5 Finite difference method -1 The Taylor series(1)(2)First-oder differenceForward differenceii+1i-1Backward differencei+1ii-1CentralDifferenceturunkan juga!ii+1i-1
6 Finite difference method -3 High-oder finite differenceTo reduce the truncation error of finite difference method, high-order finite difference method is frequently employedh(1)ii+1i-2i-1i+2Taylor series of five points are,(2)(3)(4)(5)(5)By substituting (2)～(5) into Eq.(1), we have,
7 Finite difference method -4 (7)The coefficient for the first derivative must be 1 and the others must be zero . By solving 5 linear equations, we get,(8)The order of precision is 4th.
8 Finite difference method -5 Comparison of finite difference methodf(x)=x3, Δx=1,x=2 , analytical solution f’(2)=12x f(x)182764Forward finite differenceBackward finite differenceCentral difference4th order finite Central differenceExercise: Calculate above four types of finite differences under condition of f(x)=x, Δx=1, x=2