Runge-Kutta methods (RK2/RK4)

At a Glance

• Frequency: 4 sub-parts across 4 of 13 years (2014, 2015, 2019, 2022)
• Priority tier: T3
• Marks (count): 10 (2), 15 (2)
• Average solve time: ~10 min
• Difficulty mix: medium 4
• Section: B | Dominant type: computation

Why This Chapter Matters

RK4 appears in Section B at 10–15 marks in four of the last nine years, always as a pure computation: apply the four-stage formula to advance the ODE solution one or two steps. There is no theory to derive — the formula is the entire job. The skill is working through the arithmetic carefully at 4 decimal places without sign or rounding errors. The 2022 question has two steps (final answer accumulates error over two applications).

Minimum Theory

RK4 formula. For $dy/dx=f(x,y)$ with $y(x_0)=y_0$ and step size $h$: $k_1=h\,f(x_n,y_n),$ $k_2=h\,f\!\left(x_n+\tfrac{h}{2},\;y_n+\tfrac{k_1}{2}\right),$ $k_3=h\,f\!\left(x_n+\tfrac{h}{2},\;y_n+\tfrac{k_2}{2}\right),$ $k_4=h\,f(x_n+h,\;y_n+k_3),$ $y_{n+1}=y_n+\tfrac{1}{6}(k_1+2k_2+2k_3+k_4).$

Weights: $1:2:2:1$ (sum $=6$). Node pattern: $k_1$ at $(x_n,y_n)$; $k_2,k_3$ at the midpoint $x_n+h/2$; $k_4$ at the endpoint $x_n+h$.

Key structure: $k_2$ uses $k_1/2$; $k_3$ uses $k_2/2$; $k_4$ uses $k_3$ (full step, not $k_3/2$).

Question Archetypes

runge-kutta (4 question(s); 2014, 2015, 2019, 2022)

Recognition Cues — Explicit ODE $y'=f(x,y)$ with one initial condition; asked to find $y$ at a specific $x$ using “Runge-Kutta method of fourth order.”

Solution Template

1. Identify $f(x,y)$, $x_0$, $y_0$, $h$.
2. Compute $k_1$ at $(x_0,y_0)$.
3. Compute $k_2$ at $(x_0+h/2,\;y_0+k_1/2)$.
4. Compute $k_3$ at $(x_0+h/2,\;y_0+k_2/2)$.
5. Compute $k_4$ at $(x_0+h,\;y_0+k_3)$.
6. Update: $y_1=y_0+\tfrac{1}{6}(k_1+2k_2+2k_3+k_4)$.
7. Repeat for a second step if required, using $y_1$ and $x_1=x_0+h$ as the new starting point.

Worked Example 1 (one step)

2019 Paper 2, 2019-P2-Q5d (10 marks)

Using RK4, solve $dy/dx=(y^2-x^2)/(y^2+x^2)$, $y(0)=1$, at $x=0.2$ with $h=0.2$.

$f(x,y)=(y^2-x^2)/(y^2+x^2)$, $x_0=0$, $y_0=1$, $h=0.2$.

$k_1=0.2\cdot f(0,1)=0.2\cdot1=0.2000$.

$k_2=0.2\cdot f(0.1,1.1)=0.2\cdot(1.21-0.01)/(1.21+0.01)=0.2\cdot(1.20/1.22)=0.2\cdot0.9836=0.1967$.

$k_3=0.2\cdot f(0.1,1.0984)=0.2\cdot(1.2064-0.01)/(1.2064+0.01)\approx0.2\cdot0.9836=0.1967$.

$k_4=0.2\cdot f(0.2,1.1967)=0.2\cdot(1.4321-0.04)/(1.4321+0.04)=0.2\cdot0.9457=0.1891$.

$y_1=1+\tfrac{1}{6}(0.2000+2\cdot0.1967+2\cdot0.1967+0.1891)=1+\tfrac{1}{6}(1.1759)\approx1.1960$.

$\boxed{y(0.2)\approx1.1960.}$

Worked Example 2 (two steps)

2022 Paper 2, 2022-P2-Q8b (15 marks)

Using RK4, solve $dy/dx=x+y^2$, $y(0)=1$, at $x=0.2$ with $h=0.1$.

Step 1 ($x=0\to0.1$):

$k_1=0.1(0+1)=0.1000$. $k_2=0.1(0.05+1.05^2)=0.1\cdot1.1525=0.1153$. $k_3=0.1(0.05+1.0576^2)\approx0.1\cdot1.1686=0.1169$. $k_4=0.1(0.1+1.1169^2)\approx0.1\cdot1.3474=0.1347$.

$y_1=1+\tfrac16(0.1000+0.2306+0.2338+0.1347)=1+\tfrac16(0.6991)=1+0.1165=1.1165$.

Step 2 ($x=0.1\to0.2$):

$k_1=0.1(0.1+1.1165^2)=0.1\cdot1.3466=0.1347$. $k_2=0.1(0.15+1.1839^2)\approx0.1\cdot1.5516=0.1552$. $k_3=0.1(0.15+1.1941^2)\approx0.1\cdot1.5759=0.1576$. $k_4=0.1(0.2+1.2741^2)\approx0.1\cdot1.8233=0.1823$.

$y_2=1.1165+\tfrac16(0.1347+0.3104+0.3152+0.1823)=1.1165+\tfrac16(0.9426)=1.1165+0.1571=1.2736$.

$\boxed{y(0.2)\approx1.2736.}$

Common Traps

• $k_4$ uses $k_3$, not $k_3/2$. The endpoint slope uses the full step increment $y_n+k_3$, while the midpoint slopes use half increments $k_1/2$ and $k_2/2$.
• Keep $\ge4$ decimal places throughout. Rounding $k_2$ or $k_3$ to 3 places shifts the final answer in the 4th decimal. Work with 4+ places and round only at the end.
• Weights: $1:2:2:1$, not $1:1:1:1$. The middle two stages $k_2,k_3$ each get weight 2.
• Each $k_i$ already includes $h$. Don’t multiply by $h$ again in the update formula.

Marks-Aware Writing

Show all four $k$-values explicitly (each as a formula and a decimal). Display the weighted average $k_1+2k_2+2k_3+k_4$, divide by 6, and add to $y_n$. For a two-step question, repeat all four stages for the second step. The examiners give process marks for the correct formula applied correctly — getting a slightly wrong numerical answer because of rounding still earns most marks if the method is shown correctly.

Practice Set

• 2014-P2-Q6c (10 m) — — Hint: one RK4 step with $h=0.2$ from $x=0.6$ to $x=0.8$.
• 2015-P2-Q7b (10 m) — — Hint: one or two RK4 steps on an interval around $x=2$.