Artificial boundary conditions for the linearized Benjamin–Bona–Mahony equation

Abstract

We consider various approximations of artificial boundary conditions for linearized Benjamin–Bona–Mahony BBM equation. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the $\mathcal {Z}$-transform of an holomorphic function). In this paper, we derive explicit transparent boundary conditions both continuous and discrete for the linearized BBM equation. The equation is discretized with the Crank Nicolson time discretization scheme and we focus on the difference between the upwind and the centered discretization of the convection term. We use these boundary conditions to compute solutions with compact support in the computational domain and also in the case of an incoming plane wave which is an exact solution of the linearized BBM equation. We focus on and prove consistency, stability and convergence of the numerical scheme and provide many numerical experiments to show the efficiency of our transparent boundary conditions.

A The \(\mathcal {Z}\)-transform

The discrete analogue of the Laplace transform is the so called $\mathcal {Z}$-transform. In this section, we list some results used in this paper concerning this transformation. More precise details on the $\mathcal {Z}$-transform and its properties are given in [17].

Definition 2

Let $(u_n)_{n\in \mathbb {N}}\in \mathbb {C}^\mathbb {N}$. We define the $\mathcal {Z}$-transform of the sequence $(u_n)$, denoted $\widehat{u}$ by:

$$\begin{aligned} \forall z\in \mathbb {C}, \vert z\vert > R,\qquad \mathcal {Z}(u_n)=\widehat{u}(z) =\sum _{n=0}^\infty \frac{u_n}{z^n}, \end{aligned}$$

where 1 / R is the convergence radius of the series $\sum _{n=0}^\infty u_n z^n$.

Note that in the previous definition, the radius R depends a priori on the sequence $(u_n)$. We now state a couple of analogue results of the Laplace transform properties in the context of the $\mathcal {Z}$-transform. One has the following discrete differentiation Theorem:

Theorem 4

When these quantities are well defined, one has:

$$\begin{aligned} \mathcal {Z}(u_{n+1}) = z\widehat{u}(z)-zu_0. \end{aligned}$$

We also have the following behavior of the $\mathcal {Z}$-transform regarding the convolution:

Theorem 5

Let $(u_n),(v_n)\in \mathbb {C}^\mathbb {N}$. If $\widehat{u}(z)$ exists for $\vert z\vert >R_u$ and $\widehat{v}(z)$ exists for $\vert z\vert > R_v$, then $\mathcal {Z}(u_n\star v_n)$ exists for $\vert z\vert > \max (R_u,R_v)$ and one has:

$$\begin{aligned} \forall \vert z\vert >\max (R_u,R_v),\qquad \mathcal {Z}(u_n\star v_n) = \widehat{u}(z)\widehat{v}(z). \end{aligned}$$

By a classical series-integral commutation, one gets the following Plancherel Theorem:

Theorem 6

Let $(u_n),(v_n)\in \mathbb {C}^\mathbb {N}$. If $\widehat{u}(z)$ exists for $\vert z\vert >R_u$ and $\widehat{v}(z)$ exists for $\vert z\vert > R_uR_v$ with $R_u<r< 1/R_v$, then, one has, for all $r\in [R_u;R_v]$:

$$\begin{aligned} \sum _{n=0}^\infty u_n\overline{v_n} = \frac{1}{2\pi }\int _0^{2\pi } \widehat{{u}}(re^{i\varphi }) \overline{\widehat{{v}}(e^{i\varphi }/r)}d\varphi . \end{aligned}$$

If $R_u<1$ and $R_v<1$, then $r=1$ can be chosen in the above equality.

We now state the following inversion result (which is a consequence of Cauchy Integral Theorem):

Theorem 7

Let $(u_n)$ be a complex sequence and $\widehat{u}$ its $\mathcal {Z}$-transform with radius of convergence $R_u$. Then, one gets the following relation:

$$\begin{aligned} \forall n\in \mathbb {N},\qquad u_n = \frac{1}{2i\pi }\int _{C(0,R)} \widehat{u}(z)z^{n-1}dz \end{aligned}$$

where $R > R_u$ and C(0, R) denotes the circle of center 0 and radius R.