Efficient boundary corrected Strang splitting

Abstract

Strang splitting is a well established tool for the numerical integration of evolution equations. It allows the application of tailored integrators for different parts of the vector field. However, it is also prone to order reduction in the case of non-trivial boundary conditions. This order reduction can be remedied by correcting the boundary values of the intermediate splitting step. In this paper, three different approaches for constructing such a correction in the case of inhomogeneous Dirichlet, Neumann, and mixed boundary conditions are presented. Numerical examples that illustrate the effectivity and benefits of these corrections are included.

Comparison of efficiency among different techniques to avoid order reduction with Strang splitting

“…On the other hand, applying numerical differentiation to get local order 3 causes less problems with ACR than with EO. Moreover, ACR has the advantage that you do not have to worry about the calculation of q which cost, although small according to [14], has not even been included for the comparisons here.…”

“…In one dimension, this was achieved just by integrating twice the linear functionq(t) and that was done analytically for every value of t ∈ [t n , t n + 1 ). However, in two dimensions, that cannot be done any more and the elliptic problems (14) should be numerically solved, not only for every value t n , but even theoretically for every t ∈ [t n , t n + 1 ). In contrast, notice that numerical differentiation with ACR (12) just requires approximating Af (t n , u(t n )) at each step and no elliptic problem must be numerically solved.…”

“…In any case, we would like to remark here that the cost of calculating the function q with EO2 is not considered in this article and that ACR does never need to calculate such a function q. Besides, although some techniques have been proposed to calculate q efficiently [14], they have not either been tested in complicated domains.…”

“…Then, splitting the problem as described in the preliminaries of this article, the order reduction can be avoided, as it is justified in [16] with a similar analysis to that in [15]. The calculation of q can also be done analytically in one dimension and simple domains in two dimensions (as it is explained in Section 3 of this article); and even numerically in a cheap way in more complicated domains, according to [14, 16], although it is not in fact applied to such problems there. In this article, we will concentrate on the technique in [16] for 1‐dimensional and 2‐dimensional simple domains.…”

An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation

“…Then we just need to solve the sequence of a linear tri-diagonal system, and the whole process of programming is also simplified. SSM has been applied successfully in a diffusion-reaction problem [20]. To the best of our knowledge, no high-order CFDS combined with POD and SSM (E-CFDS6-SSM) aimed to solve MDPE efficiently has been developed so far.…”