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Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model

  • Ying He
  • Jie ShenEmail author
Original Paper
  • 39 Downloads

Abstract

We consider in this paper numerical approximation of a nonlinear fluid-structure interaction (FSI) model with a fixed interface. We construct a new class of pressure-correction schemes for the FSI problem, and prove rigorously that they are unconditionally stable. These schemes are computationally very efficient, as they lead to, at each time step, a coupled linear elliptic system for the velocity and displacement in the whole region and a discrete Poisson equation in the fluid region.

Keywords

Fluid-structure interaction Pressure correction Stability analysis 

Mathematics subject classification:

74F10 76D05 65M12 35Q30 

1 Introduction

Fluid-structure interaction (FSI) plays an important role in many scientific/engineering applications, e.g., design of engineering systems, blood flow in human arteries, etc. It has been extensively studied in recent years both analytically and computationally (cf. [6, 9, 11, 18] and the references therein).

There are mainly three approaches, monolithic, partitioned and semi-implicit projection, for solving FSI problems numerically. The partitioned approach (cf., for instance, [2, 4, 10, 24]) solves the fluid and structure dynamics separately with explicit interface conditions. While each subproblem can be solved efficiently by existing algorithms, the explicit treatment of the interface condition may lead to instability in the presence of strong added-mass effect [5] and requires very restrictive time step constraint. In contrast, the monolithic approach (cf., for instance, [19, 20, 23]) simultaneously solves the fluid and structure dynamics coupled by the implicit interface conditions. This type of schemes usually have good stability properties, but at each time step, a nonlinear coupled system has to be solved and, due to the presence of the pressure in the coupled system, it is usually difficult to design an effective iterative scheme to solve the nonlinear coupled system. On the other hand, the semi-implicit projection approach was first proposed in [12]. It decouples the computation of fluid velocity from that of the pressure and structure displacement by using a projection method. This method appears to have some computational advantage over the partitioned or monolithic approaches (cf., for instance, [1, 3, 12]).

In this paper, we shall construct a different class of semi-implicit projection schemes which decouple the computation of pressure from that of the velocity and structure displacement. Our schemes will be computationally very efficient. More precisely, in the first step of our schemes, we solve a coupled, but elliptic, system for an intermediate fluid velocity and the structure displacement; then in the second step, we solve a Poisson equation for the fluid pressure and obtain the fluid velocity with a simple correction. Furthermore, we shall also prove rigorously that these schemes are unconditionally stable.

For fluid problems, an effective approach to decouple the computation of the pressure from that of the velocity is to use a projection-type method, originally proposed by Chorin and Temam in the late 1960s [7, 28]. A comprehensive review on various projection-type methods can be found in [14]. However, a main difficulty in the design of a projection method for the FSI problem is to assign a boundary condition for the pressure at the interface. It is well known that a proper boundary condition for the pressure Poisson equation in a projection-type method, at the Dirichlet part of the boundary, is the homogeneous Neumann boundary condition. Indeed, most existing projection-type schemes for the FSI problem also use, explicitly or implicitly, the Neumann-type boundary condition for the pressure Poisson equation at the Dirichlet part of the boundary as well as at the interface. However, imposing a Neumann-type boundary condition for the pressure at the interface appears to affect, to a certain degree, the stability of the scheme, and we are not aware of any proof of unconditional stability for this type of projection schemes, only a conditional stability has been proved in [12] for a linear FSI problem. In a previous paper [17], the authors constructed an unconditionally stable scheme for a linear FSI problem. The aim of this paper is to extend it to a nonlinear FSI problem.

In [13], the authors proposed and analyzed pressure-correction schemes for Navier–Stokes equations with open boundary where the usual stress-free boundary condition is applied. It is shown that the proper boundary condition at the open boundary is of Dirichlet type instead of Neumann type. Two schemes are constructed in [13], one is based on the standard pressure correction which leads to poor accuracy at the open boundary, while the other is based on the rotational pressure correction and with a proper Dirichlet boundary condition at the open boundary. It is shown in [13] that both the standard and rotational pressure-correction projection schemes, when applied to the time-dependent Stokes problem, are unconditionally stable, but the rotational version leads to much better accuracy. Since one of the matching interface conditions for the FSI problem is related to the stress, it makes sense to extend the approach in [13] for problems with open boundary to the FSI problem.

Besides the difficulty associated with the pressure boundary condition on the interface, another major difficulty is to prove the unconditional stability of the rotational pressure-correction scheme for the nonlinear FSI problem. The original stability proof of the rotational pressure-correction scheme in [15] was only valid for Stokes problems. An essential step of the proof was to take the “discrete time derivative” of the scheme. Unfortunately, this proof cannot be extended to the nonlinear case. In [8], the authors constructed an unconditionally stable rotational velocity-correction scheme for the Navier–Stokes equations. However, they only provided a stability proof for the linear Stokes equations, while showing numerically that the scheme was unconditionally stable. In [25], the author proposed a Gauge–Uzawa approach for the rotational pressure-correction scheme of the Navier–Stokes equations, and proved that the scheme was unconditionally stable. We shall extend the approach in [25] for the Gauge–Uzawa scheme of the Navier-Stokes equations to the rotational pressure-correction schemes for the FSI problem.

To fix the idea, we consider in this paper a simple model of the FSI problem where the movement of the interface is assumed infinitesimal so the interface is treated as fixed. This nonlinear FSI problem captures many of the essential difficulties of the more general FSI problems with moving interface, and its well-posedness has been studied in [22].

The rest of the paper is organized as follows. In the next section, we describe the governing equations for our FSI model, formulate its weak form and the energy dissipation law. In Sect. 3, we construct a standard and rotational pressure-correction scheme for the FSI problem and prove their unconditional stability. Then, in Sect. 4, we describe a generic approach for spatial discretization as well as a Fourier–Legendre method for a special case of a periodic channel. We present some numerical results in Sect. 5 to validate our numerical schemes and to demonstrate their temporal accuracy. Some concluding remarks are given in Sect. 6.

2 Governing Equations

We consider the following model for interaction of a viscous fluid with an elastic body in a two- or three-dimensional bounded domain \(\Omega\), with the fluid region \(\Omega _f\), the solid region \(\Omega _s\) and the interface \(\Gamma _c,\) so we have \(\Omega =\Omega _f\cup \Omega _s\cup \Gamma _c\). We also denote \(\Gamma _f=\partial \Omega _f\backslash \Gamma _c\) and \(\Gamma _s=\partial \Omega _s\backslash \Gamma _c\) (cf. Fig.  1).
Fig. 1

Geometry discription for fluid-structure problem

We assume that the interface undergoes infinitesimal displacements, i.e., \(\Gamma _c\) is fixed. The more complicated situation with moving interface will be considered in a forthcoming paper.

In the fluid region \(\Omega _f\), we have the Navier–Stokes equations:
$$\begin{aligned}&\rho _fu_{t}-\text { div } \epsilon (u)+(u \cdot \nabla )u+\nabla p =\rho _ff_1&\mathrm{{in}} \quad \Omega _{f}\times (0,T), \end{aligned}$$
(2.1a)
$$\begin{aligned}&\text { div } u=0&\mathrm{{in}}\quad \Omega _{f}\times (0,T), \end{aligned}$$
(2.1b)
$$\begin{aligned}&u=0&\mathrm{{on}} \quad \Gamma _{f}\times (0,T), \end{aligned}$$
(2.1c)
$$\begin{aligned}&u|_{t=0}=u_0&\mathrm{{in}} \quad \Omega _{f}, \end{aligned}$$
(2.1d)
where u denotes the fluid velocity, p the fluid pressure, \(u_0\) is the given initial velocity, \(f_1\) is the given body force per unit mass, \(\epsilon (u)=\frac{\mu }{2}(\nabla u+\nabla u ^{\rm T})\) is the strain tensor, and \(\rho _f\) and \(\mu\) are the constant fluid density and viscosity.
In the solid region \(\Omega _s\), we have the wave equation for linear elasticity:
$$\begin{aligned}&\rho _sw_{tt}-\text { div } \sigma (w) =\rho _sf_2&\mathrm{{in}} \quad \Omega _{s}\times (0,T), \end{aligned}$$
(2.2a)
$$\begin{aligned}&w=0&\mathrm{{on}} \quad \Gamma _{s}\times (0,T), \end{aligned}$$
(2.2b)
$$\begin{aligned}&w(\cdot ,0)=w_0\quad&\mathrm{{in}} \quad \Omega _{s}, \end{aligned}$$
(2.2c)
$$\begin{aligned}&w_{t}(\cdot ,0)=w_1\quad&\mathrm{{in}} \quad \Omega _{s}, \end{aligned}$$
(2.2d)
where w denotes the displacement of the solid, \(w_0\) and \(w_1\) are the given initial data, and \(\sigma (w)\) is the elastic stress tensor, given by
$$\begin{aligned} \sigma _{ij}(w)=\lambda \sum _{k=1}^{3}\epsilon _{kk}(w)+2\mu _2\epsilon _{ij}(w),\,\lambda ,\,\mu _2 \le 0,\;\text {with }\; \epsilon _{kj}(w)=\frac{1}{2} (\partial _k w_j+\partial _j w_k), \end{aligned}$$
\(f_2\) is the given loading force per unit mass, \(\lambda\) and \(\mu _2\) are the Lamé constants, and \(\rho _s\) is the constant solid density.
Across the fixed interface \(\Gamma _c\) between the fluid and solid, the velocity and the stress vector are required to be continuous, i.e.,
$$\begin{aligned} w_t=u \quad {\text{on}}\;\Gamma _c\times (0,T) \end{aligned}$$
(2.3)
and
$$\begin{aligned} \sigma (w)\cdot \mathbf{{ n}}=\epsilon (u)\cdot \mathbf{{n}}-p\mathbf{{n}}-\frac{1}{2}(u\cdot \mathbf{{n}})u \quad {\text{on}}\; \Gamma _c\times (0,T), \end{aligned}$$
(2.4)
where \(\mathbf {n}\) denotes the outward normal vector along \(\Gamma _c\) w.r.t. \(\Omega _s\). For instance, if \(\Gamma _c=\{(x,y)|y=0\}\), then \(\mathbf{{n}}=(0,1)\).
For simplicity, we take in this paper \(\rho _f=\rho _s=1\), \(f_1=f_2=0\). We further take \(\lambda =1\) and \(\mu _2=0\) which implies \(\text { div } \sigma (w)=\Delta w\), and the interface condition (2.4) reduces to
$$\begin{aligned} \frac{\partial w}{\partial \mathbf { n}}=\mu \frac{\partial u}{\partial \mathbf {n}}-p\mathbf{{n}}-\frac{1}{2}(u\cdot \mathbf{{n}})u \quad {\text{on}}\; \Gamma _c\times (0,T). \end{aligned}$$
(2.5)
To derive a weak formulation for (2.1)–(2.2), we need to introduce some notations. Let us denote by \(H^k(\Omega )\) and \(H^k_0(\Omega )\) (for \(k\ge 0\)) the standard Sobolev spaces, equipped with the standard norm \(\Vert \cdot \Vert _{k,\Omega }\). In particular, we denote \(L^2(\Omega )=H^0(\Omega )\) with the associated norm \(\Vert \cdot \Vert\). We will use \(\mathbf{{H}^k}(\Omega _f)\) to denote the vector-valued Sobolev spaces. We also denote
$$\begin{aligned} H^1_{0,\Gamma _f}(\Omega _f)=\{v\in H^1(\Omega _f): v|_{\Gamma _f}=0\},\quad H^1_{0,\Gamma _s}(\Omega _s)=\{v\in H^1(\Omega _s): v|_{\Gamma _s}=0\}. \end{aligned}$$
Then, a weak solution (upw) for (2.1)–(2.2) will satisfy
$$\begin{aligned}&(u_{t}+(u \cdot \nabla )u,\varphi )_{\Omega _f}+(\mu \nabla u,\nabla \varphi )_{\Omega _f}-(p,\text {div}\varphi )_{\Omega _f}\nonumber \\&+\left(\mu \frac{\partial u}{\partial \mathbf{{n}}}-p\cdot \mathbf{{n}} ,\varphi \right)_{\Gamma _{c}}=0,\quad \forall \varphi \in \mathbf{H}^1_{0,\Gamma _f}(\Omega _f), \end{aligned}$$
(2.6a)
$$(\text {div} \, u,q)_{\Omega _f}=0, \quad \forall q\in L^2(\Omega _f), $$
(2.6b)
$$\begin{aligned}&(w_{tt},\psi )_{\Omega _s}+(\nabla w,\nabla \psi )_{\Omega _s}-\left( \frac{\partial w}{\partial \mathbf{n}},\psi \right) _{\Gamma _{c}}=0, \quad \forall \psi \in \mathbf{H}^1_{0,\Gamma _s}(\Omega _s), \end{aligned}$$
(2.6c)
with the interface conditions (2.3) and (2.5) on \(\Gamma _c\) .
Using (2.5), we can reformulate the above as
$$\begin{aligned}&(u_{t}+(u\cdot \nabla ) u,\varphi )_{\Omega _f}+(\mu \nabla u,\nabla \varphi )_{\Omega _f}-(p,\text {div}\varphi )_{\Omega _f} +\frac{1}{2}((u\cdot \mathbf{{n}})u,\varphi )_{\Gamma _{c}}\nonumber \\& +\left( \frac{\partial w}{\partial \mathbf {n}},\varphi \right) _ {\Gamma _{c}}=0,\quad \forall \varphi \in \mathbf{H}^1_{0,\Gamma _f}(\Omega _f), \end{aligned}$$
(2.7a)
$$\begin{aligned}&(\text {div} \; u,q)_{\Omega _f}=0, \quad \forall q\in L^2(\Omega _f), \end{aligned}$$
(2.7b)
$$\begin{aligned}&(w_{tt},\psi )_{\Omega _s}+(\nabla w,\nabla \psi )_{\Omega _s}-\left( \frac{\partial w}{\partial n},\psi \right) _{\Gamma _{c}}=0, \quad \forall \psi \in \mathbf{H}^1_{0,\Gamma _s}(\Omega _s), \end{aligned}$$
(2.7c)
with \(u=w_t\) on the interface \(\Gamma _c\).
Setting \(\varphi =u, \psi =w_{t}\) in  (2.7a) and (2.7c), using the identity (note that \(\mathbf {n}\) is the inward normal along \(\Gamma _c\) w.r.t. \(\Omega _f\))
$$\begin{aligned} ((u \cdot \nabla )v,v)_{\Omega _f}=-\frac{1}{2}((u\cdot \mathbf{{n}})v,v)_{\partial \Omega _f} \text { if } \text {div}\, u=0, \end{aligned}$$
(2.8)
and summing up the two resultant equations, we obtain
$$\begin{aligned} \frac{1}{2}\partial _{t}\Vert u\Vert ^2_{\Omega _f}+\mu \Vert \nabla u\Vert ^2_{\Omega _f}+ \frac{1}{2}\partial _{t}\Vert w_t\Vert ^2_{\Omega _f}+ \frac{1}{2}\partial _{t}\Vert \nabla w\Vert ^2_{\Omega _s}=0, \end{aligned}$$
or equivalently
$$\begin{aligned} \partial _{t}\left\{ \Vert u\Vert ^2_{\Omega _f}+ \Vert w_t\Vert ^2_{\Omega _f}+\Vert \nabla w\Vert ^2_{\Omega _s}\right\} =-2\mu \Vert \nabla u\Vert ^2_{\Omega _f}\le 0, \end{aligned}$$
(2.9)
where
$$\begin{aligned} \Vert u\Vert ^2_{\Omega _f}+ \Vert w_t\Vert ^2_{\Omega _f}+\Vert \nabla w\Vert ^2_{\Omega _s}:=E(u,w,w_t) \end{aligned}$$
(2.10)
is the total energy of the FSI system.

For the well-posedness of the system (2.7), we refer to [22].

3 Time Discretization

For FSI problems, it is very important to design numerical schemes which have good, preferably unconditional, stability property. Usually, this is achieved by fully coupled, implicit schemes which require solving, at each time step, a coupled, nonlinear, saddle-point system.

We construct in this section time discretization schemes based on the standard and rotational pressure-correction approach for (2.7). These schemes are unconditionally stable and lead to, at each time step, a coupled, linear elliptic system in \(\Omega\) and a pressure Poisson equation in \(\Omega _f\), which can be efficiently solved by standard numerical methods. The stability analysis for each scheme is carried out in this section.

3.1 Standard Pressure-Correction Scheme

We first construct a first-order scheme for the FSI problem based on the standard pressure-correction scheme for the Navier–Stokes problem with the open boundary condition [13]:

Step 1 Given \((u^n,p^n,w^n)\), compute \({\tilde{u}}^{n+1}\in \mathbf{H}^1_{0,\Gamma _f}(\Omega _f)\) and \(w^{n+1}\in \mathbf{H}^1_{0,\Gamma _s}(\Omega _s)\) by solving
$$\begin{aligned}&\left( \frac{{\tilde{u}}^{n+1}-u^{n}}{\Delta t},\varphi \right) _{\Omega _f} +(\mu \nabla {\tilde{u}}^{n+1},\nabla \varphi )_{\Omega _f} +\frac{1}{2}((u^n\cdot \mathbf{{n}}){\tilde{u}}^{n+1},\varphi )_{\Gamma _{c}}\nonumber \\& +((u^{n}\cdot \nabla ){\tilde{u}}^{n+1},\varphi )_{\Omega _f} -(p^{n},\text {div}\varphi )_{\Omega _f} +\left( \frac{\partial w^{n+1}}{\partial \mathbf{{n}}},\varphi \right) _{\Gamma _{c}}=0, \quad \forall \varphi \in \mathbf{H}^1_{0,\Gamma _f}(\Omega _f), \end{aligned}$$
(3.1a)
$$\begin{aligned}&{\tilde{u}}^{n+1}=\frac{w^{n+1}-w^{n}}{\Delta t}\quad {\text{on}} \quad \Gamma _{c}, \end{aligned}$$
(3.1b)
$$\begin{aligned}&\left( \frac{w^{n+1}-2w^{n}+w^{n-1}}{\Delta t^2},\psi \right) _{\Omega _s}+(\nabla w^{n+1},\nabla \psi )_{\Omega _s}\nonumber \\&\quad -\left( \frac{\partial w^{n+1}}{\partial \mathbf{{n}}},\psi \right) _{\Gamma _{c}}=0,\quad \forall \psi \in \mathbf{H}^1_{0,\Gamma _s}(\Omega _s). \end{aligned}$$
(3.1c)
This is a coupled, linear elliptic system for \(({\tilde{u}}^{n+1}, w^{n+1})\), with the coupling condition at the interface \(\Gamma _c\). Hence, it can be efficiently solved, for example, by a standard domain decomposition approach (cf., for instance, [26, 29]).
Step 2 Compute \(u^{n+1}\in {\mathbf{H}}^1(\Omega _f)\) and \(p^{n+1}\in {\mathbf{H}}^1(\Omega _f)\) by solving
$$\begin{aligned}&\frac{u^{n+1}-{\tilde{u}}^{n+1}}{\Delta t}+\nabla (p^{n+1}-p^n)=0, \end{aligned}$$
(3.2a)
$$\begin{aligned}&\text { div } u^{n+1}=0 \quad {\text{in}} \quad \Omega _{f}, \end{aligned}$$
(3.2b)
$$\begin{aligned}&u^{n+1}\cdot \mathbf{{n}}|_{\Gamma _{f}}=0 \text { and } p^{n+1}|_{\Gamma _c}=p^n|_{\Gamma _c}. \end{aligned}$$
(3.2c)
We observe that a Dirichlet boundary condition is imposed for \(p^{n+1}\) on the interface \(\Gamma _c\), as opposed to the usual Neumann boundary condition in a pressure-correction formulation. This is due to the interface condition (2.5) which is similar to the open boundary condition considered in [13].
We denote \({\mathbf{H}}^1_{0,\Gamma _c}(\Omega _f)=\{q\in {\mathbf{H}}^1(\Omega _f)\backslash {\mathbb {R}}, q|_{\Gamma _c}=0\}\). Then, the above system is equivalent to: Find \((p^{n+1}-p^n)\in {\mathbf{H}}^1_{0,\Gamma _c}(\Omega _f)\) such that
$$\begin{aligned}&(\nabla (p^{n+1}-p^n),\nabla q)=-\frac{1}{\Delta t}(\nabla \cdot {\tilde{u}}^{n+1} , q),\quad \forall q\in {\mathbf{H}}^1_{0,\Gamma _c}(\Omega _f), \end{aligned}$$
(3.3a)
$$\begin{aligned}&u^{n+1}={\tilde{u}}^{n+1}-{\Delta t}\nabla (p^{n+1}-p^n). \end{aligned}$$
(3.3b)
Hence, we only have to solve a Poisson equation at this step.

For the above scheme, we have the following result:

Theorem 3.1

The scheme (3.1)–(3.3), with \(p^{0}|_{\Gamma _c}=0\), is unconditionally stable. More precisely, if we define the discrete energy
$$\begin{aligned} E^n=\Vert u^n\Vert +\Vert \delta _tw^n\Vert ^2+\Vert \nabla w^n\Vert ^2+(\Delta t)^2\Vert \nabla p^n\Vert ^2, \end{aligned}$$
(3.4)
then we have, for all \(n\ge 0\),
$$\begin{aligned} E^{n+1}-E^n+ \Vert {\tilde{u}}^{n+1}-u^{n}\Vert ^{2}+2\mu \Delta t\Vert \nabla \tilde{u}^{n+1}\Vert ^2+\Delta t^2\Vert \delta ^2_{tt}w^{n+1}\Vert ^{2}+\Delta t^2\Vert \nabla ( \delta _tw^{n+1})\Vert ^{2}\le 0. \end{aligned}$$

Proof

To simplify the notations, we define, for any sequence \(\{u^{k}\}\), the discrete time derivatives \(\delta _tu^{n+1}:=\frac{u^{n+1}-u^n}{\Delta t}\) and \(\delta ^2_{tt}u^{n+1}:=\frac{\delta _tu^{n+1}-\delta _tu^n}{\Delta t}=\frac{u^{n+1}-2u^n+u^{n-1}}{\Delta t^2}\).

Taking \(\varphi =2{\tilde{u}}^{n+1}\) in (3.1a), \(\psi =2\delta _tw^{n+1}\) in (3.1c), and taking the inner product of (3.2a) with \(q=2\Delta t \nabla p^n\), summing up the three relations, we obtain
$$\begin{aligned}&\frac{1}{\Delta t} \{ \Vert {\tilde{u}}^{n+1}\Vert ^{2}-\Vert u^{n}\Vert ^{2}+\Vert {\tilde{u}}^{n+1}-u^{n}\Vert ^{2}\}+2\Vert \nabla {\tilde{u}}^{n+1}\Vert ^{2}-2(p^n, \text { div } {\tilde{u}}^{n+1})_{\Omega _f}\nonumber \\&\quad +\frac{1}{\Delta t}\{\Vert \delta _t w^{n+1}\Vert ^{2}-\Vert \delta _t w^{n}\Vert ^{2}+\Vert \delta _t w^{n+1}-\delta _t w^{n}\Vert ^{2}\}\nonumber \\&\quad +\frac{1}{\Delta t}\{\Vert \nabla w^{n+1}\Vert ^{2}-\Vert \nabla w^{n}\Vert ^{2}+\Delta t^2\Vert \nabla \delta _tw^{n+1}\Vert ^{2}\} =0. \end{aligned}$$
(3.5)
Rewrite (3.2a) as
$$\begin{aligned} \frac{u^{n+1}}{\sqrt{\Delta t}}+\sqrt{\Delta t}\nabla p^{n+1}=\frac{{\tilde{u}}^{n+1}}{\sqrt{\Delta t}}+\sqrt{\Delta t}\nabla p^n. \end{aligned}$$
(3.6)
Taking the inner product with itself from both sides and integrating by parts, thanks to \(p^k|_{\Gamma _c}=0\) for all k (due to \(p^{0}|_{\Gamma _c}=0\)), and \({\tilde{u}}^{n+1}\cdot \mathbf{n}|_{\Gamma _f}=0={u}^{n+1}\cdot \mathbf{n}|_{\Gamma _f}\), we obtain
$$\begin{aligned} \frac{1}{\Delta t}\Vert u^{n+1}\Vert ^2+\Delta t\Vert \nabla p^{n+1}\Vert ^2=\frac{\Vert {\tilde{u}}^{n+1}\Vert ^2}{\Delta t}+\Delta t\Vert \nabla p^{n}\Vert ^2-2( p^n,\text {div} {\tilde{u}}^{n+1})_{\Omega _f}. \end{aligned}$$
(3.7)
Summing up (3.5) and (3.7), we obtain
$$\begin{aligned}&\frac{1}{\Delta t} \{ \Vert u^{n+1}\Vert ^{2}-\Vert u^{n}\Vert ^{2}+\Vert {\tilde{u}}^{n+1}-u^{n}\Vert ^{2}\}+2\Vert \nabla {\tilde{u}}^{n+1}\Vert ^{2}\nonumber \\&\quad +\frac{1}{\Delta t}\{\Vert \delta _t w^{n+1}\Vert ^{2}-\Vert \delta _t w^{n}\Vert ^{2}+\Vert \delta _t w^{n+1}-\delta _t w^{n}\Vert ^{2}\}\nonumber \\&\quad +\frac{1}{\Delta t}\{\Vert \nabla w^{n+1}\Vert ^{2}-\Vert \nabla w^{n}\Vert ^{2}+\Delta t^2\Vert \nabla \delta _t w^{n+1}\Vert ^{2}\} \nonumber \\&\quad +\Delta t\{\Vert \nabla p^{n+1}\Vert ^{2}-\Vert \nabla p^{n}\Vert ^{2}\}=0, \end{aligned}$$
which implies the desired result.

We recall that due to the artificial Dirichlet boundary condition for the pressure in (3.2c), a higher-order discretization for the velocity will not increase the accuracy. Hence, to obtain a higher-order scheme, one needs to resort to the rotational pressure-correction (cf. [13]).

3.2 Rotational Pressure-Correction Schemes

3.2.1 First-Order Scheme

We start by constructing a first-order scheme.

Step 1 Given (\(u^n\), \(v^n\), \(w^n\), \(p^n\)), compute \({\tilde{u}}^{n+1}\in \mathbf{H}^1_{0,\Gamma _f}(\Omega _f)\) and \(w^{n+1}\in \mathbf{H}^1_{0,\Gamma _s}(\Omega _s)\) by solving
$$\begin{aligned}&\left( \frac{{\tilde{u}}^{n+1}-u^{n}}{\Delta t},\varphi \right) _{\Omega _f} +(\mu \nabla {\tilde{u}}^{n+1},\nabla \varphi )_{\Omega _f}+\frac{1}{2}((u^{n}\cdot { \mathbf {n}}){\tilde{u}}^{n+1},\varphi )_{\Gamma _{c}}\nonumber \\&\quad +((u^{n}\cdot \nabla ){\tilde{u}}^{n+1},\varphi )_{\Omega _f} -(p^{n},\text {div}\varphi )_{\Omega _f} +\left( \frac{\partial w^{n+1}}{\partial \mathbf {n}},\varphi \right) _{\Gamma _{c}}=0, \quad \forall \varphi \in \mathbf{H}^1_{0,\Gamma _f}(\Omega _f), \end{aligned}$$
(3.8a)
$$\begin{aligned}&{\tilde{u}}^{n+1}=\frac{w^{n+1}-w^{n}}{\Delta t}\quad {\text{on}} \quad \Gamma _{c}, \end{aligned}$$
(3.8b)
$$\begin{aligned}&\left( \frac{{w}^{n+1}-2{w}^{n}+w^{n-1}}{{\Delta t}^2},\psi \right) _{\Omega _s}+(\nabla w^{n+1},\nabla \psi )_{\Omega _s}\nonumber \\&\quad -\left( \frac{\partial w^{n+1}}{\partial \mathbf{{n}}},\psi \right) _{\Gamma _{c}}=0,\quad \forall \psi \in \mathbf{H}^1_{0,\Gamma _s}(\Omega _s). \end{aligned}$$
(3.8c)
Step 2 Compute \(u^{n+1}\in {\mathbf{H}}^1(\Omega _f)\) and \(p^{n+1}\in {\mathbf{H}}^1(\Omega _f)\) by solving
$$\begin{aligned}&\frac{u^{n+1}-{\tilde{u}}^{n+1}}{\Delta t}+\nabla (p^{n+1}-p^n+\lambda \mu \,\text {div}\, \tilde{u}^{n+1})=0 \quad {\text{in}} \quad \Omega _{f}\nonumber \\&\text { div } u^{n+1}=0 \quad {\text{in}} \quad \Omega _{f}\nonumber \\&u^{n+1}\cdot \mathbf{{n}}|_{\Gamma _{f}}=0 \text { and } p^{n+1}|_{\Gamma _c}=(p^n-\lambda \mu \,\text {div} \,\tilde{u}^{n+1})|_{\Gamma _c}, \end{aligned}$$
(3.9a)
where \(\lambda \in (0,\frac{2}{d})\) (with d being the space dimension) is a preselected parameter. We note that when \(\lambda =0\), the scheme reduces to the standard pressure-correction scheme.

We observe that the main difference of the rotational scheme (3.8)–(3.9) with the standard scheme (3.1)–(3.3) is the additional term \(\lambda \mu \,\text {div}\, \tilde{u}^{n+1}\) in (3.9a). This term replaces the artificial Dirichlet B.C. \(p^{n+1}|_{\Gamma _c}=p^{n}|_{\Gamma _c}\) by an improved B.C. \(p^{n+1}|_{\Gamma _c}=(p^n-\lambda \mu \text {div} \tilde{u}^{n+1})|_{\Gamma _c}\). On the other hand, the numerical procedures for the two schemes are essentially identical.

The proof of unconditional stability for the rotational scheme is much more difficult. The original stability proof of the rotational pressure-correction scheme in [15] was carried out only for Stokes problems, and an essential step of the proof was to take the “discrete time derivative” of the scheme. Unfortunately, this proof cannot be extended to the nonlinear case. However, we can prove that the above rotational scheme is unconditionally stable using a similar procedure to that in the proof below for the second-order rotational scheme. We omit the details for the sake of brevity.

3.2.2 Second-Order Scheme

We observe that it is not straightforward to construct a second-order version of (3.8)–(3.9) using the usual backward difference formula (BDF). Hence, we first introduce an additional variable \(v=w_{t}\) and rewrite the FSI equations as
$$\begin{aligned}&u_{t}-\mu \Delta u+(u \cdot \nabla )u+\nabla p =0&\mathrm{{in}} \quad \Omega _{f}\times (0,T), \end{aligned}$$
(3.10a)
$$\begin{aligned}&\text {div}\; u=0&\mathrm{{in}} \quad \Omega _{f}\times (0,T), \end{aligned}$$
(3.10b)
$$\begin{aligned}&v_{t}-\Delta w=0&\mathrm{{in}} \quad \Omega _{s}\times (0,T), \end{aligned}$$
(3.10c)
$$\begin{aligned}&w_{t}-v=0&\mathrm{{in}} \quad \Omega _{s}\times (0,T), \end{aligned}$$
(3.10d)
with the boundary condition
$$\begin{aligned}&u=0\quad&\mathrm{{on}} \quad \Gamma _{f}\times (0,T), \end{aligned}$$
(3.11a)
$$\begin{aligned}&w=0\quad&\mathrm{{on}} \quad \Gamma _{s}\times (0,T), \end{aligned}$$
(3.11b)
$$\begin{aligned}&u =v\quad&\mathrm{{on}} \quad \Gamma _{c}\times (0,T), \end{aligned}$$
(3.11c)
$$\begin{aligned}&\frac{\partial w}{\partial \mathbf {n}}=\mu \frac{\partial u}{\partial \mathbf {n}}-p\mathbf{{n}}-\frac{1}{2}(u\cdot \mathbf{{n}})u\quad&\mathrm{{on}} \quad \Gamma _{c}\times (0,T), \end{aligned}$$
(3.11d)
and the initial condition
$$u(\cdot ,0)=u_0 \quad {\text{in}} \quad \Omega _{f},$$
(3.12a)
$$w(\cdot ,0)=w_0\quad {\text{in}} \quad \Omega _{s}, $$
(3.12b)
$$v(\cdot ,0)=w_1\quad {\text{in}} \quad \Omega _{s}. $$
(3.12c)

We can now construct a second-order rotational pressure-correction scheme as follows:

Step 1 Given (\(u^n\), \(w^n\), \(v^n\), \(p^n\)), compute \({\tilde{u}}^{n+1}\in \mathbf{H}^1_{0,\Gamma _f}(\Omega _f)\) and \(v^{n+1}, w^{n+1}\in \mathbf{H}^1_{0,\Gamma _s}(\Omega _s)\) by solving
$$\begin{aligned}&\left( \frac{3{\tilde{u}}^{n+1}-4u^{n}+u^{n-1}}{2\Delta t},\varphi \right) _{\Omega _f} +(\mu \nabla {\tilde{u}}^{n+1},\nabla \varphi )_{\Omega _f} +\frac{1}{2}\left( \left( \left( 2u^{n}-u^{n-1}\right) \cdot { \mathbf {n}}\right) {\tilde{u}}^{n+1},\varphi \right) _{\Gamma _{c}}\nonumber \\&\quad +((2u^{n}-u^{n-1})\cdot \nabla {\tilde{u}}^{n+1},\varphi )_{\Omega _f} -(p^{n},\text {div}\varphi )_{\Omega _f}\nonumber \\&\quad +\left( \frac{\partial w^{n+1}}{\partial \mathbf {n}},\varphi \right) _{\Gamma _{c}}=0, \quad \forall \varphi \in \mathbf{H}^1_{0,\Gamma _f}(\Omega _f), \end{aligned}$$
(3.13a)
$$\begin{aligned}&{\tilde{u}}^{n+1}={v}^{n+1}\quad {\text{on}} \quad \Gamma _{c}, \end{aligned}$$
(3.13b)
$$\begin{aligned}&\frac{3w^{n+1}-4w^{n}+w^{n-1}}{2\Delta t}-{v}^{n+1}=0\quad {\text{in}} \quad \Omega _{s}, \end{aligned}$$
(3.13c)
$$\begin{aligned}&\left( \frac{3{v}^{n+1}-4{v}^{n}+v^{n-1}}{2\Delta t},\psi \right) _{\Omega _s}+(\nabla w^{n+1},\nabla \psi )_{\Omega _s}\nonumber \\&\quad -\left( \frac{\partial w^{n+1}}{\partial \mathbf{{n}}},\psi \right) _{\Gamma _{c}}=0,\quad \forall \psi \in \mathbf{H}^1_{0,\Gamma _s}(\Omega _f). \end{aligned}$$
(3.13d)
Step 2 Compute \((u^{n+1},p^{n+1})\) by solving
$$\begin{aligned}&\frac{3(u^{n+1}-{\tilde{u}}^{n+1})}{2\Delta t}+\nabla (p^{n+1}-p^n+\lambda \mu \text {div} \tilde{u}^{n+1})=0 \quad {\text{in}} \quad \Omega _{f},\nonumber \\&\text { div } u^{n+1}=0 \quad {\text{in}} \quad \Omega _{f},\nonumber \\&u^{n+1}\cdot \mathbf{{n}}|_{\Gamma _{f}}=0 \text { and } p^{n+1}|_{\Gamma _c}=(p^n-\lambda \mu \text {div} \tilde{u}^{n+1})|_{\Gamma _c}, \end{aligned}$$
(3.14a)
where \(\lambda \in (0,\frac{2}{d})\) is a preselected parameter.
Several remarks are in order:
  • One observes that all the terms, except the pressure, are discretized with a second-order BDF or Adams–Bashforth formula. We recall that a first-order treatment of the pressure term, coupled with second-order treatment for other terms, can lead to second-order accuracy for the velocity [14].

  • It is clear that, at each time step, the numerical procedure for solving (3.13)–(3.14) is essentially the same as for the first-order scheme (3.1)–(3.3).

  • In [25], the author proved the unconditional stability for a Gauge–Uzawa scheme of the Navier–Stokes equations. A useful idea in [25] is to introduce a sequence \(\{q^n\}\) defined by
    $$\begin{aligned} q^n=\lambda \mu \text {div}\; \tilde{u}^n+q^{n-1}\text { with } q^{-1}=q^0=0. \end{aligned}$$
    (3.15)
    We shall also use this sequence in our stability proof below.

Theorem 3.2

The scheme (3.13)–(3.14), with \(p^{-1}|_{\Gamma _c}=p^{0}|_{\Gamma _c}=0\), is unconditionally stable. More precisely, if we define the discrete energy as
$$\begin{aligned} \begin{aligned} E^{n+1}&=\Vert u^{n+1}\Vert ^2+\Vert 2u^{n+1}-u^{n}\Vert ^2+\Vert v^{n+1}\Vert ^2 +\Vert 2v^{n+1}-v^{n}\Vert ^2+\Vert \nabla w^{n+1}\Vert ^2\\&\quad +\Vert 2\nabla w^{n+1}-\nabla w^{n}\Vert ^2+2\Delta t\Vert q^{n+1}\Vert ^2+\frac{4\Delta t^2}{3}\Vert \nabla (p^{n+1}+q^{n+1})\Vert ^2, \end{aligned} \end{aligned}$$
(3.16)
then we have
$$\begin{aligned} E^{n+1}+\Delta t^4\Vert \delta _{tt}u^{n+1}\Vert ^2+\Delta t^4\Vert \delta _{tt}v^{n+1}\Vert ^2+\Delta t^4\Vert \delta _{tt}w^{n+1}\Vert ^2+(2-d\lambda )2\Delta t\mu \Vert \nabla \tilde{u}^{n+1}\Vert ^2\le E^n. \end{aligned}$$

Proof

For any sequence \(\{u^n,\tilde{u}^n\}\), we have
$$\begin{aligned} \begin{aligned}&\left( \frac{3{\tilde{u}}^{n+1}-4u^{n}+u^{n-1}}{2\Delta t},4\Delta t\tilde{u}^{n+1}\right) _{\Omega _f}=2(3{\tilde{u}}^{n+1}-4u^{n}+u^{n-1},\tilde{u}^{n+1})_{\Omega _f}\\&\quad =6({\tilde{u}}^{n+1}-u^{n+1},\tilde{u}^{n+1})_{\Omega _f}+2(3{u}^{n+1}-4u^{n}+u^{n-1},\tilde{u}^{n+1}-u^{n+1})_{\Omega _f}\\&\qquad +2(3{u}^{n+1}-4u^{n}+u^{n-1},u^{n+1})_{\Omega _f}. \end{aligned} \end{aligned}$$
(3.17)
Let \(I^n_1(u)\), \(I^n_2(u)\) and \(I^n_3(u)\) be the last three terms in the right-hand side. Using the algebraic identities
$$\begin{aligned} 2(a^{k+1},a^{k+1}-a^k)=|a^{k+1}|^2-|a^{k}|^2+|a^{k+1}-a^k|^2 \end{aligned}$$
(3.18)
and
$$\begin{aligned} \begin{aligned}&2(a^{k+1},3a^{k+1}-4a^k+a^{k-1})\\&\quad =|a^{k+1}|^2+|2a^{k+1}-a^{k}|^2+|a^{k+1}-2a^k+a^{k-1}|^2-|a^k|^2-|2a^k-a^{k-1}|^2, \end{aligned} \end{aligned}$$
(3.19)
we find
$$\begin{aligned} \begin{aligned}&I^n_1(u)=3\Vert \tilde{u}^{n+1}\Vert ^2-3\Vert u^{n+1}\Vert ^2+3\Vert \tilde{u}^{n+1}-u^{n+1}\Vert ^2,\\&I^n_3(u)=\Vert u^{n+1}\Vert ^2+\Vert 2u^{n+1}-u^{n}\Vert ^2+\Vert u^{n+1} -2u^n+u^{n-1}\Vert ^2-\Vert u^n\Vert ^2-\Vert 2u^n-u^{n-1}\Vert ^2. \end{aligned} \end{aligned}$$
(3.20)
Using the first equation in (3.14a), we have
$$\begin{aligned} I^n_2(u)=-\frac{4\Delta t}{3} (3{u}^{n+1}-4u^{n}+u^{n-1},\nabla (p^{n+1}-p^n+\lambda \mu \text {div}\; \tilde{u}^{n+1}))_{\Omega _f}=0. \end{aligned}$$
Taking \(\varphi =4\Delta t\tilde{u}^{n+1}\) in (3.13a), and using (2.8) and the above relation, we obtain
$$\begin{aligned} I^n_1(u)+I^n_3(u)+4\Delta t\mu \Vert \nabla \tilde{u}^{n+1}\Vert ^2-4\Delta t(p^n,\text {div }\tilde{u}^{n+1})_{\Omega _f}+4\Delta t\left( \frac{\partial w^{n+1}}{ \partial n},\tilde{u}^{n+1}\right) _{\Gamma _c}=0. \end{aligned}$$
(3.21)
Taking \(\psi =4\Delta t v^{n+1}\) in (3.13d), using (3.13b) and (3.13c), we find
$$\begin{aligned} \begin{aligned} I^n_3(v)+\tilde{I}^n_3(w)-4\Delta t\left( \frac{\partial w^{n+1}}{ \partial n},\tilde{u}^{n+1}\right) _{\Gamma _c}=0, \end{aligned} \end{aligned}$$
(3.22)
where, by (3.20),
$$\begin{aligned} \begin{aligned} \tilde{I}^n_3(w)&=2(\nabla (3{w}^{n+1}-4w^{n}+w^{n-1}),\nabla w^{n+1})_{\Omega _f}\\&=\Vert \nabla w^{n+1}\Vert ^2+\Vert 2\nabla w^{n+1}-\nabla w^{n}\Vert ^2+\Vert \nabla w^{n+1}-2\nabla w^n+\nabla w^{n-1}\Vert ^2\\&\quad -\Vert \nabla w^n\Vert ^2-\Vert 2\nabla w^n-\nabla w^{n-1}\Vert ^2. \end{aligned} \end{aligned}$$
(3.23)
Using (3.15), we can rewrite (3.14a) as
$$\begin{aligned} \frac{\sqrt{3}u^{n+1}}{\sqrt{\Delta t}}+\frac{2\sqrt{\Delta t}}{\sqrt{3}}\nabla (2p^{n+1}+q^{n+1})=\frac{\sqrt{3}{\tilde{u}}^{n+1}}{\sqrt{\Delta t}}+\frac{2\sqrt{\Delta t}}{\sqrt{3}}\nabla (p^n+q^n). \end{aligned}$$
Taking the inner product with itself from both sides of the above equation, integrating parts and using (3.15) and the fact that \((p^k+q^k)|_{\Gamma _c}=\cdots =(p^0+q^0)|_{\Gamma _c}=0\), we obtain
$$\begin{aligned} \begin{aligned}&\frac{3}{\Delta t}\Vert u^{n+1}\Vert ^2+\frac{4\Delta t}{3}\Vert \nabla (p^{n+1}+q^{n+1})\Vert ^2-\frac{3\Vert {\tilde{u}}^{n+1}\Vert ^2}{\Delta t}-\frac{4\Delta t}{3}\Vert \nabla (p^{n}+q^{n})\Vert ^2\\& =-4( p^{n}+q^{n},\text {div }{\tilde{u}}^{n+1})_{\Omega _f} =-4( p^{n},\text {div }{\tilde{u}}^{n+1})_{\Omega _f}-\frac{4}{\lambda \mu }( q^{n},q^{n+1}-q^n)_{\Omega _f}\\& =-4( p^{n},\text {div }{\tilde{u}}^{n+1})_{\Omega _f}+\frac{2}{\lambda \mu }\{\Vert q^n\Vert ^2-\Vert q^{n+1}\Vert ^2+\Vert q^{n+1}-q^n\Vert ^2\}. \end{aligned} \end{aligned}$$
(3.24)
Multiplying the above by \(\Delta t\) and adding it to (3.21), we obtain
$$\begin{aligned} \begin{aligned} 0&=I^n_1(u)+I^n_3(u)+4\Delta t\mu \Vert \nabla \tilde{u}^{n+1}\Vert ^2+I^n_3(v)+I^n_3(w) + 3\Vert u^{n+1}\Vert ^2\\&\quad +\frac{4\Delta t^2}{3}\Vert \nabla (p^{n+1}+q^{n+1})\Vert ^2-3\Vert {\tilde{u}}^{n+1}\Vert ^2-\frac{4\Delta t^2}{3}\Vert \nabla (p^{n}+q^{n})\Vert ^2\\&\quad -\frac{2}{\lambda \mu }\Delta t\{\Vert q^n\Vert ^2-\Vert q^{n+1}\Vert ^2+\Vert q^{n+1}-q^n\Vert ^2\}. \end{aligned} \end{aligned}$$
(3.25)
Thanks to (3.15), we have
$$\begin{aligned} \frac{2}{\lambda \mu }\Vert q^{n+1}-q^n\Vert ^2=2{\lambda \mu }\Vert \text {div }\tilde{u}^{n+1}\Vert ^2\le 2{\lambda \mu } d\Vert \nabla \tilde{u}^{n+1}\Vert ^2, \end{aligned}$$
where we have used the well-known Korn’s inequality \(\Vert \text {div }\tilde{u}^{n+1}\Vert ^2\le d\Vert \nabla \tilde{u}^{n+1}\Vert ^2\) with \(d=2\) or 3 being the space dimension.
Finally, using the above inequality, (3.20) and (3.23) in (3.25), we find
$$\begin{aligned} \begin{aligned}&E^{n+1}-E^{n}\\& =-\Vert u^{n+1}-2u^n+u^{n+1}\Vert ^2-\Vert v^{n+1}-2v^n+v^{n+1}\Vert ^2-\Vert \nabla (w^{n+1}-2w^n+w^{n+1})\Vert ^2\\&\quad -4\Delta t\mu \Vert \nabla \tilde{u}^{n+1}\Vert ^2+2d\lambda \Delta t\mu \Vert \text {div }\tilde{u}^{n+1}\Vert ^2\\& \le -\Vert u^{n+1}-2u^n+u^{n+1}\Vert ^2-\Vert v^{n+1}-2v^n+v^{n+1}\Vert ^2-\Vert \nabla ( w^{n+1}-2w^n+w^{n+1})\Vert ^2\\&\quad -(2-d\lambda )2\Delta t\mu \Vert \nabla \tilde{u}^{n+1}\Vert ^2, \end{aligned} \end{aligned}$$
which implies the desired result.

Remark 3.3

With the stability results established in this section, it is also possible to derive similar error estimates for these schemes as in [13].

4 Galerkin-Type Spatial Discretization and Implementation

We briefly describe a general procedure to implement the time discretization schemes constructed in the last section. Let \(\mathbf{X}_h\subset {\mathbf{H}}^1_{0,\Gamma _f}(\Omega _f)\), \(M_h\subset {\mathbf{H}}^1(\Omega _f)\), \(M^0_h=\{q\in M_h:\, q|_{\Gamma _c}=0\}\) and \(\mathbf{W}_h\subset \mathbf{H}^1_{0,\Gamma _s}(\Omega _s)\) be some finite dimensional approximation spaces, with \((\mathbf{X}_h,M_h)\) preferably satisfying the Babuska–Brezzi inf-sup condition. We also denote \(\mathbf{Y}_h=\mathbf{X}_h+\nabla M_h^0\). We note that one can generalize the stability proofs for the semi-discretized schemes in the last section to their full discretized versions using the above discrete settings; we refer to [14] for more detail in this regard.

To fix the idea, we take the scheme (3.8)–(3.9) as an example. The other schemes can be treated by using exactly the same procedure.

4.1 A General Setup

A Galerkin approximation of the scheme (3.8)–(3.9) is as follows:

Step 1 Let \(\tilde{w}_h^{n+1}=\delta _tw_h^{n+1}\). Then we look for \((u^{n+1}_h,\tilde{w}^{n+1}_h)\in \mathbf{X}_h\times \mathbf{W}_h\) such that
$$\begin{aligned}&\alpha (\tilde{u}^{n+1}_h,\varphi _h)_{\Omega _f}+(\nabla \tilde{u}^{n+1}_h,\nabla \varphi _h)_{\Omega _f}+((u_h^{n}\cdot \nabla ) {\tilde{u}}_h^{n+1},\varphi _h)_{\Omega _f}\nonumber \\& +\frac{1}{2}((u_h^{n}\cdot { \mathbf {n}}){\tilde{u}}_h^{n+1},\varphi _h)_{\Gamma _c}+\beta \left( \frac{\partial \tilde{w}^{n+1}_h}{\partial \mathbf{{n}}},\varphi _h\right) _{\Gamma _c}=<f^{n}_h,\varphi _h>_{\Omega _f},\quad \forall \varphi _h\in \mathbf{X}_h, \end{aligned}$$
(4.1a)
$$\begin{aligned}&\tilde{u}^{n+1}_h=\tilde{w}^{n+1}_h \quad \text {at } \Gamma _c, \end{aligned}$$
(4.1b)
$$\begin{aligned}&\alpha (\tilde{w}^{n+1},\psi _h)_{\Omega s}+\beta (\nabla \tilde{w}^{n+1},\nabla \psi _h)_{\Omega _s}-\beta \left( \frac{\partial \tilde{w}^{n+1}}{\partial \mathbf{{n}}},\psi _h\right) _{\Gamma _c}\nonumber \\&\quad =<g_h,\psi _h>_{\Omega _s},\quad \forall \varphi _h\in \mathbf{W}_h, \end{aligned}$$
(4.1c)
where \(\alpha =\frac{1}{\Delta t}, \beta = \Delta t\), and
$$\begin{aligned} <f^n_h,\varphi _h>_{\Omega _f}:=\alpha (u_h^n,\varphi _h)_{\Omega _f}+( p_h^n,\text {div}\varphi _h)_{\Omega _f}-\left( \frac{\partial w^{n}_h}{\partial \mathbf{{n}}},\varphi _h\right) _{\Gamma _c}, \end{aligned}$$
(4.2)
and
$$\begin{aligned} <g^n_h,\psi _h>_{\Omega _s}:=\alpha (\tilde{w}_h^n,\psi ^n_h)_{\Omega _s}-(\nabla w_h^n,\nabla \psi _h)_{\Omega _s}+ \left( \frac{\partial w^{n}_h}{\partial \mathbf{{n}}},\psi _h\right) _{\Gamma _c}. \end{aligned}$$
(4.3)
Define
$$\begin{aligned} {\hat{u}}^{n+1}_h(\mathbf{x})= & {} \left\{ \begin{array}{ll}\tilde{u}^{n+1}_h(\mathbf{x}),&\quad{}\text {if }{} \mathbf{x}\in \Omega _f,\\ \tilde{w}^{n+1}_h(\mathbf{x}),&\quad{}\text {if }\mathbf{x}\in \Omega _s;\end{array}\right. \\ {\hat{\beta }}(\mathbf{x}):= & {} \left\{ \begin{array}{ll}1,&\quad{}\text {if }\mathbf{x}\in \Omega _f,\\ \beta ,&\quad{}\text {if }{} \mathbf{x}\in \Omega _s;\end{array}\right. \\ b(u,v,\varphi ):= & {} ((u\cdot \nabla )\tilde{v},\varphi )_{\Omega _f}+ \left( \frac{1}{2}(u\cdot { \mathbf {n}})\tilde{v},\varphi \right) _{\Gamma _c}; \end{aligned}$$
and
$$\begin{aligned} \mathbb {X}_h=\left\{\hat{u}_h\in \mathbf{H}^1(\Omega ): \; \hat{u}_h|_{\Omega _f}\in \mathbf{X}_h,\; \hat{u}_h|_{\Omega _s}\in \mathbf{W}_h\right\}. \end{aligned}$$
Then, we can rewrite (4.1) as: Find \({\hat{u}}^{n+1}_h\in \mathbb {X}_h\) such that
$$\begin{aligned} \begin{aligned}&\alpha ({\hat{u}}^{n+1}_h,\phi _h)+({\hat{\beta }}\nabla {\hat{u}}^{n+1}_h,\nabla \phi _h)+b(u_h^n,{\hat{u}}^{n+1}_h,\phi _h)\\& =<{f}^n_h,\phi _h>_{\Omega _f}+<g^n_h,\phi _h>_{\Omega _s},\quad \forall \phi _h \in \mathbb {X}_h. \end{aligned} \end{aligned}$$
(4.4)
Thus, the equation (4.4) can be viewed as a two-domain approximation to a linear elliptic problem with the discontinuous coefficient \(\hat{\beta }\). Note that from (4.1b), \({\hat{u}}^{n+1}_h(\mathbf{x})\) is continuous at \(\Gamma _c\). Hence, one can efficiently solve the coupled linear system using a standard domain decomposition approach. In particular, in the two-dimension case, one can form the Schur-complement to solve the unknown at the interface first, and then solve for the velocity in the fluid region and displacement in the solid region separately (cf., for instance, [26, 29] and a simple example in the next subsection).
Step 2 Find \(\phi _h^{n+1}\in M^0_h\) such that
$$\begin{aligned} (\nabla \phi _h^{n+1},\nabla q_h)_{\Omega _f}=\frac{1}{\Delta t}({\tilde{u}}_h^{n+1},\nabla q_h )_{\Omega _f},\quad \forall q_h\in M^0_h; \end{aligned}$$
(4.5)
and compute \(u_h^{n+1}\in \mathbf{Y}_h\) and \(p_h^{n+1}\in M_h\) by
$$\begin{aligned} \begin{aligned} u_h^{n+1}&={\tilde{u}}_h^{n+1}-{\Delta t}\nabla \phi _h^{n+1},\\ p_h^{n+1}&=p_h^n+\phi _h^{n+1}-\lambda \mu Q_h\text {div}{\tilde{u}}_h^{n+1}, \end{aligned} \end{aligned}$$
(4.6)
where \(Q_h\) is an \(L^2\)-projection operator onto \(M_h\).

We note that (4.5) is just a discrete Poisson equation in \(\Omega _f\) with the homogeneous Dirichlet boundary condition on \(\Gamma _c\), and (4.6) involves only a projection, so they can be efficiently solved.

4.2 An Example with a Fourier–Legendre Approximation

As an example, we consider a two-dimensional periodic channel with \(\Omega _f=(0,2\pi )\times (0,1)\), \(\Omega _s=(0,2\pi )\times (-1,0)\), so \(\Omega =(0,2\pi )\times (-1,1)\), \(\Gamma _f=\{(x,y)|\,x\in (0,2\pi ), y=1\}\), \(\Gamma _c=\{(x,y)|\,x\in (0,2\pi ), y=0\}\) and \(\Gamma _s=\{(x,y)|\,x\in (0,2\pi ), y=-1\}\). We denote \(\mathbf{I^+}\,,\mathbf{{I^-}}\,,\mathbf{{I}}\) by \(\mathbf{{I^+}}=[0,1]\), \(\mathbf{{I^-}}=[-1,0]\) and \(\mathbf{I}=[-1,1]\). We assume that all functions are periodic in the x-direction.

Let \(h=(M,N),\) where M is the number of equally spaced points in the x-direction, and \(N+1\) is the number of Legendre–Gauss–Lobatto points in the y-direction of \(\Omega _f\) and \(\Omega _s\). For simplicity, we use the same number of points in the y-direction of \(\Omega _f\) and \(\Omega _s\). Let \(P_N\) be the set of all polynomials of degree less than or equal to N. We set
$$\begin{aligned} \begin{aligned} X_h&=\left\{ v_h=\sum _{k=-M/2}^{M/2} v_k(y){\text{e}}^{{\rm i}kx} \text { with } v_k(\cdot )\in P_N,\; v_k(1)=0\right\} , \; \mathbf{X}_h=X_h\times X_h,\\ W_h&=\left\{ w_h=\sum _{k=-M/2}^{M/2} w_k(y){\text{e}}^{{\rm i}kx} \text { with } w_k(\cdot )\in P_N,\; w_k(-1)=0\right\} , \; \mathbf{W}_h=W_h\times W_h,\\ M_h&=\left\{ q_h=\sum _{k=-M/2}^{M/2} q_k(y){\text{e}}^{{\rm i}kx} \text { with } q_k(\cdot )\in P_{N-1}\right\} ,\\ M^0_h&=\{q_h\in M_h: \, q_h|_{y=0}=0\}, \; \mathbf{Y}_h=\mathbf{X}_h +\nabla M^0_h,\\ X^0_N&=\{v\in H^1(I): v|_\mathbf{I^+},v|_\mathbf{I^-}\in P_N,\; v(-1)=v(1)=0\},\; \mathbf{X}^0_N=X^0_N\times X^0_N. \end{aligned} \end{aligned}$$
(4.7)
For the sake of efficiency and to take full advantage of periodicity in the x-direction, we shall treat the nonlinear convective term in (4.1) explicitly. To this end, we modify (4.2) to
$$\begin{aligned} <f^n_h,\varphi _h>_{\Omega _f}:=\alpha (u_h^n,\varphi _h)_{\Omega _f}+( p_h^n,\text {div}\varphi _h)_{\Omega _f}-\left( \frac{\partial w^{n}_h}{\partial \mathbf{{n}}},\varphi _h\right) _{\Gamma _c}-b(u^n_h,u^n_h,\varphi _h). \end{aligned}$$
(4.8)
With this modification we expand all the functions in discrete Fourier series, e.g.,
$$\begin{aligned} \left( {\hat{u}}_h^{n+1},{f}_h^n,{g}_h^n\right) =\sum _{m=-M/2}^{M/2}\left( u^{n+1}_{m}(y),f^n_{m}(y),g^n_{m}(y)\right) {\text{e}}^{{\rm i}mx}. \end{aligned}$$
(4.9)
The system (4.4) reduces to: For \(m=-M/2,\cdots ,0,1,\cdots , M/2\), find \(u^{n+1}_m\in \mathbf{X}^0_N\) such that
$$\begin{aligned} (\alpha _m u^{n+1}_m,\phi )_\mathbf{I}+\left( {\hat{\beta }} {{\text{d}} u^{n+1}_m \over {\text{d}}y} ,{{\text{d}} \phi \over {\text{d}}y}\right) _\mathbf{I}=(f^n_m,\phi )_\mathbf{I^+} +(g^n_m,\phi )_\mathbf{I^-} ,\quad \forall \phi \in \mathbf{X}^0_N, \end{aligned}$$
(4.10)
where
$$\begin{aligned} \alpha _m=\left\{ \begin{array}{ll}\alpha +m^2, &{} \quad \text {if }y \in \mathbf{I^+},\\ \alpha +\beta m^2, &{} \quad \text {if } y\in \mathbf{I^-}. \end{array} \right. \end{aligned}$$
Next we construct a set of basis functions for \(X^0_N\).
We define, for \(i=0,1,...,N-2\),
$$\begin{aligned} \hat{\varphi }_i(y)= & {} \left\{ \begin{array}{ll}L_{k}(2y-1)-L_{k+2}(2y-1),&\quad{}\text {if }y\in \mathbf{I^+},\\ 0,&\quad{}\text {if }y\in \mathbf{I^-};\end{array}\right. \\ \hat{\varphi }_{N-1+i}(y)= & {} \left\{ \begin{array}{ll}0,&\quad{}\text {if }y\in \mathbf{I^+},\\ L_{k}(1+2y)-L_{k+2}(1+2y),&\quad{}\text {if }y\in \mathbf{I^-};\end{array}\right. \end{aligned}$$
and the basis function at the interface is
$$\begin{aligned} \hat{\varphi }_{2N-2}=\left\{ \begin{array}{ll}1-y,&\quad{}\text {if }y\in \mathbf{I^+},\\ 1+y,&\quad{}\text {if }y\in \mathbf{I^-}.\end{array}\right. \end{aligned}$$
Then,
$$\begin{aligned} X^0_{N}=\text {span}\left\{ \hat{\varphi }_0,\hat{\varphi }_1, \cdots ,\hat{\varphi }_{2N-2}\right\} . \end{aligned}$$
(4.11)
Then, writing
$$\begin{aligned} u^{n+1}_{m}(y)=\sum _{k=0}^{2N-2} \hat{u}^{n+1}_{m,k}\hat{\varphi }_{k}(y),\;\hat{f}_{m,k}^n =(f^n_m,\hat{\varphi }_k)_{\mathbf{I}^+}+(g^n_m,\hat{\varphi }_k)_{\mathbf{I}^-}, \end{aligned}$$
and taking \(\varphi =\hat{\varphi }_k\) in (4.10), we can derive the following linear system:
$$\begin{aligned} \begin{array}{llll} \left( \alpha \left[ \begin{array}{lll} M_{11}&{}0&{}m_{13}\\ 0&{}M_{22}&{}m_{23}\\ m_{31}^{\rm T}&{}m_{32}^{\rm T}&{}m_{33}\end{array}\right] +\left[ \begin{array}{lll} S_{11}&{}0&{}s_{13}\\ 0&{}S_{22}&{}s_{23}\\ s_{31}^{\rm T}&{}s_{32}^{\rm T}&{}s_{33}\end{array}\right] \right) &{}\left[ \begin{array}{l} {\bar{u}}_1\\ {\bar{u}}_2\\ {\bar{u}}_3 \end{array}\right] &{}=&{} \left[ \begin{array}{l} {\bar{f}}_1\\ {\bar{f}}_2\\ {\bar{f}}_3 \end{array}\right] , \end{array} \end{aligned}$$
(4.12)
where \(\bar{u}_1=(\hat{u}^{n+1}_{m,0},\hat{u}^{n+1}_{m,1},\cdots ,\hat{u}^{n+1}_{m,N-2})^{\rm T}\), \(\bar{u}_2=(\hat{u}^{n+1}_{m,N-1},\hat{u}^{n+1}_{m,N},\cdots ,\hat{u}^{n+1}_{m,2N-3})^{\rm T}\) and \(\bar{u}_3=u^{n+1}_{m,2N-2}\), similarly for \(\bar{f}_1\), \(\bar{f}_2\) and \(\bar{f}_3\); \(M_{ij}\) and \(S_{ij}\) are block mass and stiffness matrices. We recall that \(M_{ii}\) (\(i=1,2\)) are penta-diagonal and \(S_{ii}\) (\(i=1,2\)) are diagonal (cf. [16, 27]). So the linear system can be easily solved by the Schur-complement approach, More precisely, solve first \(\bar{u}_3\) using a block Gaussian elimination, and then solve \(\bar{u}_1\) and \(\bar{u}_2\) separately.

It is clear that (4.5) will reduce to a sequence of one-dimensional problems in \(\mathbf{I}^+\) which can be easily solved by a Legendre-spectral method.

5 Numerical Results

To examine the correctness and accuracy of the proposed numerical schemes, we consider the following non-homogeneous problem:
$$\begin{aligned}&u_{t}-\Delta u+(u \cdot \nabla )u+\nabla p =f&{\text{in}} \quad \Omega _{f}\times (0,T), \end{aligned}$$
(5.1a)
$$\begin{aligned}&\text {div} \; u=0&{\text{in}}\quad \Omega _{f}\times (0,T), \end{aligned}$$
(5.1b)
$$\begin{aligned}&w_{tt}-\Delta w=g&{\text{in}} \quad \Omega _{s}\times (0,T), \end{aligned}$$
(5.1c)
with the boundary condition
$$\begin{aligned}&u=0\quad&\mathrm{{on}} \quad \Gamma _{f}\times (0,T), \end{aligned}$$
(5.2a)
$$\begin{aligned}&w=0\quad&\mathrm{{on}} \quad \Gamma _{s}\times (0,T), \end{aligned}$$
(5.2b)
$$\begin{aligned}&u =w_{t}\quad&\mathrm{{on}} \quad \Gamma _{c}\times (0,T), \end{aligned}$$
(5.2c)
$$\begin{aligned}&\frac{\partial w}{\partial n}=\frac{\partial u}{\partial n}-p\mathbf{n}-\frac{1}{2}(u\cdot \mathbf{{n}})u+h&\mathrm{{on}} \quad \Gamma _{c}\times (0,T), \end{aligned}$$
(5.2d)
where \(\Omega _f=(0,2\pi )\times (0,1)\), \(\Omega _s=(0,2\pi )\times (-1,0)\) with periodic boundary conditions in the x-direction.
We set the exact solution to be
$$\begin{aligned} \begin{aligned}&u=(-\sin (\pi t)\cos (x)\sin (y-1),\sin (\pi t)\sin (x)(\cos (y-1)-1)),\\&p=\sin (\pi t)\cos (x)\cos (y),\\&w=(-\cos (\pi t)\cos (x)\sin (y-1),-\cos (\pi t)\sin (x)(\cos (y+1)-1)). \end{aligned} \end{aligned}$$
(5.3)
The functions fgh can then be computed accordingly.

We employ the Fourier–Legendre method presented in the last section and choose (MN) large enough so that the errors are dominated by the time discretization. In the following examples, we choose \(\lambda =0.5\), which is a preselected parameter introduced in (3.9a) and (3.14a).

In Fig. 2, we plot the \(L^2\)-errors for the pressure and for the velocity and displacement with the second-order standard and rotational pressure-correction schemes. We observe that the rotational scheme performs much better than the standard scheme.
Fig. 2

\(\Delta t=0.01\), T=2; second-order scheme; top: standard; bottom: rotational

In Fig. 3, we plot the convergence rate of the second-order rotational scheme. We consider ending time \(T=2\) and vary the step size from \(\Delta t=0.1\) to \(\Delta t=0.000\,1\). We observe that the \(L_2\) errors for the fluid velocity, the structure displacement and the pressure all converge at a rate close to 3 / 2. Due to the Dirichlet boundary condition used for the pressure at the interface, the second-order rotational scheme does not achieve full second-order accuracy for the velocity. This is consistent with the error estimates derived and convergence rates observed for Stokes equations with open boundary in [13].
Fig. 3

\(L_2\) Error for second-order rotational scheme

Next, we examine the energy stability of our schemes by solving the homogeneous (with f, g and h being zero) FSI problem with the same initial conditions as in the last example. We take the second-order rotational scheme as an example and plot in Fig. 4 the discrete energy for the cases with \(\Delta t=0.01\) and \(\Delta t=0.05\). We observe that the discrete energy indeed decays monotonically.
Fig. 4

Energy decay for time step 0.01 and 0.05

6 Conclusions

We constructed in this paper standard and rotational pressure correction schemes for the FSI problem with a fixed interface and proved rigorously that they are unconditionally energy stable. These schemes are new and fundamentally different from existing schemes for the FSI problem. Besides the unconditional stability, they are also computationally very efficient: at each time step, they lead to (i) a coupled linear elliptic system for the velocity and displacement, with the coupling condition at the interface between the fluid and solid regions, which can be efficiently solved by using a standard domain decomposition (with two domains) approach; and (ii) a discrete Poisson equation in the fluid region.

We validated these schemes by using a Fourier–Legendre spatial discretization for the FSI problem in a periodic channel. In particular, our numerical results indicate that the convergence rates of the second-order rotational scheme for the velocity, pressure and displacement in \(L^2\)-norm are close to 3/2-order.

Although we only considered the FSI problem with fixed interface, we believe that the essential approaches used here in constructing our numerical schemes can be extended to the FSI problem with moving interface [21], which we plan to address in a future endeavor.

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Copyright information

© Shanghai University 2019

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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