4. Finite Difference Formulas in 2D¶
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 \(nj-1\):
\[y_j = y_0 + j \Delta y\]
- Define \(u_{i,j} = u(x_i,y_j)\)
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
\[\left . {{\partial u} \over {\partial x}} \right \vert_{i,j} = {{u_{i+1,j}-u_{i,j}} \over {\Delta x}} + O(\Delta x)\]
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
\[\left . {{\partial u} \over {\partial x}} \right \vert_{i,j} = {{u_{i+1,j}-u_{i-1,j}} \over {2 \Delta x}} + O(\Delta x^2)\]
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
\[\left . {{\partial^2 u} \over {\partial x^2}} \right \vert_{i,j} = {{u_{i+1,j}-2u_{i,j}+u_{i-1,j}} \over {\Delta x^2}} + O(\Delta x^2)\]