A High-Order Ghost Method for Solving Moving …people.duke.edu/~xz101/Xianyi_Zengs_Home_Page/...7...

FRG Seminar, A High-Order Ghost Method for Moving Boundary Condition Problem

Xianyi Zeng 03/15/2010

A High-Order Ghost Method for Solving Moving Boundary

Condition Problem �

FRG Seminar 03/15/2010

Xianyi Zeng

Outline of Presentation

Part 1: Overview •  Background and Potential Applications •  Procedure Outline

Part 2: Modified GFM: Matched Ghost Value Part 3: Applying Correct Boundary Condition Part 4: Case Study, 1D Wave Equation Part 5: Conclusion and Future Work

Background and Potential Applications

l  In many fluid-structure problems, using a fixed mesh is more favorable: l  Interfaces have large deformations l  Interfaces’ topology changes l  Flapping wing

Background and Potential Applications

l  Many methods have been developed: l  Embedded method l  Immersed method l  Fictitious domain method l  Ghost fluid method l  Etc

l  Some of these method are successful for fluid-fluid interaction. But for fluid-structure interaction, they are at best first order accurate at the interface, and most of them are even not consistent (with the governing equation) at interface.

Procedure Outline

l  The simple wave equation with moving-boundary condition is used as a model problem for Fluid-structure interaction problem.

l  Assuming all real nodes have correct values, ghost fluid method is studied, and modified to give high-order accuracy at the moving interface.

l  For non-conforming mesh, the correct boundary condition is derived analytically.

Part 1: Overview

Part 2: Modified GFM: Matched Ghost Value •  Original GFM and Motivation for Modification

•  Modification by Operator Matching Part 3: Applying Correct Boundary Condition Part 4: Case Study, 1D Wave Equation Part 5: Conclusion and Future Work

Part 1: Overview

Part 2: Modified GFM: Matched Ghost Value •  Original GFM and Motivation for Modification •  Modification by Operator Matching Part 3: Applying Correct Boundary Condition Part 4: Case Study, 1D Wave Equation Part 5: Conclusion and Future Work

Original Ghost Method and Motivation for Modification

l  Ghost fluid method: l  Solve problems with multiple domains and different governing

equations. l  Solve problems on a fixed mesh

l  1D Illustration: l  From time step to .

real ghost

nt 1nt +

Interface

l  GFM introduces two essential steps: l  Identifying the ghost grid points and real grid points l  Defining proper fluid values for the ghost grid points.

l  On populating ghost values, original GFM suggests: l  Constant extrapolation l  Linear extrapolation

l  These approaches are not always consistent at the interface. l  Illustrated by 1D wave equation using constant extrapolation

and central difference scheme.

1GI Iu u+ =

1 12GI I Iu u u+ −−=

l  1D wave equation:

l  The last real grid point is . l  Constant extrapolation: . l  Central difference at last real grid point:

l  Modified equation solved near the interface:

l  Inconsistent! (Unless CFL number: ) l  Global accuracy is at best first order

0t xu au+ =Ix

1GI Iu u+ =

02 4t x

a uu a λ⎛ ⎞+ + =⎜ ⎟⎝ ⎠

1 1 02

GI I Idu u uadt x

+ −−+ =

2I I Idu u uadt x

−−⇒ + =

l  Remark 1: l  For this problem, if linear extrapolation is used, the local

truncation error suggests consistency at the method. l  A more complicated scheme will be shown later, that when

linear extrapolation is used, the method is still first order accurate.

l  Remark 2: l  If backward difference is used in this case, the ghost values play

no role in updating real grid points. l  The effect of ghost values are dependent on the underlying

scheme.

l  Conclusion: In order to achieve high order of accuracy, the analysis suggests that defining ghost values should depend on underlying scheme.

Part 1: Overview

Part 2: Modified GFM: Matched Ghost Value •  Original GFM and Motivation for Modification

•  Modification by Operator Matching Part 3: Applying Correct Boundary Condition Part 4: Case Study, 1D Wave Equation Part 5: Conclusion and Future Work

Modification by Operator Matching

l  At a specific time step, two ways to update the status are compared: l  Use standard method, on the real domain only. This means there

should be a underlying scheme for inner grid points, and special treatments for boundary grid points.

l  Use ghost method, which means the same underlying scheme is to be used on both real and ghost domain. There is no special treatment for interface points between the two domains.

l  For a single time step, we know how to achieve high order of accuracy for the first approach.

l  For a single time step, the ghost method will be high order accurate, if it produces the same results as the standard method.

l  Suppose the real grid points are given by l  Given a standard method, which can be expressed as:

l  There should be different treatments for inner points and boundary points:

l  Depending on the stencil of , there can be more than one boundary operators ( ).

l  Ghost method use ghost values: and only :

1,I Ix x−…

1 ( )n ni iu u i I+ = ≤L

0( ) ( ) away fromn ni i i Iu u x x=L L

( ) ( ) adjacent to n b ni i i Iu u x x=L L

,n Gu 0L1 0 ,( )n n G

i i u i Iu + = ≤L

l  Idea of operator matching: l  We want to have the ghost values be defined in such a way that,

both methods should give the same results. This gives a system of equations for the ghost values:

l  Example: central difference l  Suppose one use central difference for inner points and

backward difference for boundary point:

0 ,( ) ( ) adjacent to n G b ni i i Iu u x x=L L

0 , ,1 1

1( ) ( )2

n G n n G nI I I Iu u u uλ + −= − −L

1( ) ( )b n n n nI I I Iu u u uλ −= − −L

,1 12n G

I Iu u u+ −= −}⇒

Part 1: Overview Part 2: Modified GFM: Matched Ghost Value

Part 3: Applying Correct Boundary Condition •  Model Problem: Wave Equation with Moving Boundary •  Boundary Conditions for 1D Wave Equation •  BC for 2D Wave Equation •  Summary on The Approach

Part 4: Case Study, 1D Wave Equation Part 5: Conclusion and Future Work

Part 1: Overview

Part 2: Modified GFM: Matched Ghost Value Part 3: Applying Correct Boundary Condition •  Model Problem: Wave Equation with Moving Boundary •  Boundary Conditions for 1D Wave Equation •  Boundary Conditions for 2D Wave Equation •  Summary on The Approach

Model Problem: Wave Equation with Moving Boundary

l  1D wave equation for function :

l  Defined on the “irregular” space-time domain:

l  Initial condition:

l  “Moving” boundary condition:

l  are known functions (forced motion).

0u uat x∂ ∂

+ =∂ ∂

0 ( ), 0x d t Tt≤ ≤ ≤ ≤

0(0, ) ) (0)( 0xu x u x d= ≤ ≤

( , ( )) ( ) 0u t d t g t t T= ≤ ≤

( , )u t x

0 ( ), ( ), ( )u x d t g t

l  Well - posedness of the problem:

l  Consistency between initial condition and boundary condition:

l  Direction of wave propagation:

l  Speed of incoming-information at right boundary:

0 (( (0))0) dg u=

'( ) 0d t a t T> ≤ ≤

l  Discretization:

l  The domain [0, L] is discretized into N equispaced grid points

l  It is assumed that for any time, and N is large enough, such that there are always sufficient grid points between the boundary and endpoints of the whole domain [0, L].

l  At time t, the index for last grid point in real domain is :

0iLx i NN

x i xΔ ≤ ≤ Δ= =

0 ( )d t L< <

( )d t

1( )t tI Ix d t x +<≤

Part 1: Overview

Boundary Conditions for 1D Wave Equation

l  At time step , the solver does not see the boundary We need to apply the boundary condition at , using Taylor series expansion:

l  Easy BC: only the first term is used, i.e. set:

l  This is also the traditional way of applying the boundary condition.

2 3, ) ( , ( )) ( , ( 1)) ( )( , ( )2

( )nn n n n x n n n n nI xxx u tu t u d td t t d t u xt Oδ δ= + Δ+ +

( )nn I nx d tδ = −

nt ( )d tnIx

( , ( )) ( )n

nI n n nu t d t gu t= =

l  Alternative BC: we can use, at least the first two term in the Taylor series expansion:

l  The value of second term can be obtained analytically, by take the temporal derivative on:

l  The calculation procedure:

( ) ( , ( ))n

nI n n x n nu g t u t d tδ= +

( , ( )) ( )u t d t g t=

( , ( )) '( ) ( , ( )) '( )t xu t d t d t u t d t g t+ =

( , ( )) '( ) ( , ( )) '( )x xau t d t d t u t d t g t− + =

⇒( )( , ( ))( )xg tu t d td t a

ʹ′=

ʹ′ −

l  First Alternative BC: second order accurate approximation to the true value:

l  Since it is always satisfied that

The equation above is well-defined.

( )( )( )n

n n nI n

u g tg td t aδ ʹ′

= +ʹ′ −

'( ) 0d t a t T> ≤ ≤

l  Second Alternative BC l  Use the first three terms in the Taylor series expansion, which

requires the value of:

l  This value can be obtained analytically:

( , ( ))xx n nu t d t

( , ( )) ( )u t d t g t=

( '( ) ) ( , ( )) '( )xd t a u t d t g t− =

''( )( ''( ) ) ''( ) '( )( , ( ))( ( ) )xx

g t d t a d t g tu t d td t a− −

=ʹ′ −

⇒ 2( '( ) ) ( , ( )) ''( ) ''( )( , ( ))xx xd t a u t d t d t u t g td t− + =

l  Second Alternative BC: third order accurate approxi - mation to the true value: l  This approach can be continued, to obtain an

approximation to the true value of , to arbitrary order of accuracy.

l  Expression becomes very complicated for higher order approximations.

( ) ''( )( ''( ) ) ''( ) '( )·( ) 2 ( ( ) )n

n n n n n n n nI n

g t g t d t a d t g tu g td t a d t aδ δʹ′ − −

= + +ʹ′ ʹ′− −

,( )nn Iu t x

Part 1: Overview

l  The method above can be easily extended to 2D wave equation :

l  Moving boundary condition:

l  s gives parameterization of the boundary curve:

l  and are known functions, satisfying several conditions for well - posedness.

0u ua bt x

uy∂∂ ∂

+ +∂

=∂ ∂

( , ( , ), ( , )) ( , ) 0u t x s t y s t g s t t T= ≤ ≤% %

0 ( , ), ( , ), ( , )u x y x s t y s t% %

: ( ( , ), ( , ))t x s t y s tΓ % %

( , )g s t

l  Suppose, at time step , the boundary value at a real node , is to be computed.

l  Choose arbitrary point on boundary curve at this time step: , which is close to .

l  The Taylor series expansion: 0 0( , ) nx y ∈Γ% %

0 0( , )x ynt

0 0( , )x y

0 0 0 0 0 0 0

2 20 0 0 0 0 0

0, , ) ( , ) ( , )

( , , ( , , )

1 1, , ( , , ) ( , , )2 2high order terms

n n x n y y n

xx n y yy n x y xy n

x yu t x y u x y u t x y

x y u t x y t

u u xt

δ δ δ δ

% % % % % %

0 0x x xδ = − %

0 0y y yδ = − %

l  Similar as 1D case, the analytical values for the derivatives can be calculated, for first order derivatives:

l  For second order derivatives, define:

t t t tx

s s s s

g y b x a y bu

g y x y− − −

=% % %% % %

t t t t

s s s sy

x a g x a y bu

x g x y− − −

=% % %% % %

( ) 2( )( ) ( )( ) ( ) ( ) ( )

t t t t

s t s t s t s t

s s s s

x a x a y b y bx x a x y b y x a y y

x yJ b

− − − −

= − − + − −

% % % %% % % % % % % %

% %% %

l  We have: 2

2( )( ) ( )( ) ( ) ( )

tt tt x tt y t t t

xx ts ts x ts

y s t s t s t

x y s s sss

g x u y u x a y b y bg x u y u x y b y x a y y bg x u y u x y y

u J −− − − − −

= − − − + − −

− −

% % % % %% % % % % % % %% % %% %

( ) ( )( ) ( )t tt tt x tt y t

xy s t ts ts x ts y s t

s ss ss x ss y s

x a g x u y u y bx x a g x u y u y y bx

u Jg x u y u y

− − − −

= − − − −

− −

% % % %% % % % % %

% % % %

( ) 2( )( )( ) ( ) ( )

ss ss s

t t t tt tt x tt y

s t s t s t ts ts x

s s s x y

x a x a y b g x u y ux x a x y b y x a g x u y ux x y g x u y u

u J −− − − − −

= − − + − − −

− −

% % % % %% % % % % % % %

% %% % %

Part 1: Overview

Summary on Applying Boundary Conditions

1.  This approach uses the mathematical boundary condition to produce exact evaluations of Taylor series.

2.  Only one point on the boundary is needed, to calculate approximate boundary value for a grid point near the true boundary. And this holds for both 1D and 2D (and similarly 3D) cases.

3.  When exact values for the boundary conditions are not available, the expressions can be used to derive approximated boundary conditions easily.

Part 1: Overview

Part 2: Modified GFM: Matched Ghost Value Part 3: Applying Correct Boundary Condition Part 4: Case Study, 1D Wave Equation •  Problem Description •  Lax - Wendroff Method •  2nd Order Upwinding Scheme + RK2 •  3rd Order Spatial Discretization + RK3

Part 5: Conclusion and Future Work

Part 1: Overview

Case Study: Problem Description

l  1D wave equation for function :

l  Moving boundary condition:

l  initial condition:

l  The whole computational domain is defined to be [0, 2]:

2 0u ut x∂ ∂

− =∂ ∂

( , ( )) ( )( ) 1.6 0.1sin(4 )( ) 1.2cos(4 )

u t d t g td t tg t t

( , )u t x

(0, ) 1.2 0 1.6u x x= − ≤ ≤

Case Study: Problem Description

l  The problem has the analytical solution: l  In the application of various methods in solving the

MBC problem, the CFL number is fixed to be . l  Number of equispaced grid points are: 41, 81, 161, 321,

641. l  Three measures of errors will be calculated and plotted: 1.  Absolute point wise error at the last real grid point 2.  norm of error on the whole real domain 3.  norm of the error on the whole real domain

1.6( ')

1.2 2( , )

( ') 2 1.6, 2 ' 2x t

u t xg t d t t xx tt− +⎧

= ⎨+ + = +>

0.2λ =

Part 1: Overview

Lax-Wendroff Method

l  Lax-Wendroff inner-point and boundary-point operators are given by:

l  One ghost value, at , should be defined, such that

we have the operator matching condition satisfied: l  This implies linear extrapolation:

0 21 1 1 1

1 1( ) ( ) ( 2 )2 2

( ) ( )

n n n n n n ni i i i i i i

b n n n ni i i i

u u u u u u u

u u u u

+ − + −

= − − + − +

= − −

1nIx +

, 2 ,1 1 1 1 1

1 1( ) ( 2 ) ( )2 2n n n n n n n n n

n n G n n G n n n n nI I I I I I I I Iu u u u u u u u uλ λ λ+ − + − −− − + − + = − −

n G n nI I Iu u u+ −= −

Lax-Wendroff Method

l  Solutions after 80 time steps with 41 grid points, with four different settings are compared:

l  Two choices of applying boundary condition at :

1.  Easy BC (first order accurate). 2.  First Alternative BC (second order accurate).

l  Two choices of populating ghost values: 1.  Constant extrapolation. 2.  Matched GV (in this case, equivalent to linear extrapolation).

Lax-Wendroff Method

Convergence rates in point wise error at last real grid point:

Lax-Wendroff Method

Convergence rates in norm of error: 1L

Lax-Wendroff Method

Convergence rates in norm of errors: L∞

Part 1: Overview

2nd Order Upwinding Scheme + RK2

l  Using 2nd order upwinding discretization in space. For a forward Euler step, the operator , is defined to be:

l  The 2nd order accuracy in time is achieved by using RK2:

,211 1

1( ) ( 4 3 )2

( ) ( )

( ) ( )n n n n

n n n n

n n n n ni i i i i

b n n n nI I I I

u u u u u

u u u u

= − − + −

= − −

(1) (2) (1)

( ) ( )1 ( )2

u u u u

l  The operator to be matched is given by: l  Four ghost values are obtained: l  This is linear extrapolation, to a stencil of four grid

points.

1 12 2

t tΔ Δ= + oL I L L

n G n nI I I

l  There are six different settings: l  Two choices of applying boundary condition at :

1.  Easy BC (first order accurate). 2.  First Alternative BC (second order accurate).

l  Three choices of populating ghost values: 1.  Constant extrapolation. 2.  Linear extrapolation (to one ghost node). 3.  Matched GV (in this case, linear extrapolation to four ghost

nodes).

Part 1: Overview

3rd Order Spatial Discretization + RK3

l  3rd order accurate scheme, using spatial discretization, which for a forward Euler step, has the operator defined to be:

l  At inner points, one downwinding node and two

upwinding nodes are used. l  BDF2 and central difference are used at boundary nodes.

1( ) ( 6 3 2 )61( ) (3 4 )21( ) ( )2

n n n n n

n n n n

n n n n n ni i i i i i

b n n n n nI I I I I

b n n n nI I I I

u u u u u u

u u u u

λ− −

− −

= − − + − −

= − −

l  RK3 is used in time: l  The operator to be matched is given by:

1 2 3 13 3 4 4

t t tΔ Δ Δ+⎛ ⎞= + ⎜ ⎟⎝ ⎠o oL I L I L L

(2) (1)

( )3 1 ( )4 41 2

The operator matching process gives six ghost values:

,1 1 2

,2 1 2

,3 1 3

,4 1 2 3

,5 1 2 3 4

3 30 18

99 321 108

n n n n

n n n n n

n n n n n n

n G n n nI I I I

n G n n n nI I I I I

n G n n n n nI I I I I I

n GI I

+ − −

+ − − −

+ − − − −

= − + +

1 2 3 42007 602100 556n n n n n

n n n n nI I I Iu uu u− − − −++−+

l  For 3rd order schemes, we can safely allow a 2nd order error in boundary terms, and the formula above can be simplified to:

,1 1 2

,2 1 2

28 8 1

n n n n

n G n n nI I I I

+ − −

l  There are six different settings: l  Two choices of applying boundary condition at :

1.  Easy BC (first order accurate). 2.  Second Alternative BC (third order accurate).

l  Three choices of populating ghost values: 1.  Constant extrapolation. 2.  Linear extrapolation. 3.  Simplified matched GV.

Part 1: Overview

Conclusions and Future Work

l  Conclusions l  Original ghost fluid method can be inconsistent at the interface l  Matched ghost values are introduced to achieve high order of

accuracy l  For wave equation under forced motion, on non-conforming

mesh, the correct boundary values on mesh points can be calculated to high order of accuracy, both in one-dimensional and multi-dimensional cases.

l  Numerical experiments confirms that second and third order of accuracy can be achieved by using the “matched ghost values” and “alternative boundary conditions”.

Conclusions and Future Work

l  Future Work l  Operator matching for 2D wave equations l  Operator matching for “partial” boundary conditions, like Euler

equations with transmission condition in 1D and 2D.

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