Spectral-fractional step Runge–Kutta discretizations for initial boundary value problems with time dependent boundary conditions

Abstract

Abstract. In this paper we develop a technique for avoiding the order reduction caused by nonconstant boundary conditions in the methods called splitting, alternating direction or, more generally, fractional step methods. Such methods can be viewed as the combination of a semidiscrete in time procedure with a special type of additive Runge-Kutta method, which is called the fractional step Runge-Kutta method, and a standard space discretization which can be of type finite differences, finite elements or spectral methods among others. Spectral methods have been chosen here to complete the analysis of convergence of a totally discrete scheme of this type of improved fractionary steps. The numerical experiences performed also show the increase of accuracy that this technique provides.

A domain decomposition method based on the iterative operator splitting method

“…The operators A and B are spatially discretized operators and given as in our model equation (1) in Section 2, i.e. they correspond to the discretized in space convection and diffusion operators (matrices).…”

“…where A denotes, for example, a second-order partial differential operator Au = −∇D∇u + v∇u + cu for the given coefficients D ∈ R + , v ∈ R n , c ∈ R + , and n is the dimension of the space or a first-order partial differential equation as given in our model equation (1) in Section 2. The underlying domains Ω 1 and Ω 2 are convex and Lipschitzian and do not influence the following analysis.…”

Consistency of iterative operator‐splitting methods: Theory and applications

“…In the first example we deal with a partial differential equation that is time-dependent (see [23]). …”

A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems

“…Fortunately, this order reduction can be avoided if we modify the boundary conditions for u n+1/2 and u n+1 using the technique proposed in [1] for general fractional step methods. Such technique ensures that if we choose the following boundary conditions:…”