FIG EJ SU UQJ 2

Ordinary Differential Equations (ODEs) Solutions
📌 ODE studies can be Zero-Dimensional (transient), differentiated only with respect to time ($t$).
📌 One-dimensional steady-state problems involve only one independent variable, typically $x$.
📌 Solutions can be found using analytical integration tricks or numerical methods like Euler's method or Runge-Kutta schemes.

Finite Difference Method (FDM) for ODEs
📌 FDM relies on forward, backward, and central differencing to approximate derivatives (slope) at discrete points ($i-1, i, i+1$).
📌 The second derivative approximation involves $\frac{1}{\Delta x^2}$ and terms derived from Taylor series expansion.
📌 Truncation error arises from neglecting higher-order terms in the Taylor expansion, a common source of error in CFD alongside roundoff and modeling errors.
📌 For the example $d^2u/dx^2 = 0$ over $x \in [0, 1]$ with $u(0)=1$ and $u(1)=0$, the numerical solution $u_I = -1/4$ (at $x=1/2$) matches the analytical solution $u(x) = x^2 - x$ when $\Delta x = 1/2$.
📌 Numerical solutions yield values only at selected locations (grid points), unlike exact solutions which provide values for any $x$.

Partial Differential Equations (PDEs) and Grid Generation
📌 PDEs governing fluid flow (e.g., continuity, momentum) are typically second-order (due to diffusion) and nonlinear (due to convective terms like $u \frac{\partial u}{\partial x}$).
📌 Numerical methods (FDM, Finite Volume Method (FVM), Finite Element Method (FEM)) require grid generation (discretization) of the geometry in $x, y, z$ dimensions to apply conservation equations locally.
📌 Interest in gridless methods is growing to bypass the complex and often necessary step of grid generation required by finite methods.

Comparison of Finite Methods
📌 Finite Difference Method (FDM):
* Oldest method, traceable to the early 20th century, popularized mid-20th century CFD.
* Best suited for structured grids in simple geometries (e.g., pipes or ducts).
* Uses the differential form of conservation equations, meaning it cannot capture phenomena involving discontinuities like shock waves.
📌 Finite Volume Method (FVM):
* Present form popularized in the late 1990s.
* Uses the integral form of conservation equations derived from first principles.
* Can handle both structured and unstructured meshes and effectively capture shocks.
📌 Finite Element Method (FEM):
* More recent for fluid flow (late 1970s), highly popular in structural mechanics (e.g., torsion problems).
* Suffers from issues regarding the conservation property, vital for fluid dynamics.
📌 All three methods convert the PDEs into a system of linear algebraic equations solved simultaneously to find unknowns ($P, u, v, w$) at each grid point.

Key Points & Insights
➡️ CFD fundamentally involves converting continuous PDEs governing flow into discrete algebraic equations using schemes like FDM, FVM, or FEM.
➡️ FDM relies on the Taylor series expansion to approximate derivatives, introducing truncation error by discarding high-order terms.
➡️ The Finite Volume Method (FVM) is preferred in modern CFD for fluid flow as it uses the integral form, allowing it to correctly handle non-continuous flow features like shock waves.
➡️ Grid generation is a mandatory precursor for FDM, FVM, and FEM because these finite methods require the geometry to be discretized before applying conservation laws over elements or grid points.

📸 Video summarized with SummaryTube.com on Mar 11, 2026, 14:58 UTC