5  1-D Diffusion Equation

The diffusion equation in 1D is:

\[\frac{\partial u}{\partial t} = \nu \frac{\partial^2 u}{\partial x^2}\]

The equation has second-order derivative, which we first learn how to implement in the code. \(\nu\) is the value of viscosity.

5.1 Discretizing \(\frac{\partial^2 u}{\partial x^2}\)

Descretizing the second-order derivative w/ the Central Difference Scheme: a combination of Forward Difference and Backward Difference of the first derivative.

\[u_{i+1} = u_i + \Delta x \frac{\partial u}{\partial x}\bigg|_i + \frac{\Delta x^2}{2} \frac{\partial ^2 u}{\partial x^2}\bigg|_i + \frac{\Delta x^3}{3!} \frac{\partial ^3 u}{\partial x^3}\bigg|_i + O(\Delta x^4)\]

\[u_{i-1} = u_i - \Delta x \frac{\partial u}{\partial x}\bigg|_i + \frac{\Delta x^2}{2} \frac{\partial ^2 u}{\partial x^2}\bigg|_i - \frac{\Delta x^3}{3!} \frac{\partial ^3 u}{\partial x^3}\bigg|_i + O(\Delta x^4)\]

Neglecting \(O(\Delta x^4)\) or higher(very small, so neglect-able..)

\[u_{i+1} + u_{i_1} = 2u_i + \Delta x^2 \frac{\partial^2 u}{\partial x^2}\bigg|_i\]

then put it together w/ the diffusion equation:

\[\frac{u_i^{n+1} - u_i^n}{\Delta t} = \nu\frac{u_{i+1}^n + u_{i-1}^n - 2u_i^n}{\Delta x^2}\]

then programmatic equation to solve \(u\) is:

\[u_i^{n+1} = \nu\frac{\Delta t}{\Delta x^2}(n_{i+1}^n + u_{i-1}^n - 2u_i^n) + u_i^n\]

(def init-params
  {:mode  :diffusion
   :nx    42
   :nt    20
   :nu    0.3
   :sigma 0.2})

source: notebooks/steps/step_03.clj