A Second‐Order Diagonally Implicit Runge‐Kutta Time‐Stepping Method

Abstract This paper presents a subset of the family of diagonally implicit Runge‐Kutta (DIRK) time‐stepping methods for finite‐difference models of parabolic (diffusion‐like) equations. It includes the first‐order‐accurate Euler implicit (backward in time, DIRK1) and the second‐order Crank‐Nicolson (DIRK2) methods as special cases. It combines a series of DIRK1 partial time steps so as to eliminate additional power terms of the time step δt, in the local error, by the number of partial steps used. This offers a large increase in computational efficiency, going from a DIRK1 to a DIRK2 method, and improves on the Crank‐Nicolson method with a better choice of Runge‐Kutta parameters. For a linear diffusion example, an optimal‐parameter DIRK2 method offers the same accuracy as the Euler implicit method at two orders of magnitude larger time step, with an order of magnitude better accuracy than the Crank‐Nicholson method. In a highly nonlinear horizontal unsaturated water flow example, using eight simulated medium to coarse soils, a DIRK2 method produces either an average maximum accuracy improvement of 4.7 times over the Euler implicit method, without Newton or Picard iteration, or from 3.8 to 48 times faster computer run times for the same accuracy.