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

Splitting-methods based on Approximate Matrix Factorization and Radau-IIA formulas for the time integration of advection diffusion reaction PDEs

“…In this paper, we follow the techniques developed in [9,14] for an efficient implementation of the Radau IIA formula due to the large dimensions of the ODEs under consideration and to the presence of stiffness. Thus, as a first approach to solve for the stage values (Y n ), we apply an Inexact Newton Iteration as introduced in [14,Sect.…”

“…The resulting semi-explicit method was successfully tested on several interesting nonlinear 2D-3D advection diffusion reaction PDEs. In particular, the method was shown to be competitive [2,9] when compared to the well-known codes VODPK [3,4,6,15], RKC [17,19] and its implicitexplicit counterpart IRKC [16,18].…”

“…Here, we will focus in the stability aspects whereas in [8] the convergence details for PDEs were treated in length. The methods here considered allow to reach convergence order three just in two iterations (q = 2) whereas the method previously considered in [9] needs three iterations (q = 3) to get order of convergence three in ODE sense (and only reaches order 2.25 in PDE sense). The remainder of this paper is organized as follows.…”

“…We assume that some standard convergent method based on Finite Differences or Finite Volumes is used for the spatial discretization of the PDEs and the Boundary Conditions. A variable-stepsize integrator of order three (in ODE sense) with good stability properties based on performing three Newton-AMF iterations for the 2-stage Radau IIA method was developed in [9]. The resulting semi-explicit method was successfully tested on several interesting nonlinear 2D-3D advection diffusion reaction PDEs.…”

“…In [8] a family of methods, including the method proposed in [9] as a very particular case, was introduced. These methods make use of an implicit Runge-Kutta method as starting point, which must have good stability properties and a high convergence order (relative to the number of stages).…”