9  Step 7: 2-D Diffusion

And here is the 2D-diffusion equation:

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

You will recall that we came up with a method for discretized second order derivatives in Step 3, when investigating 1-D diffusion. We are going to use the same scheme here, with our forward difference in time and two second-order derivatives.

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

Once again, we reorganize the discretized equation and solve for \(u_{ij}^{n+1}\)

\[\begin{split} u_{i,j}^{n+1} = u_{i,j}^n &+ \frac{\nu \Delta t}{\Delta x^2}(u_{i+1,j}^n - 2 u_{i,j}^n + u_{i-1,j}^n) \\ &+ \frac{\nu \Delta t}{\Delta y^2}(u_{i,j+1}^n-2 u_{i,j}^n + u_{i,j-1}^n) \end{split}\]

9.0.1 Initial Conditions

(def nx 31)

(def ny 31)

(def nt 10)

viscosity

(def nu 0.05)

CFL number

(def sigma 0.2)

(def dx (/ 2 (- nx 1)))

(def dy (/ 2 (- ny 1)))

(def init-params
  {:nx    nx
   :ny    ny
   :nu    nu
   :nt    nt
   :sigma sigma
   :dx    dx
   :dy    dy
   :dt    (* sigma dx dy (/ 1 nu))})

Create spatial grid

(def grid-start 0)

(def grid-end 2)

(def spatial-arr (two-d/create-array-2d
                   (assoc init-params
                     :x-start grid-start :x-end grid-end
                     :y-start grid-start :y-end grid-end)))

Create the initial u with the initial condition:

\[u(x,y) = \begin{cases} \begin{matrix} 2\ \text{for} & 0.5 \leq x, y \leq 1 \cr 1\ \text{for} & \text{everywhere else}\end{matrix}\end{cases}\]

(def array-u (two-d/create-init-u init-params spatial-arr))

initial plotting goes:

9.0.2 Iterating in 2-D w/ diffusion equation

at \(nt=10\)

at \(nt=14\)

at \(nt=50\)

source: notebooks/steps/step_07.clj