Convergence of a Decoupled Splitting Scheme for the CHNS System
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CONVERGENCE OF A DECOUPLED SPLITTING SCHEME FOR
THE CAHN–HILLIARD–NAVIER–STOKES SYSTEM
CHEN LIU∗, RAMI MASRI†,AND BEATRICE RIVIERE‡
Abstract. This paper is devoted to the analysis of an energy-stable discontinuous Galerkin algo-
rithm for solving the Cahn–Hilliard–Navier–Stokes equations within a decoupled splitting framework.
We show that the proposed scheme is uniquely solvable and mass conservative. The energy dissi-
pation and the 𝐿∞stability of the order parameter are obtained under a CFL condition. Optimal
a priori error estimates in the broken gradient norm and in the 𝐿2norm are derived. The stability
proofs and error analysis are based on induction arguments and do not require any regularization of
the potential function.
Key words. Cahn–Hilliard–Navier–Stokes, discontinuous Galerkin, stability, optimal error
bounds
AMS subject classifications. 65M12, 65M15, 65M60
1. Introduction. The Cahn–Hilliard–Navier–Stokes (CHNS) system serves as
a fundamental phase-field model extensively used in many fields of science and engi-
neering. The simulation of the CHNS equations is a challenging computational task
primarily because of: (i) the coupling of highly nonlinear equations; and (ii) the re-
quirement of preserving certain physical principles, such as conservation of mass and
dissipation of energy. A common approach to overcome these difficulties is to decou-
ple the mass and momentum equations, and to further split the nonlinear convection
from the incompressibility constraint [27]. The splitting scheme constructed from this
strategy only requires the successive solution of several simpler equations at each time
step. Thus, such an algorithm is both convenient for programming and efficient in
large-scale simulations. A non-exhaustive list of several computational papers on the
CHNS model include [9,2,6,19,32,20].
The analysis of semi-discrete spatial formulations with continuous and discontin-
uous Galerkin (dG) methods for solving the CHNS equations has been extensively
investigated. Without being exhaustive, we refer to the papers [10,31,8,21] for
the study of fully coupled schemes. For decoupled splitting algorithms based on pro-
jection methods, we mention a few papers [14,4,28,29]. Han and Wang in [14]
introduce a second order in time scheme and show unique solvability, but this work
does not contain any theoretical proof of convergence. Cai and Shen in [4] formulate
an energy-stable scheme and show convergence based on a compactness argument. In
this work, in order to obtain energy dissipation, the authors introduced an additional
stabilization term. Similar stabilizing strategies can be found in [28,29]. Although
this technique enforces a discrete energy law, it also introduces an extra consistency
error. A major difficulty in proving optimal convergence error rates of a numerical
scheme for the CHNS system arises from the nonlinear potential function. A widely
used regularization technique is to truncate the potential and to extend it with a qua-
dratic growth [4,27,17]. An important objective of our work is to obtain a rigorous
∗Department of Mathematics, Purdue University, 150 North University Street, West Lafayette,
Indiana 47907 (liu3373@purdue.edu).
†Department of Numerical Analysis and Scientific Computing, Simula Research Laboratory, Oslo
0164, Norway (rami@simula.no).
‡Department of Computational Applied Mathematics and Operations Research, Rice Univer-
sity, 6100 Main Street, Houston, Texas 77005 (riviere@rice.edu). Partially supported by NSF-DMS
2111459.
1
arXiv:2210.05625v1 [math.NA] 11 Oct 2022
2CHEN LIU, RAMI MASRI, AND BEATRICE RIVIERE
convergence analysis of the scheme without modifying and regularizing the poten-
tial function. To the best of our knowledge, the theoretical analysis of a decoupled
splitting scheme in conjunction with interior penalty dG discretization without any
regularization on the potential function is not available in literature.
The main contribution of this work is the stability and error analysis of a dG
discretization of a splitting scheme for the CHNS model. We prove the energy stabil-
ity, the 𝐿∞stability of the order parameter, and we derive the optimal a priori error
bounds in both the broken gradient norm and the 𝐿2norm. Our analysis is novel and
general in sense that: (i) we successfully avoid using any artificial stabilizing terms
(which introduce an extra consistency error) when discretizing the CHNS system,
and (ii) no regularization (truncation and extension) assumptions on the potential
function are needed for the analysis. The proofs are technical and rely on induction
arguments. A priori error bounds are valid for convex domains because the conver-
gence analysis utilizes dual problems. Our arguments can be extended to analyze
splitting algorithms for other type of phase-field models.
The outline of this paper is as follows. In Section 2, the CHNS mathematical
model is presented. In Section 3, we introduce the fully discrete numerical scheme.
The unique solvability of the scheme is proved in Section 4. We show that our scheme
is energy stable in Section 5, and we derive error estimates in Section 6. Numerical
experiments validating our theoretical results are presented in Section 7. Concluding
remarks follow.
2. Model problem. Let Ω⊂R𝑑(𝑑=2 or 3) be an open bounded polyhedral
domain and let 𝒏denote the unit outward normal to the boundary 𝜕Ω. In the context
of incompressible immiscible two-phase flows, we introduce a scalar field order param-
eter as a phase indicator, which is defined as the difference between mass fractions.
The unknown variables in the CHNS system are the order parameter 𝑐, the chemical
potential 𝜇, the velocity 𝒖, and the pressure 𝑝, satisfying:
𝜕𝑡𝑐−Δ𝜇+∇· (𝑐𝒖)=0 in (0, 𝑇] ×Ω,(2.1a)
𝜇= Φ0(𝑐)−𝜅Δ𝑐in (0, 𝑇] ×Ω,(2.1b)
𝜕𝑡𝒖+𝒖·∇𝒖−𝜇sΔ𝒖=−∇𝑝−𝑐∇𝜇in (0, 𝑇] ×Ω,(2.1c)
∇·𝒖=0 in (0, 𝑇] × Ω.(2.1d)
The parameter 𝜅and shear viscosity 𝜇sare positive constants. The Ginzburg–Landau
potential function Φis defined by:
(2.2) Φ(𝑐)=1
4(1−𝑐)2(1+𝑐)2.
This polynomial potential can be decomposed into the sum of a convex part Φ+and
a concave part Φ−. We have:
Φ=Φ++Φ−,where Φ+=1
4(1+𝑐4)and Φ−=−1
2𝑐2.
We supplement our model problem (2.1) with the following initial and boundary
conditions:
𝑐=𝑐0,𝒖=𝒖0on {0} × Ω,(2.3a)
∇𝑐·𝒏=0,∇𝜇·𝒏=0,𝒖=0on (0, 𝑇] × 𝜕Ω.(2.3b)
NUMERICAL ANALYSIS OF A SPLITTING SCHEME FOR CHNS SYSTEM 3
Let 𝑐0denote the average of the initial order parameter. The model problem (2.1)
satisfies the global mass conservation property:
(2.4) 1
|Ω|∫Ω
𝑐=1
|Ω|∫Ω
𝑐0=𝑐0,
as well as the energy dissipation property [30,12]. Let 𝐹denote the total energy of
the system.
(2.5) 𝐹(𝑐, 𝒖)=∫Ω
1
2|𝒖|2+∫ΩΦ(𝑐) + 𝜅
2|∇𝑐|2,d
d𝑡𝐹(𝑐, 𝒖) ≤ 0.
We end this section by briefly stating the functional setting used throughout the
paper. For a given real number 𝑝≥1, on a domain 𝒪 ∈ R𝑑, where 𝑑=2 or 3, the
standard notation for the 𝐿𝑝(𝒪) spaces is employed. Let (·,·)𝒪denote the 𝐿2inner
product over 𝒪. We also define
𝐿2
0(𝒪) ={𝜔∈𝐿2(𝒪) :(𝜔,1)𝒪=0}.
Let 𝐷𝜶denote the weak 𝜶-th partial derivative with multi-index 𝜶. For a given
integer 𝑚≥0, the Sobolev space 𝑊𝑚,𝑝 (𝒪) is defined by
𝑊𝑚,𝑝 (𝒪) ={𝜔∈𝐿𝑝(𝒪) :𝐷𝜶𝜔∈𝐿𝑝(𝒪),∀|𝜶| ≤ 𝑚}.
The usual Sobolev semi-norm | · |𝑊𝑚 ,𝑝 (𝒪) and norm k · k𝑊𝑚,𝑝 (𝒪) are employed. We
introduce the space 𝐻𝑚(𝒪) =𝑊𝑚,2(𝒪) with the associated semi-norm | · |𝐻𝑚(𝒪) =
|·|𝑊𝑚,2(𝒪) and norm k · k𝐻𝑚(𝒪) =k · k𝑊𝑚,2(𝒪). For convenience, we use (·,·) and k · k
to denote the 𝐿2inner product and the 𝐿2norm, when 𝒪is the whole computational
domain.
3. Scheme. Let 𝒯ℎ={𝐸𝑖}be a family of conforming nondegenerate (regular)
quasi-uniform meshes of the computational domain Ωwith the maximum element
diameter ℎ. The mesh consists of simplices or of parallelepiped (parallelograms for
𝑑=2). Let Γℎdenote the set of interior faces. For each interior face 𝑒∈Γℎshared
by elements 𝐸𝑖−and 𝐸𝑖+, with 𝑖−<𝑖+, we define a unit normal vector 𝒏𝑒that points
from 𝐸𝑖−into 𝐸𝑖+. For a boundary face 𝑒, i. e., 𝑒=𝜕𝐸𝑖−∩𝜕Ω, the normal 𝒏𝑒is taken
to be the unit outward vector to 𝜕Ω. We also denote by 𝒏𝐸the unit normal vector
outward to the element 𝐸. We introduce the broken Sobolev spaces, 𝑠≥1,
𝐻𝑠(𝒯ℎ)=𝜔∈𝐿2(Ω):∀𝐸∈ 𝒯ℎ,𝜔|𝐸∈𝐻𝑠(𝐸).
The average and jump operators of any scalar function 𝜔∈𝐻𝑠(𝒯ℎ)is defined for each
interior face 𝑒∈Γℎby
{𝜔}|𝑒=1
2𝜔|𝐸𝑖−+1
2𝜔|𝐸𝑖+,[𝜔]|𝑒=𝜔|𝐸𝑖−−𝜔|𝐸𝑖+, 𝑒 =𝜕𝐸𝑖−∩𝜕𝐸𝑖+.
If 𝑒belongs to the boundary 𝜕Ω, the jump and average of 𝜔coincide with its trace
on 𝑒. The related definitions of any vector quantity in 𝐻𝑠(𝒯ℎ)𝑑are similar [26]. Fix
an integer 𝑘≥1 and denote by P𝑘(𝐸)the set of all polynomials of degree at most 𝑘
on an element 𝐸. Define the following discontinuous polynomial spaces for simplicial
meshes:
𝑀𝑘
ℎ=𝜔ℎ∈𝐿2(Ω):∀𝐸∈ 𝒯ℎ,𝜔ℎ|𝐸∈P𝑘(𝐸),
𝑀𝑘
ℎ0=𝜔ℎ∈𝑀𝑘
ℎ:(𝜔ℎ,1)=0,
X𝑘
ℎ=𝜽ℎ∈𝐿2(Ω)𝑑:∀𝐸∈ 𝒯ℎ,𝜽ℎ|𝐸∈P𝑘(𝐸)𝑑.
4CHEN LIU, RAMI MASRI, AND BEATRICE RIVIERE
For meshes with parallelograms or parallelepipeds, the space Q𝑘(𝐸), namely the space
of tensor product polynomials of degree at most 𝑘on an element 𝐸, is used instead of
P𝑘(𝐸)in the above definitions. We now present the dG pressure projection algorithm
for solving (2.1) with initial and boundary conditions (2.3). Uniformly partition [0, 𝑇]
into 𝑁Tintervals with length equal to 𝜏and for any 1 ≤𝑛≤𝑁Tlet 𝑡𝑛=𝑛𝜏and
let 𝛿𝜏be the temporal backward finite difference operator 𝛿𝜏𝑐𝑛
ℎ=(𝑐𝑛
ℎ−𝑐𝑛−1
ℎ)/𝜏. The
scheme consists of four sequential steps.
Given (𝑐𝑛−1
ℎ,𝒖𝑛−1
ℎ) ∈ 𝑀𝑘
ℎ×X𝑘
ℎ, compute (𝑐𝑛
ℎ,𝜇𝑛
ℎ) ∈ 𝑀𝑘
ℎ×𝑀𝑘
ℎ, such that for all 𝜒ℎ∈𝑀𝑘
ℎ
and for all 𝜑ℎ∈𝑀𝑘
ℎ
(𝛿𝜏𝑐𝑛
ℎ,𝜒ℎ)+ 𝑎diff (𝜇𝑛
ℎ,𝜒ℎ)+ 𝑎adv (𝑐𝑛−1
ℎ,𝒖𝑛−1
ℎ,𝜒ℎ)=0,(3.1) Φ+0(𝑐𝑛
ℎ)+ Φ−0(𝑐𝑛−1
ℎ),𝜑ℎ+𝜅𝑎diff (𝑐𝑛
ℎ,𝜑ℎ)−(𝜇𝑛
ℎ,𝜑ℎ)=0.(3.2)
Second, given (𝑐𝑛−1
ℎ,𝜇𝑛
ℎ,𝒖𝑛−1
ℎ, 𝑝𝑛−1
ℎ) ∈ 𝑀𝑘
ℎ×𝑀𝑘
ℎ×X𝑘
ℎ×𝑀𝑘−1
ℎ, compute 𝒗𝑛
ℎ∈X𝑘
ℎ, such
that for all 𝜽ℎ∈X𝑘
ℎ
(3.3) 1
𝜏(𝒗𝑛
ℎ−𝒖𝑛−1
ℎ,𝜽ℎ)+ 𝑎𝒞(𝒖𝑛−1
ℎ,𝒖𝑛−1
ℎ,𝒗𝑛
ℎ,𝜽ℎ)+ 𝜇s𝑎𝒟(𝒗𝑛
ℎ,𝜽ℎ)
=𝑏𝒫(𝜽ℎ, 𝑝𝑛−1
ℎ) + 𝑏ℐ(𝑐𝑛−1
ℎ,𝜇𝑛
ℎ,𝜽ℎ).
Next, given 𝒗𝑛
ℎ∈X𝑘
ℎ, compute 𝜙𝑛
ℎ∈𝑀𝑘−1
ℎ0, such that for all 𝜑ℎ∈𝑀𝑘−1
ℎ0
𝑎diff (𝜙𝑛
ℎ,𝜑ℎ)=−1
𝜏𝑏𝒫(𝒗𝑛
ℎ,𝜑ℎ).(3.4)
Finally, given (𝒗𝑛
ℎ, 𝑝𝑛−1
ℎ,𝜙𝑛
ℎ) ∈ X𝑘
ℎ×𝑀𝑘−1
ℎ×𝑀𝑘−1
ℎ0, compute (𝒖𝑛
ℎ, 𝑝𝑛
ℎ) ∈ X𝑘
ℎ×𝑀𝑘−1
ℎ,
such that for all 𝜒ℎ∈𝑀𝑘−1
ℎand for all 𝜽ℎ∈X𝑘
ℎ
(𝑝𝑛
ℎ,𝜒ℎ)=(𝑝𝑛−1
ℎ,𝜒ℎ)+(𝜙𝑛
ℎ,𝜒ℎ)− 𝜎𝜒𝜇s𝑏𝒫(𝒗𝑛
ℎ,𝜒ℎ),(3.5)
(𝒖𝑛
ℎ,𝜽ℎ)=(𝒗𝑛
ℎ,𝜽ℎ) + 𝜏𝑏𝒫(𝜽ℎ,𝜙𝑛
ℎ).(3.6)
For the approximation of the initial values, let 𝒖0
ℎbe the 𝐿2projection of 𝒖0and let
𝑐0
ℎbe the elliptic projection of 𝑐0, namely 𝑐0
ℎsatisfies
(3.7) 𝑎diff (𝑐0
ℎ−𝑐0,𝜒ℎ)=0,∀𝜒ℎ∈𝑀𝑘
ℎ,with constraint (𝑐0
ℎ−𝑐0,1)=0.
In addition, we set 𝑝0
ℎ=𝜙0
ℎ=0 and 𝒗0
ℎ=𝒖0
ℎ. The parameter 𝜎𝜒is a (user-specified)
positive number that can be chosen between 0 and 1/(4𝑑).
The forms 𝑎diff and 𝑎𝒟are the SIPG discretizations of the scalar and vector
Laplace operator, −Δ𝜔and −Δ𝒗, respectively. Let ˜
𝜎≥1,𝜎≥1 be given penalty
parameters. We define
𝑎diff (𝜔,𝜒)=Õ
𝐸∈𝒯ℎ∫𝐸
∇𝜔·∇𝜒−Õ
𝑒∈Γℎ∫𝑒{∇𝜔·𝒏𝑒}[𝜒](3.8)
−Õ
𝑒∈Γℎ∫𝑒{∇𝜒·𝒏𝑒}[𝜔] + ˜
𝜎
ℎÕ
𝑒∈Γℎ∫𝑒[𝜔][𝜒],∀𝜔,𝜒∈𝐻2(𝒯ℎ),
𝑎𝒟(𝒗,𝜽)=Õ
𝐸∈𝒯ℎ∫𝐸
∇𝒗:∇𝜽−Õ
𝑒∈Γℎ∪𝜕Ω∫𝑒{∇𝒗 𝒏𝑒} · [𝜽](3.9)
−Õ
𝑒∈Γℎ∪𝜕Ω∫𝑒{∇𝜽𝒏𝑒} · [𝒗]+ 𝜎
ℎÕ
𝑒∈Γℎ∪𝜕Ω∫𝑒[𝒗]·[𝜽],∀𝒗,𝜽∈𝐻2(𝒯ℎ)𝑑.
NUMERICAL ANALYSIS OF A SPLITTING SCHEME FOR CHNS SYSTEM 5
The dG form 𝑎𝒞:𝐻2(𝒯ℎ)𝑑×𝐻2(𝒯ℎ)𝑑×𝐻2(𝒯ℎ)𝑑×𝐻2(𝒯ℎ)𝑑→Rof the convection term
𝒗·∇𝒗is
𝑎𝒞(𝒘,𝒗,𝒛,𝜽)=Õ
𝐸∈𝒯ℎ∫𝐸(𝒗·∇𝒛) · 𝜽+∫𝜕𝐸𝒘
−|{𝒗} · 𝒏𝐸| (𝒛int −𝒛ext) · 𝜽int
(3.10)
+1
2Õ
𝐸∈𝒯ℎ∫𝐸(∇·𝒗)𝒛·𝜽−1
2Õ
𝑒∈Γℎ∪𝜕Ω∫𝑒[𝒗·𝒏𝑒]{𝒛·𝜽}.
Here, the set 𝜕𝐸𝒘
−is the inflow part of 𝜕𝐸, defined by 𝜕𝐸𝒘
−=𝒙∈𝜕𝐸:{𝒘(𝒙)} · 𝒏𝐸<
0, and the superscript int (resp. ext) refers to the trace of the function on a face of
𝐸coming from the interior of 𝐸(resp. coming from the exterior of 𝐸on that face).
In addition, if the face lies on the boundary of the domain, we take the exterior trace
to be zero. The discretization of the linear advection term ∇· (𝑐𝒗)is done with the
dG form 𝑎adv :𝐻2(𝒯ℎ) × 𝐻2(𝒯ℎ)𝑑×𝐻2(𝒯ℎ) → R:
(3.11) 𝑎adv(𝑐, 𝒗,𝜒)=−Õ
𝐸∈𝒯ℎ∫𝐸
𝑐𝒗·∇𝜒+Õ
𝑒∈Γℎ∫𝑒{𝑐}{𝒗·𝒏𝑒}[𝜒].
The dG form 𝑏ℐ:𝐻2(𝒯ℎ)×𝐻2(𝒯ℎ)×𝐻2(𝒯ℎ)𝑑→Rof the interface term −𝑐∇𝜇is equal
to 𝑎adv with switched arguments:
𝑏ℐ(𝑐, 𝜇,𝜽)=𝑎adv(𝑐, 𝜽,𝜇).(3.12)
Finally, for the discretization of the gradient and divergence terms, such as −∇𝑝,
−∇𝜙, and ∇·𝒗, we introduce the dG bilinear form 𝑏𝒫:𝐻2(𝒯ℎ)𝑑×𝐻1(𝒯ℎ) → R:
𝑏𝒫(𝜽, 𝑝)=Õ
𝐸∈𝒯ℎ∫𝐸
𝑝∇·𝜽−Õ
𝑒∈Γℎ∪𝜕Ω∫𝑒{𝑝}[𝜽·𝒏𝑒].(3.13)
With Green’s theorem, an equivalent expression for 𝑏𝒫is:
𝑏𝒫(𝜽, 𝑝)=−Õ
𝐸∈𝒯ℎ∫𝐸
𝜽·∇𝑝+Õ
𝑒∈Γℎ∫𝑒{𝜽·𝒏𝑒}[𝑝].(3.14)
The broken space 𝐻1(𝒯ℎ)and discrete space 𝑀𝑘
ℎare equiped with the semi-norm
|·|DG.
|𝜔|2
DG =Õ
𝐸∈𝒯ℎk∇𝜔k2
𝐿2(𝐸)+˜
𝜎
ℎÕ
𝑒∈Γℎk[𝜔]k2
𝐿2(𝑒),∀𝜔∈𝐻1(𝒯ℎ).
Note, | · |DG is a norm on 𝐻1(𝒯ℎ) ∩ 𝐿2
0(Ω). The vector space 𝐻1(𝒯ℎ)𝑑is equiped with
the following norm:
k𝒗k2
DG =Õ
𝐸∈𝒯ℎk∇𝒗k2
𝐿2(𝐸)+𝜎
ℎÕ
𝑒∈Γℎ∪𝜕Ωk[𝒗]k2
𝐿2(𝑒),∀𝒗∈𝐻1(𝒯ℎ)𝑑.
We now recall several properties satisfied by the dG forms. The forms 𝑎diff and 𝑎𝒟
are coercive. There exist ˜
𝜎0and 𝜎0such that for all ˜
𝜎≥˜
𝜎0and 𝜎≥𝜎0, there exist
𝐾𝛼>0 and 𝐾𝒟>0 independent of ℎsuch that
𝐾𝛼|𝜔ℎ|2
DG ≤𝑎diff (𝜔ℎ,𝜔ℎ),∀𝜔ℎ∈𝑀𝑘
ℎ,(3.15)
𝐾𝒟k𝒗ℎk2
DG ≤𝑎𝒟(𝒗ℎ,𝒗ℎ),∀𝒗ℎ∈X𝑘
ℎ.(3.16)
摘要：
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CONVERGENCEOFADECOUPLEDSPLITTINGSCHEMEFORTHECAHN{HILLIARD{NAVIER{STOKESSYSTEMCHENLIU,RAMIMASRIy,ANDBEATRICERIVIEREzAbstract.Thispaperisdevotedtotheanalysisofanenergy-stablediscontinuousGalerkinalgo-rithmforsolvingtheCahn{Hilliard{Navier{Stokesequationswithinadecoupledsplittingframework.Weshowthatth...
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分类：图书资源
价格：10玖币
属性：43 页
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时间：2025-04-24
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