Part-II: Splitting Methods for IncompressibleFlows

Jie ShenDepartment of Mathematics

Purdue University

IMS, National Singapore University, July 29-30, 2009

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Outline

• A review of projection-type schemes

– Pressure-correction method– Velocity-correction method

• Consistent splitting method

• Role of inf-sup conditions

• Concluding remarks

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Navier-Stokes Equations(unsteady, viscous and incompressible)

∂u

∂t− ν∆u + (u · ∇)u +∇p = f , in Q = Ω× (0, T ],

divu = 0, in Q, u|t=0 = u0,

with appropriate boundary conditions:

u|Γ1 = 0, (no-slip);

u · n|Γ2 = 0, (σ · n)τ |Γ2 = 0 (free-slip);

σ · n|Γ3 = g or (−ν∇u + pI) · n|Γ3 = g (open boundary);

where σ = −ν(∇u +∇uT ) + pI.

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Coupled approaches

A generalized Stokes (or linearized Navier-Stokes) problem

1∆t

un+1 − ν∆un+1 +∇pn+1 = hn+1,

divun+1 = 0,

needs to be solved at each time step !

• Not as easily solvable as a Poisson equation (but could bemade equally efficient in many situations with a sophisticatedmultigrid method);

• Inf-sup condition is needed.

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Decoupled approach: Splitting schemes

Pioneer work: The projection method, proposed by Chorin andTemam in late 60’s, decouples the computation of pressure andvelocity — very efficient but the accuracy is not satisfactory.

Countless modifications/implementations have been pro-posed/performed in the last 40 years...

A critical issue: Splitting error (very hard to remove since thepressure is not a dynamic variable) — a loss of accuracy for thepressure and vorticity.

Objective: Design splitting schemes with a loss of accuracy aslittle as possible !

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Notations:

• Dun+1

∆t = 1∆t(βqu

n+1 −∑q−1j=0 βju

n−j): q-th order BDF approxi-mation to ut(·, tn+1).

• p∗,n+1: (q − 1)-th order extrapolation for p(·, tn+1).

q = 1 : Dvk+1 = vk+1 − vk, p∗,k+1 = 0;

q = 2 : Dvk+1 =32vk+1 − 2vk +

12vk−1, p∗,k+1 = pk,

q = 3 : Dvk+1 =16(11vk+1 − 18vk + 9vk−1 − 2vk−2), p∗,k+1 = 2pk − pk−1.

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Pressure-correction Method

1

∆t(βqun+1 −

q−1∑j=0

βjun−j)− ν∆un+1 +∇p∗,n+1 = h(tn+1),

un+1|∂Ω = 0,βq(un+1 − un+1)

∆t+∇(pn+1 − p∗,n+1) = 0,

divun+1 = 0, un+1 · n|∂Ω = 0.The second step reduces to a Pressure Poisson equation — onlya sequence of Poisson-type equations need to be solved at eachtime step.

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• The second step is nothing but the Helmholtz decomposition ofun+1, i.e., un+1 is the orthogonal projection of un+1 onto

H = u ∈ L2(Ω) : ∇·u = 0, u · n|∂Ω = 0.

• The pressure approx. satisfies an artificial Neumann B.C.

q = 1 :∂pn+1

∂n|∂Ω = 0; q = 2 :

∂pn+1

∂n|∂Ω =

∂p0

∂n|∂Ω

— numerical boundary layer and a loss of accuracy.

• Error estimates: (proved by many in various forms)

‖un − u(tn)‖0 + ∆t(‖un − u(tn)‖1 + ‖pn − p(tn)‖0) ≤ c∆t2.

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D. PRESSURE ERROR: TIME STEP=0.05, SCHEME 1.3-1.4.

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E. PRESSURE ERROR: TIME STEP=0.025, SCHEME 1.3-1.4.

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x 10-4

B. VELOCITY ERROR: TIME STEP=0.025, SCHEME 1.3-1.4.

Figure 1: Plots of the pressure and velocity errors

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Reducing the effect of the artificial Neumannpressure boundary condition

Minev, Timmermans and Van de Vosse (1996):1

∆t(βqun+1 −

q−1∑j=0

βjun−j)− ν∆un+1 +∇p∗,n+1 = h(tn+1),

un+1|∂Ω = 0;βq(un+1 − un+1)

∆t+∇φn+1 = 0,

divun+1 = 0, un+1 · n|∂Ω = 0;

Eliminating un+1 from above, one finds that the correct way to

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update pn+1 is:

pn+1 = φn+1 + p∗,n+1 − νdivun+1.

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Stability and Error Analysis

(Guermond & S., Math. Comp. ’04)

A striking fact is that a priori the following estimate holds (evenfor q = 1):

‖divun‖0 ≤ c∆t3/2, q = 1, 2,

which leads to

Improved error estimates:

‖un − u(tn)‖1 + ‖pn − p(tn)‖0 ≤ c∆t32.

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Figure 2: Plots of the pressure errors: standard form and rota-tional form

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Figure 3: Convergence rates of 2nd-order pressure-correctionschemes: standard form and rotational form with spectral method

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The large errors at the corners are due to the domain singular-ity.

Consider Ω = (0, 2π)× (−1, 1) with periodic condition in x.

Figure 4: Pressure error in a channel: (L) standard; (R) rotational

Second-order convergence for the pressure is proved for this

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case by Brown, Cortez & Minion (2001).

Remark: High-order (q ≥ 3) versions are only stable if ∆t ≥ch2 or ∆t ≥ cN−3 (for linear problems) !

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Velocity-correction method

Idea: “switch” the roles of pressure and velocity in thepressure-correction scheme.

1∆t

(βqun+1 −q−1∑j=0

βjun−j)− ν∆u∗,n+1 +∇pn+1 = h(tn+1),

divun+1 = 0, un+1 · n|∂Ω = 0,βq(un+1 − un+1)

∆t− ν∆(un+1 − u∗,n+1) = 0,

un+1|∂Ω = 0.

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The difficulty of computing ∆u∗,n+1 (in the finite-element cases)can be avoided by reformulating the first step, e.g., for q = 2, wehave:

3un+1 − 7un + 5un−1 − un−2

2∆t+∇(pn+1 − pn) = h(tn+1)− h(tn),

divun+1 = 0, un+1 · n|∂Ω = 0.

The second step is still the same:3(un+1 − un+1)

2∆t− ν∆(un+1 − un) = 0,

un+1|∂Ω = 0.

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• The first step is still a projection step and can be implementedas a pressure-Poisson equation.

• As in the standard pressure-correction scheme, the accuracy ofthe scheme is affected by the nonphysical B.C. For q = 2, wehave

∆un+1 · n|Γ = ∆un · n|Γ = · · · = ∆u0 · n|Γ,which implies that

∂pn+1

∂n= (h(tn+1) + ν∆u0) · n|∂Ω.

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Stability and Error Analysis

(Guermond & S., SINUM ’03)

Stability: It is proved that the schemes (q = 1, 2) are uncondi-tionally stable despite the explicit treatment of the viscous term.

Error estimates

‖un − u(tn)‖1 + ‖un − u(tn)‖1 + ‖pn − p(tn)‖0 ≤ c∆t,(∆t

n∑k=1

(‖uk − u(tk)‖20 + ‖uk − u(tk)‖20)

)12

≤ c(∆t)2.

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How to reduce the effect of the nonphysical B.C.?

Idea: Use the alternative formulation of the NSE, i.e. replacethe explicit term −∆u∗,n+1 in the scheme by ∇×∇× u∗,n+1 !

Rational: Since ∆u∗,n+1 = ∇divu∗,n+1−∇×∇×u∗,n+1, thishas an implicitly effect in reducing divu∗,n+1.

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Rotational form of the velocity-correction scheme

1

∆t(βqun+1 −

q−1∑j=0

βjun−j) + ν∇×∇× u∗,n+1 +∇pn+1 = h(tn+1),

divun+1 = 0, un+1 · n|∂Ω = 0,

βq(un+1 − un+1)

∆t− ν∆un+1 − ν∇×∇× u∗,n+1 = 0,

un+1|∂Ω = 0.

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This scheme leads to a consistent Neumann B.C.

∂pn+1

∂n= (h(tn+1) + ν∆un+1) · n|∂Ω.

• As before, the term ∇ × ∇ × u∗,n+1 in the first step can beeliminated.

• Stability and Error Analysis: (Guermond & S., SINUM ’03)

The key is the following a priori estimate (q = 1, 2):

‖divun‖0 ≤ c∆t32.

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• Improved error estimates:

‖un − u(tn)‖1 + ‖pn − p(tn)‖0 ≤ c∆t32.

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Figure 5: Pressure errors of 2nd-order velocity-correctionschemes: standard form and rotational form

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Schemes of Orszag, Israeli & Deville (OID ’86) andKarniadakis, Israeli and Orszag (KIO ’91)

1∆t

(βqun+1? −

q−1∑j=0

βjun−j) +∇pn+1 = h(tn+1),

divun+1? = 0,

∂pn+1

∂n|∂Ω = [−ν∇×∇× u∗,n+1 + h(tn+1)] · n|∂Ω,

βq(un+1 − un+1? )

∆t− ν∆un+1 = 0, un+1|∂Ω = 0.

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• un? is purely an intermediate quantity, not an approximation ofu(tn).

• Second-order velocity derivatives are used as boundary condi-tions for the pressure — not easy to analyze and to implement.

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Reinterpretation of the schemes of OID and KIOConsider the change of variable

un+1 = un+1? − ∆t

βq∇×∇× u∗,n+1.

• The KIO scheme is exactly, in the spacial continuous case, thevelocity-correction scheme in the rotational form.

• A rigorous stability and convergence result for the schemes ofOID and KIO is now available.

• The variational reformulation allows the use of finite elements,and can be easily analyzed in the full discrete case.

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A New Class of Splitting Schemes

Key idea: Replace the divergence-free condition by apressure-Poisson equation in the weak form:∫

Ω

∇p · ∇q =∫

Ω

(f + ν∆u− u·∇u) · ∇q, ∀q ∈ H1(Ω).

Note the difference between the usual pressure Poisson equa-tion: ∫

Ω

∇p · ∇q =∫

Ω

(f − u·∇u) · ∇q, ∀q ∈ H1(Ω),

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Standard splitting schemesNow, let p∗,k+1 be a q-th order extrapolation for p(·, tk+1).

Duk+1

∆t− ν∆uk+1 +∇p?,k+1 = gk+1, uk+1|Γ = 0,

(∇pk+1,∇q) = (gk+1 + ν∆uk+1,∇q), ∀q ∈ H1(Ω).

An equivalent formulation:

To avoid computing ∆uk+1 explicitly in the second step, we canreformulate the second-step as

(∇(pk+1 − p∗,k+1),∇q) = (Duk+1

∆t,∇q), ∀q ∈ H1(Ω).

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• This is NOT a projection scheme, for the velocity approximationuk+1 is not divergence-free.

• As in a projection scheme, one only needs to solve a set ofHelmholtz-type equations for uk+1 and a Poisson equation (inthe weak form) for pk+1.

• We observe that ∂∂n(pk+1 − p∗,k+1)|∂Ω = 0 — an artificial B.C.

which will result in a loss of accuracy.

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Consistent splitting schemes

(Guermond & S., JCP ’03)

Replacing ∆uk+1 with −∇×∇×uk+1:

Duk+1

∆t− ν∆uk+1 +∇p?,k+1 = gk+1, uk+1|Γ = 0,

(∇pk+1,∇q) = (gk+1 − ν∇×∇×uk+1,∇q), ∀q ∈ H1(Ω).

A more implementational friendly version (See also Johnston

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and Liu ’04):Duk+1

∆t− ν∆uk+1 +∇p?,k+1 = gk+1, uk+1|Γ = 0,

(∇pk+1,∇q) = (gk+1,∇q)− ν∫∂Ω

n×∇×uk+1 · ∇qdγ, ∀q ∈ H1(Ω).

A even more implementational friendly version:

Duk+1

∆t− ν∆uk+1 +∇p?,k+1 = gk+1, uk+1|Γ = 0,

(∇ψk+1,∇q) = (Duk+1

∆t,∇q), ∀q ∈ H1(Ω),

pk+1 = ψk+1 + p?,k+1 − ν∇·uk+1.

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Standard formulation Consistent splitting scheme

Figure 6: Convergence rates: second-order new splitting scheme

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Standard formulation Consistent splitting scheme

Figure 7: Pressure error field at time t = 1 in a square.

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• For q = 1, we are able to show that for the consistent schemehas full first-order accuracy for the velocity in H1-norm and thepressure in L2-norm.

• For q = 2, we are only able to show that for the standardscheme is unconditionally stable and has the same accuracyas the standard version of the pressure-correction schemes.

• Numerical results indicate that for q = 2, the consistent schemeis unconditionally stable and has full second-order accuracy forthe velocity in H1-norm and the pressure in L2-norm.

• H1 regularity on the pressure is needed for this approach. Anice mathematical analysis of this formulation is given by Liu &

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Liu & Pego ’05.

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