Explicit Central Difference Method

The explicit central difference method is an explicit second order method for approximating the solution of the second order differential equation y''(x) = f(x, y) with initial conditions
y(x₀) = y₀, y'(x₀) = y'₀.

Note that the integrand f(x,y) does not depend upon y'.

The algorithm for the explicit central difference method is derived by using the central difference approximations for y'(x) and y''(x):

and

Let x_n = x₀ + nh, y_n be the approximation to y(x_n), and f_n = f(x_n, y_n), the procedure proceeds recursively via the explicit equation for y_{n + 1} as follows:

In order to begin the recursion, two successive starting values of y are required, one of which is y₀ and the other starting value y₁ is approximated by

Particular classes of problems may have a more accurate estimate for y₁.

Richardson extrapolation may be used to increase to increase both the order and accuracy.

Function List

• void Explicit_Central_Difference_Method( double (*f)(double, double), double y[], double x0, double c, double h, int max_columns, int number_of_steps )
  
  This function uses the explicit central difference method to estimate the solution of the initial value problem, y'' = f(x,y); y(x0) = y[0] and y'(x0) = c, at x0 + nh where for
  n =1, …, number_of_steps and h is the step size. The argument max_columns is the number of step size halving + 1 used in Richardson extrapolation so that if richardson_columns = 1 then no extrapolation to the limit is performed. On input, y[0] is the value of y(x) at x = x0. On output y[n] is the value of y(x) at x = x0 + n h for
  n = 0, …, number_of_steps.
  
  

C Source Code

• The file, explicit_central_difference.c, contains a version of Explicit_Central_Difference_Method( ) written in C.