Truncation error and stability analysis of iterative and non-iterative Thomas-Gladwell methods for first-order non-linear differential equations

The consistency and stability of a Thomas-Gladwell family of multistage time-stepping schemes for the solution of first-order non-linear differential equations are examined. It is shown that the consistency and stability conditions are less stringent than those derived for second-order governing equations. Second-order accuracy is achieved by approximating the solution and its derivative at the same location within the time step. Useful flexibility is available in the evaluation of the non-linear coefficients and is exploited to develop a new non-iterative modification of the Thomas-Gladwell method that is second-order accurate and unconditionally stable. A case study from applied hydrogeology using the non-linear Richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non-iterative formulation.