1.1.2.4. Finite Difference Formulas in 2D

1.1.2.4.1. Extend 1D formulas to 2D

  • Just apply the definition of a partial derivative w.r.t. \(x\) is the variation in \(x\) holding \(y\) constant

  • Build 2D grid defined by the following:

  • For i between \(0\) and \(nx-1\):

\[x_i = x_0 + i \Delta x\]

  • For j between \(0\) and \(nj-1\):

\[y_j = y_0 + j \Delta y\]
../_images/2D.png

  • Define \(u_{i,j} = u(x_i,y_j)\)

1.1.2.4.2. Forward Differencing in 2D for 1st derivative

  • For the point \((i+1,j+1)\), Taylor series in 2D:

\[u_{i+1,j+1} = u_{i,j} + \left . \left ( \Delta x {\partial \over \partial x} + \Delta y {\partial \over \partial y} \right ) u \right \vert_{i,j} + \left . {1 \over 2} \left ( \Delta x {\partial \over \partial x} + \Delta y {\partial \over \partial y} \right )^2 u \right \vert_{i,j} + \left . {1 \over 6} \left ( \Delta x {\partial \over \partial x} + \Delta y {\partial \over \partial y} \right )^3 u \right \vert_{i,j} + \cdots + \text{h.o.t}\]

  • 1st order FD in x-direction, i.e.

    • \(\Delta y = 0\)

    • \(j+1=j\)

    • Terms of higher order than 1 are zero

\[\left . {{\partial u} \over {\partial x}} \right \vert_{i,j} = {{u_{i+1,j}-u_{i,j}} \over {\Delta x}} + O(\Delta x)\]

1.1.2.4.3. Central Differencing in 2D for 1st derivative

  • Take one foward Taylor step, one backward Taylor step

  • Subtract the forward and the backward steps

  • Re-arrange for the derivative

\[\left . {{\partial u} \over {\partial x}} \right \vert_{i,j} = {{u_{i+1,j}-u_{i-1,j}} \over {2 \Delta x}} + O(\Delta x^2)\]

1.1.2.4.4. Central Differencing in 2D for 2nd derivative

  • Take one foward Taylor step, one backward Taylor step

  • Add the forward and the backward steps

  • Re-arrange for the derivative

\[\left . {{\partial^2 u} \over {\partial x^2}} \right \vert_{i,j} = {{u_{i+1,j}-2u_{i,j}+u_{i-1,j}} \over {\Delta x^2}} + O(\Delta x^2)\]