Application of Runge–Kutta Methods to the Solution of the Heat Equation

This article studies the numerical solution of the one-dimensional heat equation by Runge–Kutta time discretization combined with a standard finite-difference discretization in space. Explicit and implicit Runge–Kutta methods are considered within the method-of-lines framework. For explicit methods, formulas for the stage vectors and the resulting stability restrictions are derived, showing how the stability constraint limits the benefit of higher temporal order. For implicit methods, the Runge–Kutta stages lead to a sparse block system whose blocks are tridiagonal matrices. A block Gauss–Seidel iteration is proposed for the efficient solution of this system, and sufficient convergence conditions are discussed. The convergence analysis is developed in more detail for two-stage and three-stage methods. Finally, a numerical example based on the fourth-order two-stage Gauss–Legendre method is compared with the Crank–Nicolson scheme and illustrates the advantage of higher-order implicit Runge–Kutta discretization when the temporal error is dominant.