Stream line Diffusion Method for Coupling of tw o Hyperbolic Conservation Law s M. Izadi August 20, 2005 I nstitute for Advance studies in Basic Sciences
The Plane • Review of Finite Element Method (FEM) • The Coupled Problem • Streamline-Diffusion Formulation • A priori Error Estimates for Sd-Method • Numerical Examples 2
� 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 3
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). 4
One Dim ensional Exam ple We consider (D) THE STRONG OR DIFFERENTIAL PROBLEM and • • By integrating twice, we can see that this problem has a unique solution 5
The Sobolev Space • Define where 6
Variational Form ulation • Multiplying both sides of (D) by any function yields + Integrating by part B.C. Integrating by part B.C. • Find u such that (VF) 7
Variational Form ulation • Note that (D) is equivalent to (VF). (*) • With the notations and (*) can be written as: Find u such that a(u,v)=L(v) for all admissible v 8
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 9
10 Example
I nterval Partition ( FEM) � Construct a finite-dimensional subspace as follows: � 11
Finite Elem ent Space � Let be a set of functions such that: � 12
Continuous piecew ise linear basis function 13
Finite Elem ent Approxim ation • The problem (VF) is reduced to ( ) • where 14
Linear system of Equations • This is equivalent to the system A =b, where 15
Properties of the Stiffness m atrix A 16
Properties of the Stiffness m atrix A 17
� The Coupled Problem � Initial Condition � Coupling Condition 18
Coupled problem � One dimensional example � or We can impose coupling condition at x=0 19
Coupled problem � o t d e e n ) t , e 0 W ( u y f i c e p s 0 = x t a 20
Coupled problem No coupling condition need � at x=0 In general � & � & 21
� Sd-Form ulation • We consider (1) with T is a given final time value and & 22
Space-tim e discretization 23
Space-tim e discretization 24
Finite elem ent spaces • Let k be a positive integer, introduce • Define the trial & test function spaces 25
Som e notations 26
Space-tim e Sd Form ulation � (2) � (3) 27
Continue… � After summing over n, we rewrite (3) as follows � (4) where 28
Continue … and finally � (5) where 29
Continue … � After summing over n, we have Functions in are continuous in x & discontinuous in t � Define 30
Continue … Summing (5) over n=0,1,…,N-1, we get the following analogue to (4) � (6) 31
Basic Stability Estim ates for the Sd-m ethod � Thm. where 32
A priori Error Estim ate for the Sd-m ethod To do this introduce interpolant of exact solution u and set Then we have 33
Continue … 34
Num erical Exam ple where a>0. This problem has the explicit solution 35
Test problem 1 36
37
38
Test problem 2 with the following initial condition and boundary condition 39
40
41
42
Recommend
More recommend
