
16	0.3288	2.5549
32	0.0686	2.2603
64	0.0158	2.1154

Differences between successive meshes with Heun.

Runge-Kutta 4

N	Maximum error $E_N$	$\log_2(E_{N/2}/E_N)$
2	4.7316	n/a
4	0.1442	5.0362
8	6.531·10⁻³	4.4646
16	3.469·10⁻⁴	4.2347
32	1.992·10⁻⁵	4.1223
64	1.192·10⁻⁶	4.0624

RK4 errors against the exact solution.