Numerical Methods

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Subsection C.1 Numerical Methods

Numerical methods hurt my head. 🤯

Comparison of Finite Difference Scheme Properties.

The following tables summarize common finite difference schemes for first and second derivatives.

Table C.1.3. First-Derivative Finite Difference Schemes

Scheme and Formulation	Error	Grid Points
Forward Difference 🔗 \begin{equation} \frac{df}{dx} \approx \frac{f(x+\Delta x) - f(x)}{\Delta x} \end{equation} 🔗	\(\mathcal{O}(\Delta x)\)	\(1\)
Backward Difference 🔗 \begin{equation} \frac{df}{dx} \approx \frac{f(x) - f(x-\Delta x)}{\Delta x} \end{equation} 🔗	\(\mathcal{O}(\Delta x)\)	\(1\)
Central Difference 🔗 Second-order accurate central difference scheme: \begin{equation} \frac{df}{dx} \approx \frac{f(x+\Delta x) - f(x-\Delta x)}{2\Delta x} \end{equation} 🔗	\(\mathcal{O}(\Delta x^2)\)	\(2\)

Table C.1.4. Second-Derivative Finite Difference Schemes

Scheme and Formulation	Error	Grid Points
Central Difference 🔗 \begin{equation} \frac{d^2f}{dx^2} \approx \frac{f(x+\Delta x) - 2f(x) + f(x-\Delta x)}{\Delta x^2} \end{equation} 🔗	\(\mathcal{O}(\Delta x^2)\)	\(2\)