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📘 Comprehensive Numerical Methods Cheat Sheet

✅ 1. Euler’s Method (1st Order ODE)

Purpose: Solve initial value problems (IVPs) of the form:

\frac{d y}{d x} = f (x, y), y (x_{0}) = y_{0}

Formula:

y_{n + 1} = y_{n} + h \cdot f (x_{n}, y_{n})

Key Elements:

Step Size (h): Determines accuracy (smaller $h$ = better precision).
Local Truncation Error: $O (h^{2})$ .
Global Error: $O (h)$ (first-order method).

Limitation: Poor accuracy for nonlinear or stiff equations.

✅ 2. Modified Euler’s Method (Predictor-Corrector)

Purpose: Improved accuracy over Euler’s method.

Steps:

Predictor (Euler’s Step):
$y_{n + 1}^{*} = y_{n} + h \cdot f (x_{n}, y_{n})$
Corrector (Trapezoidal Rule):
$y_{n + 1} = y_{n} + \frac{h}{2} [f (x_{n}, y_{n}) + f (x_{n + 1}, y_{n + 1}^{*})]$

Key Elements:

Error: $O (h^{3})$ local, $O (h^{2})$ global.
Advantage: Balances simplicity and accuracy.

✅ 3. Runge-Kutta 2nd Order (RK-2)

Variants: Midpoint Method or Heun’s Method.
General Form:

y_{n + 1} = y_{n} + h \cdot Φ (x_{n}, y_{n}, h)

Midpoint Method:

Φ = f (x_{n} + \frac{h}{2}, y_{n} + \frac{h}{2} k_{1}), k_{1} = f (x_{n}, y_{n})

Heun’s Method:

Φ = \frac{1}{2} (k_{1} + k_{2}), k_{2} = f (x_{n} + h, y_{n} + h k_{1})

Key Elements:

Error: $O (h^{3})$ local, $O (h^{2})$ global.
Use Case: Moderate accuracy requirements.

✅ 4. Runge-Kutta 4th Order (RK-4)

Purpose: High-precision solution for IVPs.

Slopes:

\begin{aligned} k_{1} & = h \cdot f (x_{n}, y_{n}), \\ k_{2} & = h \cdot f (x_{n} + \frac{h}{2}, y_{n} + \frac{k_{1}}{2}), \\ k_{3} & = h \cdot f (x_{n} + \frac{h}{2}, y_{n} + \frac{k_{2}}{2}), \\ k_{4} & = h \cdot f (x_{n} + h, y_{n} + k_{3}) . \end{aligned}

Update:

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

Key Elements:

Error: $O (h^{5})$ local, $O (h^{4})$ global.
Applications: Aerospace, physics simulations.

✅ 5. Newton’s Forward Interpolation

Purpose: Estimate values near the start of equally spaced data.
Formula:

P_{n} (x) = y_{0} + p Δ y_{0} + \frac{p (p - 1)}{2!} Δ^{2} y_{0} + \dots

where $p = \frac{x - x_{0}}{h}$ , and $Δ$ is the forward difference operator.

Key Elements:

Forward Differences:
$Δ y_{0} = y_{1} - y_{0}$ , $Δ^{2} y_{0} = Δ y_{1} - Δ y_{0}$ , etc.
Use Case: Extrapolation near $x_{0}$ .

✅ 6. Newton’s Backward Interpolation

Purpose: Estimate values near the end of equally spaced data.
Formula:

P_{n} (x) = y_{n} + p \nabla y_{n} + \frac{p (p + 1)}{2!} \nabla^{2} y_{n} + \dots

where $p = \frac{x - x_{n}}{h}$ , and $\nabla$ is the backward difference operator.

Key Elements:

Backward Differences:
$\nabla y_{n} = y_{n} - y_{n - 1}$ , $\nabla^{2} y_{n} = \nabla y_{n} - \nabla y_{n - 1}$ , etc.
Use Case: Extrapolation near $x_{n}$ .

✅ 7. Classification of 2nd Order PDEs

General Form:

A u_{x x} + B u_{x y} + C u_{y y} + lower-order terms = 0

Discriminant ( $D$ ):

D = B^{2} - 4 A C

Classification:

Elliptic ( $D < 0$ ): Boundary value problems (e.g., Laplace’s equation $u_{x x} + u_{y y} = 0$ ).
Parabolic ( $D = 0$ ): Initial value problems (e.g., Heat equation $u_{t} = α u_{x x}$ ).
Hyperbolic ( $D > 0$ ): Wave propagation (e.g., Wave equation $u_{t t} = c^{2} u_{x x}$ ).

✅ 8. Bender-Schmidt Method (Parabolic PDEs)

Purpose: Solve the 1D heat equation:

\frac{\partial u}{\partial t} = α \frac{\partial^{2} u}{\partial x^{2}}

Explicit Formula:

u_{i}^{j + 1} = r u_{i - 1}^{j} + (1 - 2 r) u_{i}^{j} + r u_{i + 1}^{j}

where $r = \frac{α Δ t}{(Δ x)^{2}}$ .

Key Elements:

Stability Condition: $r \leq \frac{1}{2}$ .
Grid Spacing: $Δ x$ (space step), $Δ t$ (time step).
Limitation: Conditionally stable.

📌 Summary of Errors & Orders

Method	Local Error	Global Error
Euler’s	$O (h^{2})$	$O (h)$
Modified Euler	$O (h^{3})$	$O (h^{2})$
RK-2	$O (h^{3})$	$O (h^{2})$
RK-4	$O (h^{5})$	$O (h^{4})$

🎯 When to Use Which?

Quick Estimate: Euler’s Method.
Balanced Accuracy-Speed: RK-2 or Modified Euler.
High Precision: RK-4.
Interpolation: Newton’s Forward/Backward.
PDEs: Bender-Schmidt for parabolic, classification guides solver choice.

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📘 Comprehensive Numerical Methods Cheat Sheet

✅ 1. Euler’s Method (1st Order ODE)

✅ 2. Modified Euler’s Method (Predictor-Corrector)

✅ 3. Runge-Kutta 2nd Order (RK-2)

✅ 4. Runge-Kutta 4th Order (RK-4)

✅ 5. Newton’s Forward Interpolation

✅ 6. Newton’s Backward Interpolation

✅ 7. Classification of 2nd Order PDEs

✅ 8. Bender-Schmidt Method (Parabolic PDEs)

📌 Summary of Errors & Orders

🎯 When to Use Which?

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