Avoiding order reduction when integrating linear initial boundary value problems with Lawson methods

Abstract

Exponential Lawson methods are well known to have a severe order reduction when integrating stiff problems. In a previous paper, the precise order observed with Lawson methods when integrating linear problems is justified in terms of different conditions of annihilation on the boundary. In fact, the analysis of convergence with all exponential methods when applied to parabolic problems has always been performed under assumptions of vanishing boundary conditions for the solution. In this paper, we offer a generalization of Lawson methods in order to approximate problems with nonvanishing and even time-dependent boundary values. This technique is cheap and allows to avoid completely order reduction independently of having vanishing or non-vanishing boundary conditions.

Avoiding order reduction when integrating linear initial boundary value problems with exponential splitting methods

“…We will give a technique to do it which requires a computational cost which is negligible compared with the total cost of the method since it just adds calculations with grid values on the boundaries, and not with the number of grid values on the total domain. In this sense, the technique is as cheap as that suggested in [4] for exponential Lawson methods and, among others, in [1,2] for other standard Runge-Kutta type methods. The idea, in a similar way as in [8,16,17], is to consider suitable intermediate boundary conditions for the split evolutionary problems.…”

A comparison of boundary correction methods for Strang splitting

“…We then introduce a correction q (which does not depend on u or t for timeinvariant Dirichlet boundary conditions) and instead of equations (2) and (3) we solve…”

“…In the absence of boundary conditions, it is second order accurate. Clearly, splitting is only viable if a procedure exists to efficiently solve the two partial flows (2) and (3). However, since the reaction is not stiff and good preconditioners are known for a large class of linear operators A, such an approach can be significantly more efficient than applying a monolithic implicit Runge-Kutta or multistep method (which requires both a nonlinear and a linear solver).…”

Avoiding order reduction when integrating nonlinear Schrödinger equation with Strang method

“…Moreover, they have been usually used when considering homogeneous boundary conditions and the analysis has been performed under that assumption. Just some recent research [5,6,13] has been done to include non-homogeneous boundary conditions. Moreover, the techniques which are suggested there manage to avoid order reduction for both homogeneous and nonhomogeneous boundary conditions.…”

“…Moreover, the techniques which are suggested there manage to avoid order reduction for both homogeneous and nonhomogeneous boundary conditions. ( [5] and [6] deal with Lawson and splitting methods when integrating linear problems and [13] with splitting methods for reaction-diffusion ones. )…”