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MAT-VET-F-20027

Examensarbete 15 hp June 2020

Simulation of Viscosity-Stratified Flow

Victor Carlsson Philip Isaac

Adina Persson

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Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Simulation of Viscosity-Stratified Flow

Victor Carlsson, Philip Isaac, Adina Persson

The aim of this project is to study the viscous Burgers' equation for the case where the viscosity is constant, but also when it contains a jump in viscosity. In the first case where the viscosity is constant, Burgers' is simply solved on a singular domain. For the case with jump in viscosity, Burgers' is solved on multiple domains with different viscosity. The different domains are then connected by applying inner boundary conditions at an interface in order to produce a singular solution.

The inner boundary conditions are imposed using three different methods;

simultaneous approximation term (SAT), projection and hybrid method, where the hybrid method is a combination of both the SAT and projection method. These methods are used in combination with a stable and high-order accurate summation by parts (SBP) finite difference approximation in MATLAB. The three methods are then compared to each other with respect to the least square error and the corresponding convergence rate to determine which method is the most preferable to use.

The methods resulting in the highest convergence rates are the projection and the hybrid methods. These methods manage to live up to the expected convergence rates for all operators with different orders of accuracy and are therefore both good methods to use. However, the best method to use is the projection method since it is much simpler to implement than the two other methods but still achieves just as good convergence rates as the hybrid method.

Examinator: Martin Sjödin Ämnesgranskare: Malin Göteman Handledare: Vidar Stiernström

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Populärvetenskaplig sammanfattning

Det huvudsakliga syftet med detta projekt var att undersöka och simulera viskösa Burgers ekvation då viskositeten dels var konstant, dels innehöll ett hopp. Flöden som till exempel består av flera fluider med olika viskositeter eller flöden där viskositeten varierar drastitskt över korta längdskalor kallas för viskositet-stratifierade flöden och är exempel på flöden med hopp i viskositeten.

Burgers ekvation är en förenklad modell av Navier-Stokes ekvationerna, som är de mest mest generella ekvationerna för att beskriva flöden. Då Burgers ekvation fortfarande lyckas introducera många av de svårigheter som uppstår vid numeriska lösningar till flödesproblem kan det vara fördelaktigt att först studera detta modellproblem innan man går vidare till de mer komplexa Navier-Stokes ekvationerna.

I första fallet, då viskositeten var konstant, löstes Burgers ekvation numeriskt på en domän. För fallet med ett hopp i viskositeten löstes Burgers ekvation numeriskt på två domäner med olika viskositet. De två olika domänerna kopplades samman genom att införa inre randvillkor över ett så kallat interface, där de två domänerna möts. De inre randvillkoren infördes med hjälp av tre olika metoder; SAT (Simultane- ous Approximation Term), projektion och en SAT-projektion hybridmetod, som är en kombination av de båda föregående metoderna. Dessa tre metoderna jämfördes sedan med varandra med avseende på felet i den numeriska lösningen och dess motsvarande konvergenshastighet för att kunna bestämma vilken av metoderna som var mest fördelaktig att använda.

Metoderna som påvisade bäst resultat var projektionsmetoden och hybridmetoden. Dessa metoder fick den eftersträvade konvergensen för alla operatorer av olika noggrannhetsordningar. SAT metoden kunde för vissa antal punkter och noggrannhetsordningar ge en lägre konvergenshastighet än den väntade.

Eftersom projektionsmetoden är så pass mycket enklare att implementera än både SAT- och hybridme- toden används med fördel denna metod.

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Contents

1 Introduction 5

2 Aim 5

3 Theory 6

3.1 The viscous Burgers’ equation . . . . 6

3.2 The SBP finite difference method . . . . 7

3.3 The energy method . . . . 7

3.4 Methods of imposing boundary conditions . . . . 7

3.4.1 SAT . . . . 8

3.4.2 Projection . . . . 8

3.4.3 Hybrid . . . . 8

4 Method 8 4.1 Burgers’ with constant coefficients . . . . 8

4.2 Burgers’ with jump in the coefficients . . . 12

4.2.1 Energy analysis . . . 12

4.2.2 SAT method . . . 14

4.2.3 Projection method . . . 15

4.2.4 Hybrid method . . . 16

5 Results 17 5.1 Solution . . . 17

5.2 Convergence test for the Burgers’ equation without jump in viscosity . . . 19

5.3 Convergence test for the Burgers’ equation with jump in viscosity . . . 20

6 Discussion 22

7 Conclusions 23

References 24

Appendices 25

A MATLAB code containing all methods of imposing inner boundary conditions 25

B SBP operators 27

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1 Introduction

Viscosity-stratified flow is an interesting form of fluid flow and is important for many applications within engineering. Some examples of where this kind of flow can be found are, the flow in the Earth’s outer core, the flow when a spacecraft re-enters the atmosphere (see figure1) and the flow of blood cells [8].

Thus, viscosity-stratified flows could for example be flows between two or more fluids with various viscosities or flows with drastically varying viscosity.

Figure 1: Example of viscosity stratification: Re-entry of spacecraft in the atmosphere. Source: Apollo Imagery, NASA, 1968

First introduced by Harry Bateman in 1915, the Burgers’ equation is a fundamental and hyperbolic partial differential equation (PDE), that is used among other areas within fluid mechanics [1]. The so- called viscous Burgers’ equation is the general form of Burgers’ equation. This equation is used to describe different types of flow and is a simplified model of the more complex Navier-Stokes equations.

Since the viscous Burgers’ equation provides a model with high enough complexity to illustrate a high amount of difficulties in non-linear equations, it is useful to first study this problem before moving on to Navier-Stokes.

There are several different methods for numerical calculations and in this project a high-order finite difference method (FDM) will be used to solve Burgers’ equation. These kind of methods are suitable for hyperbolic problems since it becomes possible to explicitly step in both time and space, avoiding having to solve a system of equations at each iteration. However, a common problem when handling high-order difference methods is obtaining stable boundary treatment. A high-order FDM that is well- proven to use for well-posed initial boundary value problems is the combination of summation by parts (SBP) operators and one of the following methods to impose boundary conditions; the simultaneous approximation term (SAT) or projection [5]. These methods will be explained in more detail in upcoming sections.

SBP is the discrete equivalent to integration by parts (IBP) and the SBP operators can briefly be ex- plained as finite difference operators, built to mimic the first and second derivatives in the continuous case. An important characteristic of the SBP operators is that they also mimic the energy in the contin- uous case [6]. Why this is of importance will be explained in coming sections.

2 Aim

The main focus of this project is to determine which method of imposing inner boundary conditions to the viscous Burgers’ equation is the best to use. This will be done both for the case where the viscosity is constant and for the case where the viscosity contains a jump. By using the three different methods; SAT,

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projection and a hybrid method, a stable and high-order accurate SBP finite difference approximation will be implemented in MATLAB. The three methods will then be compared with respect to the least square error (L2-error) and the corresponding convergence rate.

3 Theory

3.1 The viscous Burgers’ equation

The viscous Burgers’ equation on a domainΩ is given by

∂u( ¯x, t)

∂t = −∇ · u2( ¯x, t) 2

!

+ ∇ · (µ( ¯x)∇u( ¯x, t)), ¯x ∈Ω, t ≥ 0,

u( ¯x, 0)= f ( ¯x), ¯x ∈Ω, (1)

Lu( ¯x, t)= g(x), ¯x ∈ ∂Ω, t ≥ 0,

where the unknown field is referred to as u and the viscosity is referred to as µ( ¯x). The initial and boundary data is given by f ( ¯x) respectively g( ¯x) and Lu( ¯x, t) specifies the boundary conditions on the boundary ∂Ω.

For the case where the viscosity contains a jump at x= l on the one-dimensional domain Ω = [0, L] the problem can be written as

ut = −u2 2



x+ (µ(x)ux)x, x ∈Ω, t ≥ 0,

u(x, 0)= f (x), x ∈Ω, (2)

Lu(x, t)= g(x), x ∈∂Ω, t ≥ 0,

where the indices represents partial derivatives and

µ(x) = µ1, 0 ≤ x ≤ l

µ(x) = µ2, l ≤ x ≤ L . (3)

The problem can be solved by splittingΩ into two subdomains, Ω1= [0, l1] and2= [l2, L] and solving Burgers’ in both respectively:

u(1)t = u2 2



x+ (µ1ux)x

!(1)

, x ∈1, t ≥0, (4)

u(2)t = u2 2



x+ (µ2ux)x

!(2)

, x ∈2, t ≥0,

and connecting the solutions using inner Dirichlet and Neumann boundary conditions:

u(1)(l1, t) − u(2)(l2, t) = 0

| {z }

Dirichlet BC

, µ(1)u(1)x (l1, t) − µ(2)u(2)x (l2, t) = 0

| {z }

Neumann BC

. (5)

A singular solution can then be produced which contains a jump in viscosity at x= l.

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3.2 The SBP finite difference method

In this project, SBP operators are used to numerically calculate Burgers’ equation in MATLAB. SBP operators are finite difference stencils, that can be expressed as matrices consisting of a centered differ- ence scheme and boundary stencils. These SBP operators makes it possible to mimic the energy of the continuous problem in a semi-discrete setting. Some definitions are needed

xi = (i − 1)h, i= 1, 2...m, h= L

m −1, (6)

where a domain (x0 ≤ x ≤ xL) with length L is discretized into m equally spaced gridpoints and h is the gridspacing. A solution at grid point xi is given by vi and the solution vector is vT = [v1v2...vm]. We define the inner product for the discrete case as (v, w)H= vTHwwhere v, w ∈ Rmand H= HT > 0 with corresponding norm being kvk2H = vTHv. Using these defintions we can define the operators used.

Two difference operators, D1= H−1(Q+B2) and D2= H−1(−M−e1d1+endm), are used to approximate the first, respectively the second order derivatives ∂x and ∂x22 by using pth-order accurate narrow stencils.

Q is skew symmetric, thus Q + QT = 0 and M is symmetric positive definite, i.e. M = MT ≥ 0.

Furthermore, B = diag(−1, 0, ..., 0, 1), e1 = [1, 0..., 0]T and em = [0, ...0, 1]T. The finite difference approximations of the derivatives at both sides are defined by d1v  ux(x = xl) and dmv  ux(x= xr) [6].

3.3 The energy method

A general method to show stability for initial value problems (IVP) and boundary value problems (BVP) is called the energy method, which both can be used on the continuous PDE and in the discrete finite difference method. To simplify the discrete case, consider the semi-discrete problem where time is con- tinuous. This is a legitimate simplification since a stable semi-discretisation can be discretized in time by using for example a Runge-Kutta method, such that the completely discrete problem is stable as well [2] . In this project, SBP operators are used to achieve stable finite difference approximations [7, 4]. To prove stability the physical boundary conditions needs to be enforced. These are also necessary for showing consistency, which is done by using a stable and consistent FDM, which implies convergence.

In this project two ways of doing this will be presented; the SAT technique and the projection method [2].

In the continuous setting, the energy method consists of multiplying the PDE by u, adding the transpose and then using integration by parts. In the discrete setting, the energy method consists of multiplying with vTHand adding the transpose [2] .

In this project the well-posedness analysis is simplified by assuming homogenous boundary conditions and well-posedness is shown.

3.4 Methods of imposing boundary conditions

When the viscosity contains a jump the problem is divided by an interface into two domains, see equation (4). Regardless of which method is used to treat the interface, the outer boundary conditions are imposed using SAT. Only the method of imposing the inner boundary conditions in between the two domains is varied. The inner boundary conditions for the interface are applied using one of the three methods SAT, projection or hybrid.

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3.4.1 SAT

The purpose of the SAT technique is to weakly impose the boundary conditions. To do this the method uses an additional term, known as a penalty term, in the approximation of the PDE. After showing well- posedness using the energy method (as described in3.3) on a continuous PDE the energy method can be applied to the semi-discrete approximation of the PDE using SBP operators to discretize the domain and setting the boundary conditions using SAT. Then an energy estimate is obtained choosing penalty parameters so that the energy does not increase in time, showing stability [2].

3.4.2 Projection

In contrast to the SAT method, the projection method imposes the boundary conditions strongly. The method works by formulating a projection operator, P, which is used to project the solution onto the subspace where the boundary conditions are satisfied. The operator is defined as

P= I − H−1LT(LH−1LT)−1L, (7)

where I is the identity matrix and H is the inner product operator. L is the full boundary operator representing the analytical boundary conditions. Let g be the boundary data and v the numerical solution and we have

Lv= g.

Or in the homogeneous case

Lv= 0.

The equivalence P2 = P also applies together with HP = PTH. In this project the projection will only be used to apply the inner boundary condition which will be formulated similarly to homogeneous boundary conditions.

Similar to the SAT method, the projection method will interact with the SBP operators in such way that it will approximate the continuous boundary conditions. Applying the P operator will restrict the energy of the system and therefore be stable, which will be shown later in the report.

3.4.3 Hybrid

Using SAT or projection in combination with SBP are known ways of obtaining stable boundary treat- ment. In this project we will also investigate using a combination of both SAT and projection to impose boundary conditions, a SAT-projection hybrid, together with the SBP operators. The inner boundary condition will be divided between the methods, with the SAT applying a Neumann boundary condition and the projection method applying a Dirichlet boundary condition, see equation (4).

4 Method

4.1 Burgers’ with constant coefficients

Before investigating the Burgers’ equation in case of a jump in the viscosity, the simpler case with constant viscosity is studied. Here, the outer boundary conditions, which are the same in both the case with or without a jump in viscosity, are investigated as well as what the requirements of stability are.

The viscous Burgers’ equation is given by equation (2). In this case, where the viscosity is constant and

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the outer boundary conditions are Robin conditions the problem can be expressed as ut+ uux = µuxx, ¯x ∈ [0, L],

˜a0u(0, t) − µux(0, t) − g0(t)= 0, (8)

˜aLu(L, t) − µux(L, t) − gL(t)= 0, where

˜a0= u+ |u|

6 , ˜aL= u − |u|

6 .

To prove stability the first step is to perform an analysis of well-posedness for the continuous case using the energy method. Well-posedness is achieved if

d

dt||u||2 ≤ 0

Skew symmetric splitting is used to rewrite the Burgers’ equation to be able to perform the energy method. Using skew symmetric splitting and uux = u2

2



x = fx for equation (8) gives the following equation

ut+ αu2 2



x+ (1 − α) uux

|{z}

fx

= µuxx, (9)

where α ∈ R. Rewrite the equation using the inner product.

(u, ut)= −α (u, uux)

| {z }

=(uu,ux)

−(1 − α)(u, fx)+ (u, µuxx).

Then, by using that the following two expressions applies

(uu, ux)= (2 f, ux)= 2  f uL0− 2( fx, u), (u, fx)=

Z L 0

u fxdx= u f L0 Z L

0

uxf dx, leads to the following expression

(u, ut)+ (ut, u)

transpose|{z}

= −2α  f u0L− 2( fx, u) − (1 − α) u f L0+ (1 − α)(ux, f ) + (u, µuxx)+ (ut, u) =

= −2α f u − (1 − α)u f 0L

| {z }

BT

+2α( fx, u) + (1 − α)(ux, f ) + (u, µuxx)+ (ut, u).

The transpose is given by

(ut, u) = −2α(ux, f ) − (1 − α)( fx, u) + (µuxx, u), resulting in

d

dtkuk2= BT + ( fx, u)(3α − 1) + (ux, f )(−3α + 1) − 2µkuxk2, where the boundary terms (BT) are

BT =

−2α f u − (1 − α)u f + 2µ uux

|{z}

fx

L

0

.

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The condition for α is given by applying the requirement: (3α − 1)( fx, u) + (1 − 3α)(ux, f ) = 0, which leads to

3α − 1= 0 => α = 1 3. Inserting α into the boundary terms results in

BT =

"

4

3f u+ 2µ fx

#L 0

=

"

2

3u3+ 2µuux

#L 0

.

The energy estimate is then given by d

dtkuk2= 2µuux

u=u(L) 2 3u3

u=u(L)− 2µuux

u=u(0)+2 3u3

u=u(0)− 2µkuxk2. (10)

The Robin boundary conditions given in (8) can be written as ux(0)= ˜a0

µu(0), ux(L)= ˜aL

µu(L).

Inserting the Robin boundary conditions into the energy estimate results in d

dtkuk2= 2µuL

˜aL

µuL 2

3u3L− 2µu0˜a0 µu0+ 2

3u3L− 2µkuxk2 (11)

= u2L(2˜aL 2

3uL) − u20(−2˜a0+2

3u0) − 2µkuxk2.

Note that ˜a0 and ˜aL function as switches to assure all terms either equal zero or are negative. The boundary conditions are therefore well-posed.

Now, the energy method is applied for the semi-discrete case, where the time is kept continuous. Instead of using u which is the exact solution of equation (8), v which is an approximation of the solution is used. For the semi-discrete case Burgers’ equation with skew symmetric splitting is given by

vt = −αVD1v − (1 − α)

2 D1Vv+ µD2v+ SAT, (12)

where α = 13, which was calculated in the continuous case and D1 and D2 is given in3.2. To simplify the derivation, SAT will be removed and added back later. Inserting α gives

vt= −1

3V D1v −1

3D1Vv+ µD2v, (13)

where V is a diagonal matrix, with v on the diagonal.

Using the energy method on equation (12), the definitions of D1, D2and that V H= HV, results in vTHvt+ vTt HTv= −1

3vTHV D1v − 1

3vT(HV D1)Tv

| {z }

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1

3vTHD1Vv −1

3vT(HD1V)Tv

| {z }

∗∗

+ µ(vT(HD2+ (HD2)T)v

| {z }

−M−MT−2e1d1+2emdm

.

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Rewrite (*)

1

3vTHV D1v − 1

3vT(HV D1)Tv=

= −1 3(Vv)T

Q −1

2e1eT1 + 1 2emeTm

v − 1 3(Vv)T

Q −1

2e1eT1 +1 2emeTmT

v= (15)

= −1

3(Vv)T(−e1eT1 + emeTm)v −2

3(Vv)TQv=

= −1

3(v3m− v31) −2

3(Vv)TQv.

Rewrite (**)

1

3vTHD1Vv −1

3vT(HD1V)Tv=

= −1 3vT

Q −1

2e1eT1 +1 2emeTm

Vv −1 3vT

Q −1

2e1eT1 + 1 2emeTm

Vv= (16)

= −1

3vT(−e1eT1 + emeTm)Vv −2

3vTQVv=

= −1

3(v3m− v31) − 2

3vTQVv.

This results in the final expression for the energy estimate d

dtkvk2H= −1

3(v3m− v31) − 2

3(Vv)TQv −1

3(v3m− v31) − 2 3vTQVv

µ(−2vTMv+ 2vmdmv −2v1d1v)= (17)

= 2

3(v31− v3m) − 2µv1d1v+ 2µvmdmv −2µvTMv, where Q+ QT = 0 was used.

To derive stable boundary conditions SAT is added to the energy estimate. Stability is achieved if d

dt||v||2H ≤ 0.

The energy estimate is written as d

dtkvk2H= 2

3(v31− v3m) − 2µv1d1v+ 2µvmdmv+ 2vTHSAT

| {z }

2BT

− 2µvTMv

| {z }

≥0

, (18)

where SAT= SATl+ SATrand

SATl= H−1e1τl(˜a0v1µd1v − g0)

| {z }

Left outer boundary

,

SATr= H−1emτr(˜aLvmµdmv − gL)

| {z }

Right outer boundary

,

where τl, and τrare penalty parameters that must be tuned for stability.

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Let gl = gr= 0. We get

vTHSAT= vTH

H−1e1τl(˜a0v1µd1v)+ H−1rvmem+ σrµdTm)vm =

= τl˜a0v21τrµvmdmv+ τr˜aLv2mτrµvmdmv.

The boundary term (BT) is given by BT = 1

3(v31− v3m) − µv1d1v+ µvmdmv+ τl˜a0v21τlµv1d1v+ τrv3m+ σrµvTdTmvm, using that vTdmT = (dmv)T = dmvgives

BT = v211

3v1+ τl˜a0

− (1+ τl)µv1d1v+ v2m

 1

3vm+ τr˜aL

− (−1+ τl)µvmdmv.

By choosing the penalty parameters to be τl= −1 and τr= 1 all terms will be negative or equal to zero.

The scheme is therefore energy stable.

4.2 Burgers’ with jump in the coefficients

In case of a jump in the coefficients the problem can be described as in equation (1) whereΩ = [0, L] and a jump in the viscosity (µ) occurs at x= l. Hence it follows that (3) applies, with µ1= µ(1)and µ2= µ(2). As previously done in the case with no jump in viscosity, skew symmetric splitting is applied. Due to discontinuity in µ, the problem is divided into two domains, u(1) and u(2) which are both solutions to equation (1) for each respective µ. The two domains are defined as ¯x(1) ∈ [0, l1] and ¯x(2) ∈ [l2, L] which together form the entire domainΩ. The outer boundary conditions are the same as in (8) which are:

˜a0u(1)(0, t) − µ(1)u(1)x (0, t) − g0(t)= 0, ˜aLu(2)(L, t) − µ(2)u(2)x (L, t) − gL(t)= 0, (19) where

˜a0 = u(1)+ u

(1)

6 , ˜aL= u(2) u

(2)

6 .

In order to obtain a global solution on Ω the following continuity conditions must be satisfied on the inner boundary

u(1)(l1, t) − u(2)(l2, t) = 0

| {z }

Dirichlet BC

, µ(1)u(1)x (l1, t) − µ(2)u(2)x (l2, t) = 0

| {z }

Neumann BC

. (20)

4.2.1 Energy analysis

Following the same steps as in4.1and adding the equations the energy is given by d

dt kuk2

!(1)

+ d dt kuk2

!(2)

= 2µuux

u=u(0) 2 3u3

u=u(0)− 2µuux

u=u(l

1)+ 2 3u3

u=u(l

1)− 2µkuxk2

!(1)

(21) + 2µuux

u=u(l

2) 2 3u3

u=u(l

2)− 2µuux

u=u(L)+2 3u3

u=u(L)− 2µkuxk2

!(2)

.

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Separating the inner boundary and the outer boundary terms results in d

dt kuk2

!(1)

+ d dt kuk2

!(2)

= 2µuux

u=u(0) 2 3u3

u=u(0)

!(1)

+ − 2µuux

u=u(L)+ 2 3u3

u=u(L)

!(2)

| {z }

Outer boundaries

+ − 2µuux

u=u(l

1)+ 2 3u3

u=u(l

1)

!(1)

+ 2µuux

u=u(l

2)+ 2 3u3

u=u(l

2)

!(2)

| {z }

Inner boundaries

+ − 2µkuxk2

!(1)

+ − 2µkuxk2

!(2)

.

Applying the inner boundary conditions given by (20) shows that the terms cancel each other out. The given inner boundary conditions are thus necessary for energy conservation across the interface. The well-posedness of equation (1) therefore only depends on the outer boundary terms. The energy of the equation is

d dt kuk2

!(1)

+ d dt kuk2

!(2)

= 2u20˜a0 2

3u30− 2µkuxk2

!(1)

+ − 2u2L˜aL+ 2

3u3L− 2µkuxk2

!(2)

. (22)

Observe how ˜a0and ˜aLfunction as switches so that no matter the sign of u, the energy is always zero or decreasing. The PDE is therefore energy stable.

When solving (1) numerically the problem can be formulated by stacking the solutions and numerical operators. Let v(1) and v(2) be the corresponding numerical solutions which are vectors containing m elements.

"v(1) v(2)

#

t

= −"V(1) 0 0 V(2)

# "D1 0 0 D1

# "v(1) v(2)

# +

"µ(1) 0 0 µ(2)

# "D2 0 0 D2

# "v(1) v(2)

#

, (23)

V(1)= diag(v(1)), V(2) = diag(v(2)).

To simplify, define w="v(1)

v(2)

#

, W="V(1) 0 0 V(2)

#

, D1="D1 0 0 D1

# , µ =

"µ(1) 0 0 µ(2)

#

, D2="D2 0 0 D2

# .

Using the designations above, the system of equations can now be described as wt = −1

3W D1w −1

3D1Ww+µD2w.

When performing an energy analysis of the methods the energy estimate is then given by wTHwt+ wTtHw,

where

H="H 0

0 H

# .

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4.2.2 SAT method

In the case of a jump in viscosity, the SAT method is applied in much the same way as previously with no jump. However, since there are now also inner boundary conditions we also need more penalty parameters. We define equation (1) for the two domains for the SAT method as the following system

wt = −1 3

W D1w+ D1Ww¯  + µ D2w+ SAT, (24)

where SAT is defined as SAT="SAT1 SAT2

#

. SAT1and SAT2are defined as

SAT1= H−1em(1)

d (v(1)m − v(2)1 )+ σ(1)µ1(v(1)m − v(2)1 )

| {z }

Dirichlet

+ H−1µ1β(1)em(dmv(1)m − d1v(2)1 )

| {z }

Neumann

(25) + H−1e1τl( ˜αlv(1)1 µ1d1v(1)1 − gl)

| {z }

Left outer boundary

,

SAT2= H−1e1(2)d (v(2)1 − v(1)m )+ σ(2)µ2(v(2)1 − v(1)m)

| {z }

Dirichlet

+ H−1µ2β(2)e1(d1v(2)1 − dmv(1)m )

| {z }

Neumann

(26) + H−1emτr( ˜αrv(2)m µ2dmv(2)m − gr)

| {z }

Right outer boundary

,

where τd, τl, τr, σ and β are penalty parameters. τland τrare selected in the same way as in4.1. This is similiar to the ansatz from equation (5.20) in [9] but in this case there are two viscosities which are not necessarily equal. The following analysis will now consider the inner boundary conditions.

Performing an energy analysis of the ansatz gives us an energy estimate of the problem.

d dt

v

(1)

2 H+

v

(2)

2 H

! + 1

v(1)x

2 H+ 2µ2

v(2)x

2 H

!

=

= 2 3

hv(2)1 3

 v(1)m 3

i + h2µ1d(1)v(1)

m2d(2)v(2)

1

i + 2v(1)m

( τ(1)d 

v(1)m − v(2)1  + σ(1)µ1

v(1)m − v(2)1 

(27) + β(1)

"

1d(1)v(1)

m2d(2)v(2)

1

#)

+ 2v(2)1 (

τ(2)d 

v(2)1 − v(1)m  + σ(2)µ2

v(2)1 − v(1)m + β(2)

"

2d(2)v2

11d(1)v(1)

m

#) .

Let gl = gr = 0 and use the energy analysis to decide the penalty parameters to achieve stability. The stability requirements are σ(1)= σ(2) = σ, β(1), β(2), τ(1)d and τ(2)d . The following specification of σ and β(1)comes from equation (5.23) in [9]

σ = − 1

4(α(1)+ α(2)), where α(1)= H(m, m) and α(2)= H(1, 1).

References

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