MATH2089 Chap.6 Ordinary Differential Equations
Ordinary Differential Equations
Initial value problems become discrete time sequences through numerical updates. Euler's method uses the current slope and has first order global accuracy, while Heun and Runge Kutta methods combine staged slopes for higher order.
This chapter distinguishes accuracy order from stability, explains why stiff systems may favour implicit methods and shows how shooting converts a boundary value problem into repeated initial value solves plus root finding. Write an initial-value problem as a state, a time and a derivative rule. A transparent computation records the step number, current time, current state, evaluated slope and updated state.
Time and state must advance together; reusing an old slope or mixing a new state with an old time changes the method. For systems, the same structure applies to vectors, and higher-order scalar equations can be rewritten as first-order systems by introducing additional state variables. Euler uses one slope at the start of each step.
Its first-order global accuracy means that, under suitable smoothness and stability, halving the step should roughly halve the leading error at a fixed final time. Heun and Runge-Kutta methods combine several carefully located stage slopes to obtain higher order. Each stage has a prescribed intermediate time and state, so informal averaging is not a substitute for the printed tableau or update formula.
Formal order does not alone choose the step size. Stability can force an explicit method to take much smaller steps than visible solution features suggest. This behaviour is characteristic of stiff problems, where fast decaying modes constrain the method even when the solution of interest changes slowly. An implicit method requires a solve at each step but may permit a much more useful stability region.
Compare methods using both work per step and the number of steps required. Refinement is the main practical check. Repeat the solution with a smaller step, compare at the same physical time and inspect whether the difference follows the expected order. An exact solution, invariant or residual gives an additional check when available.
Higher nominal order is not evidence if the values have not entered the asymptotic regime or the system is unstable. A boundary-value problem imposes conditions at separated points. Shooting guesses the missing initial condition, integrates, measures the far-boundary mismatch and adjusts the guess through root finding.
Because every mismatch evaluation contains numerical integration error, the accepted shooting parameter must be checked again under step refinement. Report the method, step size, update structure, stopping rule and independent check rather than presenting a table of states without its numerical reasoning.
What this chapter covers
- 01
Initial value problems
- 02
Euler updates
- 03
Heun and Runge Kutta stages
- 04
Global error and refinement
- 05
Stiffness and implicit methods
- 06
Shooting for boundary values
Worked example · free
Two Euler updates
- +1At the initial point the slope is one.
- +1The first update gives y1=1.25 at t1=0.25.
- +1The new slope is 1.5.
- +1The second update gives y2=1.625 at t2=0.5.
Key terms
- Initial value problem
- A differential equation supplied with the state at a starting point.
- Stiff problem
- A problem for which some numerical methods need extremely small steps for stability.
- Shooting method
- A boundary value technique that adjusts missing initial data until the far boundary is met.
Ordinary Differential Equations FAQ
What does first order global error mean?
For a smooth stable problem, Euler error at a fixed final time is proportional to h. Halving h should therefore roughly halve the leading error in the asymptotic regime.
Why use an implicit method?
Implicit methods require a solve at each step but can remain stable with much larger steps on stiff problems. Stability can outweigh their extra per step cost.
How are higher order equations handled?
Introduce state variables to rewrite the equation as a first order system. Then apply the vector form of the chosen time stepping method with consistent component ordering.
Exam move
Build a table comparing Euler, implicit Euler, trapezoidal, Heun, RK2 and RK4 by order, stages and stability. Work the same IVP with two step sizes and estimate the observed rate. Practise converting a second order equation into a first order system before applying any code. Use a row-by-row state table so every stage has an explicit time, state and slope.
Add one stiff example where an explicit method is forced to take tiny steps and explain why higher order alone does not remove the stability issue. For shooting, define the endpoint mismatch as a scalar root function and show how one guessed initial condition produces one function evaluation. Repeat the accepted shot with a smaller integration step.
A strong timed solution names the method, step size, update rule, refinement evidence and stability reason rather than presenting unexplained software output.
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