Comparison Of Runge–Kutta, Multistep, And Crank Nicolson Schemes For Time Integration In The Numerical Solution Of The Heat Equation

Abstract

In this paper, three time-integration approaches for the numerical solution of the one dimensional heat conduction equation are compared, namely one-step Runge Kutta methods, Adams-type multistep methods, and the Crank–Nicolson scheme. As a model problem we consider the classical heat equation with Dirichlet boundary conditions, discretized in space by second-order central finite differences. The resulting system of linear ODEs is integrated in time using the fourth-order Runge Kutta method, the three step Adams Bashforth Moulton predictor–corrector pair, and the Crank–Nicolson scheme, which is equivalent to the trapezoidal rule. We analyze the local and global orders of accuracy, stability properties, and computational efficiency of these methods.