Accuracy of the Crank-Nicolson method depending on the parameter of the method r

Abstract

The method of Crank-Nicolson is powerful implicit method for numerical solving of parabolic partial differential equations. Because the method is implicit, it is unconditionally stable. We analyze the accuracy of the Crank-Nicolson method depending on the parameter r which is the ratio of the time step and the square of the spatial interval. To do this study, we consider metal rod initially heated with specific heat distribution. Temperature at the ends of the rod is prescribed at each time. For this example, close analytical solution given in Fourier series exists. Using Crank-Nicolson method, with Matlab we solve the problem for specific r. We compare this numerical solution with the exact analytical solution and obtain the error for the actual parameter r. Then we repeat the procedure for other values of the parameter r. On this way we are obtaining the dependence of the error of the method upon the parameter r.