AMF-Runge–Kutta formulas and error estimates for the time integration of advection diffusion reaction PDEs

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

“…The splitting for the function f can be advantageous over the Jacobian splitting in the sense that the order convergence order on non-linear problems is easier to analyze, but, in practice, it has the inconvenience of requiring the solution of nonlinear algebraic systems when the splitting terms f j are non-linear. The newly proposed methods [8], which are briefly described below in Section 2, do not present this drawback.…”

“…A remarkable difference of the splitting methods in [8] of most of splitting methods in the literature (see e.g. [12, Sect.…”

“…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).…”

“…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). To approximately solve for the internal Runge-Kutta stages, the methods in [8] are based on performing a small prefixed number of iterations with some kind of inexact Newton iteration including the Approximate Matrix Factorization (AMF) technique [11]. This latter approach is used to reduce the algebraic costs involved in the numerical solution of the multidimensional linear systems to the level of one-dimensional linear systems with small bandwidths.…”

“…[12, Sect. IV]) is that the splitting for current methods is usually carried out on the derivative function itself, f (t, y) = d j=1 f j (t, y), whereas in [8] the splitting is carried out in the Jacobian through the Approximate Matrix Factorization (AMF). A classical method in the literature is the Douglas method [7], which is widely used for parabolic problems and generalizes the ADI method proposed by Peaceman and Rachford [13].…”