PDE Practice - Boundary - .c
Latex Final Formulas:
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Explicit Finite Difference Method: u_i^{n+1} = u_i^n + \Delta t \left( \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{\Delta x^2} - u_i^n \cdot \frac{u_{i+1}^n - u_{i-1}^n}{2\Delta x} \right)
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Crank-Nicolson Method (Unconditional Stability, No CFL): \frac{u_i^{n+1} - u_i^n}{\Delta t} + \frac{u_i^n + u_i^{n+1}}{2} \cdot \frac{u_{i+1}^n - u_{i-1}^n + u_{i+1}^{n+1} - u_{i-1}^{n+1}}{4\Delta x} = \frac{1}{2} \left( \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{\Delta x^2} + \frac{u_{i+1}^{n+1} - 2u_i^{n+1} + u_{i-1}^{n+1}}{\Delta x^2} \right)
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Forward-Time Centered-Space (FTCS) Scheme for Linear Diffusion: u_i^{n+1} = u_i^n + \Delta t \cdot \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{\Delta x^2}
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Forward Elimination (TriDiagonal System - Thomas Algo): Modified coefficients b'i = b_i - \frac{a_i c{i-1}}{b'{i-1}}, \quad d'i = d_i - \frac{a_i d'{i-1}}{b'{i-1}}
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Backward Substitution: u_{n-1} = \frac{d'{n-1}}{b'{n-1}}, \quad u_i = \frac{d'i - c_i u{i+1}}{b'_i}, \quad \text{for } i = n-2, n-3, \ldots, 1