Coefficients of 2nd- to 10th-order central differencing for the second derivative

In solving the wave equation by finite differencing, we should use central finite-difference schemes because the wave equation and physical wave phenomena are symmetric. And near a boundary with boundary conditions, we should gradually drop the order of finite differencing for the derivative perpendicular to the boundary.

The nth order central finite differencing for the second derivative is

$${{\partial}^2 {\phi}_{i} \over \partial x^2} = {1\over \Delt... ... \sum_{k=1}^{n} w_{k} ( {\phi}_{i-k} + {\phi}_{i+k} ) \right]}$$ (1)

For the 2nd order: w0=-2.; w1=1.

For the 4th order: w0=-5./2.; w1=4./3.; w2=-1./12.

For the 6th order: w0=-2.72222; w1=1.50000; w2=-0.15000; w3=1.11111E-02.

For the 8th order: w0=-2.84722; w1=1.60000; w2=-0.20000; w3=2.53968E-02; w4=-1.78571E-03.

For the 10th order: w0=-2.92722; w1=1.66667; w2=-0.238095; w3=3.96825E-02; w4=-4.96031E-03; w5=3.17460E-04.