Coupling of tw o Hyperbolic Conservation Law s M. Izadi August - - PowerPoint PPT Presentation

I nstitute for Advance studies in Basic Sciences

Stream line Diffusion Method for Coupling of tw o Hyperbolic Conservation Law s

• M. Izadi

August 20, 2005

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The Plane

• Review of Finite Element Method (FEM)
• The Coupled Problem
• Streamline-Diffusion Formulation
• A priori Error Estimates for Sd-Method
• Numerical Examples

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W hy the Finite Elem ent Method?

Finite element method provides a greater flexibility to model complex geometries than finite difference and finite volume methods do. The construction of higher order approximation

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Basic Principles of FEM

• Finding a variational formulation of the problem:
• Integrating by parts in order to decease the

number of differentiations involved, thereby decreasing the smoothness demands on u.

• Retaining only the essential (Dirichlet ) boundary

conditions.

• Approximating the solution by a finite number of

degrees of freedom, i.e. within a finite dimensional space V.

• Choosing basis functions, e.g. in V, that are

locally supported (vanish on most of the domain).

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One Dim ensional Exam ple

• We consider

(D)

THE STRONG OR DIFFERENTIAL PROBLEM

• By integrating twice, we can see that this

problem has a unique solution and

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The Sobolev Space

• Define

where

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Variational Form ulation

• Find u such that

+

Integrating by part Integrating by part B.C. B.C.

(VF)

• Multiplying both sides of (D) by any function

yields

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Variational Form ulation

• With the notations

and

• Note that (D) is equivalent to (VF).

Find u such that a(u,v)=L(v) for all admissible v

(*) (*) can be written as:

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Uniqueness & Existence Theorem

Thm.(Lax-Milgram ) Let a(.,.) be a bilinear form on a Hilbert space equipped with and the following properties: a(.,.) is continuous, that is a(.,.) is coercive that is Further L(.) is a linear mapping on ,that is Then there exist a unique such that

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Example

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I nterval Partition ( FEM)

Construct a finite-dimensional subspace as follows:

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Finite Elem ent Space

Let be a set of functions such that:

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Continuous piecew ise linear basis function

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Finite Elem ent Approxim ation

( )

• The problem (VF) is reduced to
• where

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Linear system of Equations

• This is equivalent to the system A =b, where

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Properties of the Stiffness m atrix A

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Properties of the Stiffness m atrix A

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The Coupled Problem

Initial Condition Coupling Condition

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Coupled problem

One dimensional example

• r

We can impose coupling condition at x=0

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Coupled problem

• W

e n e e d t

• s

p e c i f y u ( , t ) a t x =

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Coupled problem

• No coupling

condition need at x=0

In general & &

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Sd-Form ulation

• We consider

with T is a given final time value and &

(1)

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Space-tim e discretization

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Space-tim e discretization

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Finite elem ent spaces

• Let k be a positive integer, introduce
• Define the trial & test function spaces

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Som e notations

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Space-tim e Sd Form ulation

• (2)

(3)

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Continue…

After summing over n, we rewrite (3) as follows

• (4)

where

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Continue …

and finally

• (5)

where

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Continue …

Functions in are continuous in x & discontinuous in t After summing over n, we have Define

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Continue …

Summing (5) over n=0,1,…,N-1, we get the following analogue to (4)

• (6)

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Basic Stability Estim ates for the Sd-m ethod Thm. where

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A priori Error Estim ate for the Sd-m ethod

To do this introduce interpolant of exact solution u

and set Then we have

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Continue …

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Num erical Exam ple

where a>0. This problem has the explicit solution

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Test problem 1

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Test problem 2

with the following initial condition and boundary condition

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