Course — Numerical methods

Introduction

This course introduces the main numerical methods used to solve the partial differential equations (PDEs) arising in fluid mechanics and physics. It covers finite differences, finite elements and finite volumes.

These notes are a living document: they grow chapter by chapter.

1. Finite differences

The idea is to approximate derivatives by differences on a grid. For a function $u(x)$, the centered approximation of the second derivative reads:

$$\frac{\partial^2 u}{\partial x^2}\bigg|_{x_i} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{\Delta x^2} + \mathcal{O}(\Delta x^2)$$

Applied to the heat equation $\partial_t u = \alpha\, \partial_{xx} u$, this gives an explicit scheme:

$$u_i^{n+1} = u_i^{n} + \frac{\alpha \Delta t}{\Delta x^2}\left(u_{i+1}^n - 2u_i^n + u_{i-1}^n\right)$$

stable under the condition $\dfrac{\alpha \Delta t}{\Delta x^2} \le \dfrac{1}{2}$.

import numpy as np

def heat_step(u, alpha, dx, dt):
    lap = (np.roll(u, -1) - 2*u + np.roll(u, 1)) / dx**2
    return u + alpha * dt * lap

2. Finite elements

The finite element method (FEM) starts from the weak formulation. For the Poisson problem $-\Delta u = f$ on $\Omega$ with $u = 0$ on $\partial\Omega$, we look for $u \in H_0^1(\Omega)$ such that:

$$\int_\Omega \nabla u \cdot \nabla v \, \mathrm{d}\Omega = \int_\Omega f\, v \, \mathrm{d}\Omega \qquad \forall\, v \in H_0^1(\Omega)$$

Discretizing over a basis of functions $\{\varphi_j\}$ leads to the linear system $\mathbf{K}\mathbf{u} = \mathbf{f}$, where $\mathbf{K}$ is the stiffness matrix.

3. Comparing the methods

Method	Complex geometries	Conservation	Implementation
Finite differences	✗	✗	Simple
Finite elements	✓	~	Advanced
Finite volumes	✓	✓	Medium

Going further

Upcoming chapters will cover finite volumes, spectral methods and schemes for multiphase flows. (coming soon)