Numerical solution to a linearized KdV equation on unbounded domain

Abstract

Exact absorbing boundary conditions for a linearized KdV equation are derived in this paper. Applying these boundary conditions at artificial boundary points yields an initial-boundary value problem defined only on a finite interval. A dual-Petrov-Galerkin scheme is proposed for numerical approximation. Fast evaluation method is developed to deal with convolutions involved in the exact absorbing boundary conditions. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method.

Discrete artificial boundary conditions for the linearized Korteweg-de Vries equation

“…Recall that for U 1 = 0 and U 2 = 1 we recover the case considered by Zheng et al [13]. Although the PDE (1.5) looks very simple, it has a lot of applications, for example, Whitham [14] used it for the modeling of the propagation of long waves in the shallow water equations, see also [15].…”

“…Now using the decay condition (2.8), the general solution (2.9), the separation property (2.12) and as solutions of (2.7) have to belong to L 2 (R), we obtain 13) which yields the following TBCs in the Laplace-transformed spacê…”

Flutter in a quasi-one-dimensional model of a collapsible channel

“…Following Zheng et al [30], we perform the Laplace transformation on (A 1) where s ∈ C with (s) > 0 is the argument in the Laplace-transformed space andĀ is the transform of A. The general solution to (A 2) looks likē…”

Artificial boundary conditions for certain evolution PDEs with cubic nonlinearity for non-compactly supported initial data

“…(3). The construction of TBCs for the linearized case was considered by Zheng et al [2] for the case of compactly supported initial data. Using the substitution wðx; tÞ ¼ uðx; tÞ À A ð0Þ l;r e iðj l;r xþj 3 l;r tÞ ; x 2 X l;r respectively ð4Þ and burrowing the results from [2], it easily follows that an equivalent formulation of the initial value problem for linearized form of (3) on the bounded domain X i is given by @ t u þ @ 3…”

“…If the evolution is in time, such conditions usually appear as a Dirichlet to Neumann map nonlocal in time. While the construction of such exact conditions for a class of initial conditions which have a compact support embedded in the computational domain is possible for linear evolution equations [1,2] and integrable nonlinear equations [3,4], the general problem can only be treated by introducing certain approximate boundary conditions dubbed as artificial boundary conditions (ABCs) [5][6][7][8][9]. To have an overview of the state-of-the-art of the subject we refer to the review article [11].…”