Differential Equations Numerical Methods for Differential Equations Syllabus

Numerical Methods for Differential Equations syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Numerical Methods for Differential Equations in Differential Equations, not a summary of it.

1. 1

   Single-Step Methods

   3 topics

   • Euler's Method
   • Improved Euler and Heun's Method
   • Runge-Kutta Methods
2. 2

   Multistep and Adaptive Methods

   3 topics

   • Adams-Bashforth and Adams-Moulton Methods
   • Predictor-Corrector Schemes
   • Adaptive Step-Size Control
3. 3

   Numerical Analysis Considerations

   3 topics

   • Stability and Stiff Equations
   • Convergence and Consistency
   • Numerical Solution of Systems and Higher-Order Equations
4. 4

   Boundary-Value Problem Methods

   2 topics

   • Shooting Method
   • Finite Difference Method

Numerical Methods for Differential Equations flashcards for Differential Equations

18 of 59 cards from the Numerical Methods for Differential Equations deck — real questions with worked answers.

1. What is the general form of the initial value problem (IVP) that one-step numerical methods aim to solve?

   $$\frac{dy}{dx} = f(x,y), \qquad y(x_{0}) = y_{0}$$ where we seek an approximation to $y(x)$ on some interval.

2. State the update formula for the (forward/explicit) Euler's Method.

   $$y_{n+1} = y_{n} + h\,f(x_{n}, y_{n})$$ where $h$ is the step size and $x_{n} = x_{0} + nh$.

3. What is the geometric interpretation of Euler's Method?

   It advances the solution along the tangent line at $(x_{n}, y_{n})$: the slope $f(x_{n},y_{n})$ is followed for one step of length $h$ to reach the next point.

4. What is the local truncation error (per step) of Euler's Method in terms of $h$?

   $O(h^{2})$. The local error per step is proportional to $h^{2}$, specifically $\frac{h^{2}}{2}y''(\xi)$.

5. What is the global (accumulated) error order of Euler's Method?

   $O(h)$. Euler's Method is a first-order method: halving $h$ roughly halves the global error.

6. State the update formula for the backward (implicit) Euler Method.

   $$y_{n+1} = y_{n} + h\,f(x_{n+1}, y_{n+1})$$ It is implicit because $y_{n+1}$ appears on both sides and must be solved for.

7. Why is the backward Euler Method preferred over forward Euler for stiff equations?

   Backward Euler is A-stable (unconditionally stable), so it remains stable for any step size $h$ on stiff problems, whereas forward Euler requires very small $h$ to stay stable.

8. State the Improved Euler / Heun's Method formulas (predictor-corrector form).

   Predictor: $\tilde{y}_{n+1} = y_{n} + h f(x_{n}, y_{n})$. Corrector: $$y_{n+1} = y_{n} + \frac{h}{2}\left[f(x_{n}, y_{n}) + f(x_{n+1}, \tilde{y}_{n+1})\right]$$

9. What is the global error order of Heun's (Improved Euler) Method?

   $O(h^{2})$ — it is a second-order method, with local truncation error $O(h^{3})$.

10. What idea makes Heun's Method more accurate than Euler's Method?

    It averages the slope at the start of the interval with the slope at the predicted endpoint, using the mean slope $\frac{1}{2}(k_{1}+k_{2})$ rather than only the initial slope.

11. State the update formula for the Midpoint (Modified Euler) Method.

    $$y_{n+1} = y_{n} + h\,f\!\left(x_{n} + \frac{h}{2},\; y_{n} + \frac{h}{2}f(x_{n}, y_{n})\right)$$ It is a second-order method that evaluates the slope at the interval midpoint.

12. Write the four stage slopes $k_{1},k_{2},k_{3},k_{4}$ of the classical fourth-order Runge-Kutta (RK4) method.

    $$k_{1}=f(x_{n},y_{n}),\quad k_{2}=f\!\left(x_{n}+\tfrac{h}{2},\,y_{n}+\tfrac{h}{2}k_{1}\right)$$ $$k_{3}=f\!\left(x_{n}+\tfrac{h}{2},\,y_{n}+\tfrac{h}{2}k_{2}\right),\quad k_{4}=f(x_{n}+h,\,y_{n}+hk_{3})$$

13. State the update (weighted-average) formula for the classical RK4 method.

    $$y_{n+1} = y_{n} + \frac{h}{6}\left(k_{1} + 2k_{2} + 2k_{3} + k_{4}\right)$$

14. What are the local and global error orders of the classical RK4 method?

    Local truncation error is $O(h^{5})$; global error is $O(h^{4})$ (fourth-order accurate).

15. How many function evaluations of $f$ per step does classical RK4 require?

    Four evaluations per step ($k_{1}$ through $k_{4}$).

16. What is the general form of an explicit $s$-stage Runge-Kutta method?

    $$y_{n+1} = y_{n} + h\sum_{i=1}^{s} b_{i} k_{i}, \qquad k_{i} = f\!\left(x_{n}+c_{i}h,\; y_{n} + h\sum_{j<i} a_{ij} k_{j}\right)$$

17. What is a Butcher tableau used to represent?

    It compactly encodes the coefficients of a Runge-Kutta method: the matrix $a_{ij}$ (nodes' internal weights), the nodes $c_{i}$, and the weights $b_{i}$, arranged as $\begin{array}{c|c} c & A \\ \hline & b^{T}\end{array}$.

18. What consistency condition must the weights $b_{i}$ of any Runge-Kutta method satisfy?

    $$\sum_{i=1}^{s} b_{i} = 1$$ (the weights must sum to 1 for at least first-order consistency).

See more Numerical Methods for Differential Equations flashcards →

Planning Numerical Methods for Differential Equations for Differential Equations

Numerical Methods for Differential Equations (Differential Equations) FAQ