Differential equations: Euler and RK4

Skill positionModule 3 of 4 · Topic 2 of 2

3.2 Differential equations: Euler and RK4

—Lab score— / 100Total XP—Streak—

Following a rate of change

A differential equation states how fast a quantity changes: $\frac{dy}{dt}=f(t,y)$ . Money growing at a continuous rate $r$ obeys $\frac{dy}{dt}=ry$ , with solution $y_0e^{rt}$ . A short rate pulled back to a long-run level $\theta$ obeys $\frac{dr}{dt}=\kappa(\theta-r)$ . Few equations have a formula for the solution, so we step through time numerically.

y_{n+1}=y_n+h\,f(t_n,y_n)

(1)

Euler's method follows the tangent for one short step $h$ : new value = old value + step × slope. It is simple but only first-order accurate: halving $h$ halves the error. Runge–Kutta of order 4 (RK4) samples the slope four times per step and blends them, $y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)$ , and its error shrinks like $h^4$ : halving $h$ divides it by 16.

Growth at rate 1, two waysSolve

y'=y

y(0)=1

up to

t=1

with

h=0.5

. Euler:

y_1=1+0.5\cdot1=1.5

y_2=1.5+0.5\cdot1.5=2.25

, against

e=2.7183

, an error of 0.4683. RK4 for one step to

t=0.5

k_1=1

k_2=1.25

k_3=1.3125

k_4=1.65625

, so

y=1+\frac{0.5}{6}(1+2.5+2.625+1.65625)=1.6484

, against the true

e^{0.5}=1.6487

Euler: $y_{n+1}=y_n+h\,f(t_n,y_n)$ , first-order accurate.
RK4: four slopes per step blended with weights 1, 2, 2, 1; fourth-order accurate.
Truncation error: what is lost by stepping instead of solving exactly.

A smaller step is not freeHalving the step doubles the number of steps. Past a point, rounding error grows while truncation error no longer matters, and some stiff equations blow up unless the step is small enough. Compare two step sizes to estimate the error before you trust a result.

Lab · what's graded

Two Euler steps15 pts
The Euler error15 pts
One RK4 step20 pts
Orders of accuracy10 pts
Halving the step15 pts
The Euler loop10 pts
Why RK4 beats Euler15 pts

0/ 7 answered

1Numeric answer15 ptsNot answered

Consider $y'=y$ with $y(0)=1$ and step $h=0.5$ . Take two Euler steps and give $y_2$ (the estimate of $y(1)$ ).

y_2 =

accepted: ± 0.001

2Numeric answer15 ptsNot answered

Consider $y'=y$ with $y(0)=1$ and step $h=0.5$ . The exact solution is $y(1)=e=2.7183$ . How far is the Euler estimate $y_2=2.25$ from it?

\text{error} =

accepted: ± 0.001

3Numeric answer20 ptsNot answered

Consider $y'=y$ with $y(0)=1$ and step $h=0.5$ . Take one RK4 step from $t=0$ and give $y_1$ , the estimate of $y(0.5)$ , to 4 decimals. For $f(t,y)=y$ : $k_1=y_n$ , $k_2=y_n+\frac{h}{2}k_1$ , $k_3=y_n+\frac{h}{2}k_2$ , $k_4=y_n+hk_3$ .

y_1 =

accepted: ± 0.0002

4Choice10 ptsNot answered

What are the orders of accuracy of Euler's method and of RK4?

5Numeric answer15 ptsNot answered

Euler's method has error 0.08 with step $h=0.1$ . About what error do you expect with $h=0.05$ ?

\text{error} =

accepted: ± 0.005

6Order the steps10 ptsNot answered

Put the steps of one Euler iteration in order.

Drag to reorder, or focus an item, press Space to lift it, the arrow keys to move it and Space to drop it.

1Update $y\leftarrow y+h\cdot\text{slope}$
2Start from the initial values $t_0$ and $y_0$
3Advance time $t\leftarrow t+h$ and repeat
4Compute the slope $f(t,y)$ at the current point

7Explain in words15 ptsNot answered

In two or three sentences, explain why RK4 is usually more accurate than Euler for the same step size and what it costs.

Graded on:It samples the slope several times per stepIts error shrinks much faster as the step gets smallerIt costs more work per step0 / 400 characters

Math modeKeyboard or LaTeX inputTask 1 of 7 · Ctrl ⏎ to check

Differential equations: Euler and RK4

Skill positionModule 3 of 4 · Topic 2 of 2

3.2 Differential equations: Euler and RK4

—Lab score— / 100Total XP—Streak—

Following a rate of change

y_{n+1}=y_n+h\,f(t_n,y_n)

(1)

Growth at rate 1, two waysSolve

y'=y

y(0)=1

up to

t=1

with

h=0.5

. Euler:

y_1=1+0.5\cdot1=1.5

y_2=1.5+0.5\cdot1.5=2.25

, against

e=2.7183

, an error of 0.4683. RK4 for one step to

t=0.5

k_1=1

k_2=1.25

k_3=1.3125

k_4=1.65625

, so

y=1+\frac{0.5}{6}(1+2.5+2.625+1.65625)=1.6484

, against the true

e^{0.5}=1.6487

Euler: $y_{n+1}=y_n+h\,f(t_n,y_n)$ , first-order accurate.
RK4: four slopes per step blended with weights 1, 2, 2, 1; fourth-order accurate.
Truncation error: what is lost by stepping instead of solving exactly.

Lab · what's graded

Two Euler steps15 pts
The Euler error15 pts
One RK4 step20 pts
Orders of accuracy10 pts
Halving the step15 pts
The Euler loop10 pts
Why RK4 beats Euler15 pts

0/ 7 answered

1Numeric answer15 ptsNot answered

Consider $y'=y$ with $y(0)=1$ and step $h=0.5$ . Take two Euler steps and give $y_2$ (the estimate of $y(1)$ ).

y_2 =

accepted: ± 0.001

2Numeric answer15 ptsNot answered

Consider $y'=y$ with $y(0)=1$ and step $h=0.5$ . The exact solution is $y(1)=e=2.7183$ . How far is the Euler estimate $y_2=2.25$ from it?

\text{error} =

accepted: ± 0.001

3Numeric answer20 ptsNot answered

y_1 =

accepted: ± 0.0002

4Choice10 ptsNot answered

What are the orders of accuracy of Euler's method and of RK4?

5Numeric answer15 ptsNot answered

Euler's method has error 0.08 with step $h=0.1$ . About what error do you expect with $h=0.05$ ?

\text{error} =

accepted: ± 0.005

6Order the steps10 ptsNot answered

Put the steps of one Euler iteration in order.

Drag to reorder, or focus an item, press Space to lift it, the arrow keys to move it and Space to drop it.

1Update $y\leftarrow y+h\cdot\text{slope}$
2Start from the initial values $t_0$ and $y_0$
3Advance time $t\leftarrow t+h$ and repeat
4Compute the slope $f(t,y)$ at the current point

7Explain in words15 ptsNot answered

In two or three sentences, explain why RK4 is usually more accurate than Euler for the same step size and what it costs.

Graded on:It samples the slope several times per stepIts error shrinks much faster as the step gets smallerIt costs more work per step0 / 400 characters

Math modeKeyboard or LaTeX inputTask 1 of 7 · Ctrl ⏎ to check

Differential equations: Euler and RK4

Following a rate of change

Lab · what's graded

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Differential equations: Euler and RK4

Following a rate of change

Lab · what's graded

Log in to run your code.