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.

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

2|𝒖|2+∫ΩΦ(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

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)

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)

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CONVERGENCEOFADECOUPLEDSPLITTINGSCHEMEFORTHECAHN{HILLIARD{NAVIER{STOKESSYSTEMCHENLIU,RAMIMASRIy,ANDBEATRICERIVIEREzAbstract.Thispaperisdevotedtotheanalysisofanenergy-stablediscontinuousGalerkinalgo-rithmforsolvingtheCahn{Hilliard{Navier{Stokesequationswithinadecoupledsplittingframework.Weshowthatth...

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Convergence of a Decoupled Splitting Scheme for the CHNS System.pdf

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Convergence of a Decoupled Splitting Scheme for the CHNS System

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