A variable time-step-size code for advection–diffusion–reaction PDEs

The numerical integration of time-dependent PDEs of Advection–Diffusion–Reaction type, for two and three spatial variables (in short, 2D and 3D problems) in the MoL framework is considered. The spatial discretization is made by using Finite Differences and the time integration is carried out by means of the L-stable, third-order formula known as the two stage Radau IIA method. The main point for the solution of the large-dimensional ODEs is not to solve for the stage values of the Radau method until convergence (because the convergence is very slow on the stiff components), but only giving a very few iterations and take as advancing solution the latter stage value computed. The iterations are carried out by using the Approximate Matrix Factorization (AMF) coupled to a Newton-type iteration (SNI) as indicated in Perez-Rodriguez et al. (2009) [10], which turns out in an acceptably cheap iteration. Some stability results for the whole process (AMF)–(SNI) and a local error estimate for an adaptive time-integration are also given. Numerical results on four standard PDEs are presented and some conclusions about our method and other well-known solvers are drawn.