10  Step 8: 2-D Burgers’ Equation

Remember, Burgers’ equations can generate discontinuous solutions from an initial condition that is smooth, i.e., can develop “shocks”. We want to see this in two dimensions now!

Here is our coupled set of PDEs:

\[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \; \left(\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2}\right)\]

\[\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = \nu \; \left(\frac{\partial ^2 v}{\partial x^2} + \frac{\partial ^2 v}{\partial y^2}\right)\]

We know how to discretize each term: we’ve already done it before!

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

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

And now, we will rearrange each of these equations for the only unknown: the two components \(u\), \(v\) of the solution at the next time step:

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

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

10.0.1 Initial Conditions

(def nx 41)

(def ny 41)

(def nt 120)

(def c 1)

(def nu 0.01)

(def sigma 0.0009)

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

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

(def init-params
  {:nx    nx
   :ny    ny
   :nt    nt
   :c     c
   :nu    nu
   :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 and v arrays w/ initial conditions: u(.5<=x<=1 && .5<=y<=1) is 2, else 1 v(.5<=x<=1 && .5<=y<=1) is 2, else 1

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

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

initial plotting w/ \(x\), \(y\), \(u\) goes:

(two-d/sim->plotly-plot-it! spatial-arr array-u)

(def sim-result (two-d/simulate {:array-u array-u
                                 :array-v array-v} (assoc init-params :mode :burgers)))

source: notebooks/steps/step_08.clj