I have read book Numerical Solutions of PDE by Morton and Mayers.
The Crank Nicolson method have the numerical scheme like this.
Now I read next page, there is a new method, namely Peaceman and Rachford method. In this book said that PR method is modification of Crank Nicolson Method, like this
What the modification of (3.11) to (3.13)? How to obtain (3.11) to (3.13)?
Best Answer
The coefficients of $U^{n+1}$ and $U^n$ both contain the term $\mu_x\mu_y \delta_x^2 \delta_y^2 / 4$ in the expression (3.13), which was absent in (3.11). It's therefore not possible to obtain (3.13) from (3.11). Nonetheless, without any additional information, I guess that this addition being of fourth order ($\delta_x^2 \delta_y^2$) with respect to discretization, it doesn't affect the convergence properties of the numerical solution.