Numerical ODEs: Euler, Taylor, RK, Multistep & Error Analysis

Key Concepts & Methods

1. Classification of ODEs

Ordinary Differential Equations (ODEs) are classified based on:

• Order: Highest derivative present (e.g., $y^{\prime }=f(x,y)$ is 1st-order, $y^{\prime \prime }=f(x,y,y^{\prime })$ is 2nd-order).
• Linearity: Linear ODEs have dependent variable $y$ and its derivatives to the first power (e.g., $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=g(x)$). Nonlinear ODEs include terms like $y^2$, $y{y}^{\prime }$, or $e^y$.
• Initial Value Problems (IVPs): Specified with initial conditions (e.g., $y(x_0)=y_0$).
• Boundary Value Problems (BVPs): Specified with boundary conditions (e.g., $y(a)=\alpha$, $y(b)=\beta$).

2. Single-Step Methods

Single-step methods compute $y_{n+1}$ using only the previous point $(x_n,y_n)$. Key methods:

A. Euler’s Method

Definition: Approximates the solution of $y^{\prime }=f(x,y)$ using the tangent line at $(x_n,y_n)$: $$y_{n+1}=y_n+hf(x_n,y_n),$$ where $h$ is the step size.

Derivation:

• Taylor expansion of $y(x+h)$ around $x$: $$y(x+h)=y(x)+h{y}^{\prime }(x)+\frac{h^2}{2}{y}^{\prime \prime }(x)+\cdots .$$
• Euler’s method truncates after the first two terms, ignoring higher-order derivatives.

Worked Example: Solve $y^{\prime }=2x+y$, $y(0)=1$ using Euler’s method with $h=0.1$ to approximate $y(0.4)$.

Approximate $y(0.4)\approx 1.5923$.

Error Analysis:

• Local Truncation Error (LTE): Error introduced in one step, proportional to $h^2$ for Euler’s method.
• Global Truncation Error (GTE): Accumulated error over $N$ steps, proportional to $h$ (i.e., $O(h)$).

Advantages/Disadvantages:

B. Taylor Series Method

Definition: Uses higher-order Taylor expansions to improve accuracy. For $y^{\prime }=f(x,y)$, the $k$-th order method is: $$y_{n+1}=y_n+h{y}^{\prime }(x_n)+\frac{h^2}{2!}{y}^{\prime \prime }(x_n)+\cdots +\frac{h^k}{k!}{y}^{(k)}(x_n).$$ Derivatives $y^{\prime \prime }(x),y^{\prime \prime \prime }(x),\dots$ are computed using $y^{\prime }=f(x,y)$.

Worked Example: Solve $y^{\prime }=3x^2$, $y(1)=1$ using the first four terms of Taylor’s series to approximate $y(2)$ with $h=1$.

1. Compute derivatives:

   • $y^{\prime }=3x^2$
   • $y^{\prime \prime }=6x$
   • $y^{\prime \prime \prime }=6$
   • $y^{(4)}=0$

2. Apply Taylor expansion at $x_0=1$: $$y(2)=y(1)+h{y}^{\prime }(1)+\frac{h^2}{2}{y}^{\prime \prime }(1)+\frac{h^3}{6}{y}^{\prime \prime \prime }(1).$$ Substitute $h=1$: $$y(2)=1+1(3(1{)}^2)+\frac{1}{2}(6(1))+\frac{1}{6}(6)=1+3+3+1=8.$$ Exact solution: $y(x)=x^3+C$. With $y(1)=1$, $C=0$, so $y(2)=8$ (exact).

Error Analysis:

• LTE: $O(h^4)$ for 4th-order Taylor method.
• GTE: $O(h^3)$.

Advantages/Disadvantages:

C. Runge-Kutta Methods (RK2, RK4)

Definition: Weighted averages of slopes to improve accuracy without computing higher derivatives.

RK2 (Heun’s Method):

$$\begin{array}{rl}{k}_1& =f(x_n,y_n),\\ k_2& =f(x_n+h,y_n+h{k}_1),\\ y_{n+1}& =y_n+\frac{h}{2}(k_1+k_2).\end{array}$$

RK4:

$$\begin{array}{rl}{k}_1& =f(x_n,y_n),\\ k_2& =f\left(x_n+\frac{h}{2},y_n+\frac{h}{2}{k}_1\right),\\ k_3& =f\left(x_n+\frac{h}{2},y_n+\frac{h}{2}{k}_2\right),\\ k_4& =f(x_n+h,y_n+h{k}_3),\\ y_{n+1}& =y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4).\end{array}$$

Worked Example (RK4): Solve $\frac{dy}{dx}+2x^{2}y=4$ with $y(0)=1$, $h=0.5$ to estimate $y(1)$.

Rewrite as $y^{\prime }=4-2x^{2}y$. Compute $y(0.5)$ first:

1. $k_1=f(0,1)=4-2(0{)}^2(1)=4$
2. $k_2=f(0.25,1+0.25\times 4)=4-2(0.25{)}^2(2)=3.75$
3. $k_3=f(0.25,1+0.25\times 3.75)=4-2(0.25{)}^2(1.9375)\approx 3.828$
4. $k_4=f(0.5,1+0.5\times 3.828)=4-2(0.5{)}^2(2.914)\approx 2.543$
5. $y(0.5)=1+\frac{0.5}{6}(4+2(3.75)+2(3.828)+2.543)\approx 1.964$

Repeat for $y(1)$ using $y(0.5)\approx 1.964$ and $h=0.5$.

Error Analysis:

• RK2: LTE $O(h^3)$, GTE $O(h^2)$.
• RK4: LTE $O(h^5)$, GTE $O(h^4)$.

Comparison Table:

3. Multistep Methods

Use multiple previous points to compute $y_{n+1}$. Example: Adams-Bashforth methods.

Adams-Bashforth 2-step (AB2):

$$y_{n+1}=y_n+\frac{h}{2}[3f(x_n,y_n)-f(x_{n-1},y_{n-1})].$$ Requires a starting value (e.g., from RK4).

Advantages/Disadvantages:

4. Higher-Order ODEs

Convert to a system of first-order ODEs. For example: $$y^{\prime \prime }=f(x,y,y^{\prime }),y(x_0)=y_0,y^{\prime }(x_0)=y_0^{\prime }.$$ Let $v=y^{\prime }$, then: $$\{\begin{array}{l}{y}^{\prime }=v,\\ v^{\prime }=f(x,y,v).\end{array}$$

Example: Solve $y^{\prime \prime }+y=0$, $y(0)=1$, $y^{\prime }(0)=0$ using RK4.

Convert to: $$\{\begin{array}{l}{y}^{\prime }=v,\\ v^{\prime }=-y.\end{array}$$

5. Error Analysis & Stability

• Local Truncation Error (LTE): Error per step (e.g., $O(h^2)$ for Euler).
• Global Truncation Error (GTE): Accumulated error over $N$ steps (e.g., $O(h)$ for Euler).
• Stability: A method is stable if errors do not grow unboundedly. Euler’s method is conditionally stable (requires small $h$ for stiff equations).
• Consistency: A method is consistent if LTE $\to 0$ as $h\to 0$.
• Convergence: A consistent method converges if GTE $\to 0$ as $h\to 0$.

Stiff Equations: ODEs with widely varying solution scales (e.g., $y^{\prime }=\lambda y+f(x)$, where $|\lambda |\gg 1$). Implicit methods (e.g., backward Euler) are preferred for stability.

6. Choosing a Method

Exam Tip

What Examiners Look For

1. Derivations:

   • Always derive formulas (e.g., Euler’s method, RK4) from Taylor expansions.
   • Show intermediate steps for derivatives in Taylor’s method.

2. Worked Examples:

   • Euler/Taylor/RK4: Compute tables step-by-step. Label columns clearly (e.g., $x_n$, $y_n$, $k_i$).
   • Error bounds: State LTE/GTE orders (e.g., "Euler’s method has LTE $O(h^2)$").
   • Initial conditions: Verify $y(x_0)$ matches the given IC.

3. Comparisons:

   • Compare methods in a table (order, error, stability, steps).
   • Discuss trade-offs (e.g., RK4 is accurate but computationally expensive).

4. Stability & Convergence:

   • Define stability and mention stiff equations.
   • For RK4, state that it is stable for non-stiff problems.

5. Common Pitfalls:

   • Euler’s method: Forgetting to update $x_n$ or misapplying $h$.
   • RK4: Incorrectly computing $k_2,k_3$ (wrong arguments).
   • Taylor series: Skipping derivative calculations or misapplying $h$.
   • Higher-order ODEs: Forgetting to convert to a system of first-order ODEs.

6. Exam Questions:

   • Direct computation: Solve using a specific method (e.g., "Use RK4 to estimate $y(1)$").
   • Derivation: Prove the formula for a method (e.g., "Derive the RK2 formula").
   • Error analysis: Discuss LTE/GTE for a given method.
   • Comparison: "Which method would you use for a stiff equation? Why?"

Sample Exam Questions & Answers

Quick Revision Checklist

• Can you derive Euler’s method from Taylor expansion?
• Do you know the RK4 coefficients by heart?
• Can you convert a 2nd-order ODE to a system of 1st-order ODEs?
• What is the difference between LTE and GTE?
• Which method is best for stiff equations? Why?

flowchart TD
    A[First-Order ODE: y' = f(x,y)] --> B[Single-Step Methods]
    A --> C[Multistep Methods]
    B --> B1[Euler: O(h)]
    B --> B2[Taylor: O(h^k)]
    B --> B3[RK2: O(h^3)]
    B --> B4[RK4: O(h^5)]
    C --> C1[Adams-Bashforth: O(h^3)]
    C --> C2[Adams-Moulton: O(h^5)]
    B1 --> D[Low Accuracy\nHigh Speed]
    B4 --> E[High Accuracy\nHigh Cost]
    C1 --> F[Moderate Accuracy\nRequires Startup]
    G[Higher-Order ODEs] --> H[Convert to System\nSolve as IVP]