Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf ·...
Transcript of Asymptotic and Numerical Analysis of Charged Particle Beamsmohammad/meetings/RGD/beams_sddg.pdf ·...
Asymptotic and Numerical Analysis of Charged ParticleBeams
Mohammad Asadzadeh
Chalmers University of Technology
SE-412 96 Goteborg, Sweden
E-mail: [email protected]
URL: http://www.math.chalmers.se/˜mohammad
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.1/20
Outline
The continuous problem — Asymptotic analysis
existence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme, and stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularity
convergence of solution for VFP to that of VP
iterative scheme, and stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme, and stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme,
� �
and
� �
stabilities
basic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme,
� �
and
� �
stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme,
� �
and
� �
stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical Analysis
A scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme,
� �
and
� �
stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary condition
The standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme,
� �
and
� �
stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methods
Discontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variablenumerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme,
� �
and
� �
stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variable �
numerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
Outline
The continuous problem — Asymptotic analysisexistence, uniqueness, regularityconvergence of solution for VFP to that of VP
iterative scheme,
� �
and
� �
stabilitiesbasic estimates and canonical representations
The discrete problem—Numerical AnalysisA scaled equation (in 2-3D) with mixed boundary conditionThe standard Galerkin (SG) and semi–streamline diffusion (SSD)finite element methodsDiscontinuous Galerkin, backward Euler and Crank-Nicolson forpenetration variable �numerical experiments
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.2/20
The Vlasov – Poisson – Fokker – Planck system
�������������������
�� �� ��� � � ��� � � � ��� � � ��� ���� � � � � �! � � � "# �
� ���� � � � � � �%$ ���� � � � ���� � � � � �& � �
� � �� " � � ' �( � � � )
*� � ) * � + � ) � ( )� + � �
� � � ���� � � " � ( �
� � � �,� -� � �� - � .0/ +� - ��� " � 1 ��� as
*�* 1 2
or
� � � �3� �4 - �
where
4 ���65 �
, external potential force,-
internal potential field
Difficulty in solving Cauchy problem in VP:
�
is singular (up with
(
).Control of
7 + 7 � ensure sufficient regularity for
�
to construct uniquesolution.
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.3/20
Technical steps
Key issue:
� �
estimates for + and
�
Difficulty:
�� � �� � � � � Fluid Eqs with too involved� �
estimates
Study: deterministic
Key idea: the maximum principle yields an estimate of� ����� � � �� *� * � �� � � � �
��� � � " � �
for � # (
and by interpolation we get error bounds for
7 + 7 � � 7 ��� + 7 � � 7 � 7� � 7 ��� � 7�
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.4/20
Existence, uniqueness, regularity, convergence
Theorem 0. (VFP) Assume that
�$ 5 ��� �$ � � � � � � � � � � � � � � � � � � �
�� *� * � � � � � * �%$ * *� �%$ * *� � �$ * � � � � � � � � �Then there is a unique solution
� �� � �
to the VFP equation satisfying
�5 � � � � � ���� �� ��� � � � � � � � � � � �
�� *� * � � � � � * � * *� � * � � � ���� � ��� � � � � � � � � � �
� � � ���� � � � � � � � � � � �� � �Theorem 0. (VP) If
� �� � �
and� �� � �� �
are solutions of VP and VFP respectively then� ����$ �� � 7 � � � �� � � " � 7 � 7�� � � � � �� � � " � 7�
7 � � � �� � � " � 7�
� � � � � �
� � � �� *� * � � � � �M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.5/20
Construction of iterative scheme
Assume that
�� ���� " �
is given and is as regular as
�
in theorem VFP
Solve the Fokker-Planck equation� � � �� � ��� � � � � �� �� � � � � � � ��� � � � � � ���
� � � � ���� � � � � � �$ ���� � �
Compute�����������
+� � � ���� " � � � � � � ��� � � " � ( � �
�� � � ��� " � � ' � ( � � � )
*� � ) * � +� � � � ) � ( )
The above FP equation has a unique solutionLions, the father, degenerate type problems
Iterating we get a solution to VFP.
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.6/20
The linear (Vlasov) Fokker–Planck equation
���������������
�� �� ��� � �� ��� � � � ��� � � �
� ���� � � � � � �$ ���� � �
�� ���� � � " � � � � � ��� � � " � � �� � �
given
Assumptions
(A1)
�%$ � � � � � � � � � � � � � � ��� � � �� � �� � � � �� � �� � �
(A2)
�� � �� �� � � � � � � � � � � �
Solution space
� � � � � � � � � � � �� � � � � � �� � � ��
�� � �� � � � �
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.7/20
Assume (A1) and (A2), thenFP has a unique solution
� � �
�� � � � � ��� � � � � �
weak solution of
� � � �� � �
and
�� � �
�%$ 5 ��� �5 � � �5 ��� positivity (Tartar)
�$ � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � ��� � � � � �
7 � � " � 7�
� 7 �$ 7�
�
$7 � ��� � 7
�( � � Maximum principle
Assume that
��
is divergent free, i.e.
� � �� � �
, then
�$ � � � � � � � � � � � � � � ��� � � � � � � � � � � � � � � � � � � � � � � �
7 � � " � 7 � � 7 �$ 7�
�$
7 � �� � 7�
( � � � �
- stability
Idea:
��
: let�� � � � � � �
, then for
# � � 7 ��� �� 7�
� �� �� � � � � � �� �� ���� � � � � � �$ ���� � �
has a unique solution
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.8/20
Standard forms
Omit the superscripts � and � �
. Consider VFP and its differentiatedform in
���� � �
. Recall that
� � � � �� �
and
� �� � �
denote the solutions of VFPand VP, respectively. Let
�� � � � ��
, then
�� �� �� � � �� � � � ��� � � �
�� � � � �� �� �� � � � �,� �� � � � � �� �� � � � � � � �
�� � � �� �� � ��� �� � � �� �� � � � � ��� � � � � � �� � � ��� ��
The second equation is a vector equation with
� � � � � � �� �3� � � �
�
is a
�( �(
matrix decomposed in( (
blocks:
� �� �(
��� � � �
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.9/20
Canonical form
Multiplying standard forms by � � � �� *� * � �� � �
to obtain
� � �� ��� � � � � � �� *� * � � ��� � � � ��� � ��
� � �
� ��
�������������������������������
�� � � � �� *� * � � � � � �
� �� � � � � � � �� *� * � � �� � � � � � �
� �� � � � � � � � *� * �� *� * � � � � � � � � � � �
� ��� � � �� � �
���������������������������������������
�� � � �� *� * � �� � �� �
� � � � ��
�� � �
� � � � �� �� � �
� � � � �� *� * � � � � � � � �
� � � �
�� � � � �� *� * � �� � � � � � �� �
� �� � � ��
� � � �� �
� �� � � �� � � � �� �
� � � � � � �� *� * � �� � � ��� ��
� �� � � � � � �� � �� *� * � � � � � �3� �
Estimate each
� �� , for (
� �� � �
) and use Grönwall type inequalities, ...
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.10/20
The final form of pencil beam equations
The Fokker-Planck equation�����������������������
��� �3� � � � �� � � �� � ���
�
� ����
�� � � � � �� �
�� � � �
� �� �
� � �� � � �
� � � � )� ��� � � �� � � � � ) � � � � � � � � � � �� / � � � � � � �
� � �� )� �� � � �� � � ��� � � � � � ���Forward peakedness allow: to project the FP operator from acting onthe right half of the unit sphere
� �
into the tangent plane at the point�� � ��� � �
to
� �
. In this way we get
the Fermi equation:�������
� ��
�
�� �
� ) � �
� � �� � � �
�
� �� � � � �� � � � ��� � � � � �� � � � � �
� � � � )� ��� �� � � � � ) � � � ) � � � � � � � � � with
� �� � � � ��
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.11/20
Pencil beam models in two (space) dimensions
In 2 space dimensions; we introduce the current
�� � �� � �
, the scalingvariable �� � � �� � � � � / � � � / � � �
, and the scaled forward current
�� � � � ��� )� � �� � � � � � �� � � � ,For this
�
, we get the canonical form of pencil beam equations:�������
� ��
�
�� �
� ) � � � �� ��
�� ��� �
� � ��� )� � � � � � ) � � � � � � )� � � ��
The operator
�
for the Fokker-Planck equation is:�������
� ��
� �� � � �
�� �
�� � � � � � �
� � � � �� � � � � � � � � � �� � � � � � � �
whereas the corresponding Fermi equation has simply
� �� �
� � � �
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.12/20
Numerical domain/Canonical form of a Fermi model
Fermi equation, in a slab of thickness
�
, � � �� � � � ��� � �, with symmetric
cross section
��� � � ��� ��� � � � � ) $ � ) $ � � � � $ � � $ � :��������
��������� � �� � �� � � in
� � �� �� �
��
���� )� � � $ � � ��� for
���� ) � � �� �� �
� � � � � � � � � �� � � � for � � � �� �
� � ��� � � � � ��� on
� ��� � � ���� � � � � �� � � � � � � �� � � � �
�� � �� � ��� � �
and
� � is the outward unit normal to
�
at
��� � � � � �
.
This equation is interpreted as:
time-dependent ( � viewed as time variable),
degenerate (convection in ), diffusion in �),
forward-backward ( � changes sign),
convection dominating ( is small),
convection-diffusion problem.
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.13/20
The (numerical) phase-space domain
� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
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� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
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� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � �
z
y
x
u = 0
u = 0
z
z
n=(0,−1,0)
n=(0,1,0)
z
y0
0
u(x,y,z)=00
0−
u(0,y,z)=f(y,z)
u(x,y,z)=0
Figure 0: 2D-ModelM. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.14/20
Fully discrete strategy
For convection dominated problems, having hyperbolic nature, thestandard Galerkin (SG) method converges with the rate
� �� � �, (versus
� �� � � � �
for elliptic and parabolic problems), provided that the exactsolution is in the Sobolev space
� � � �
.To speed up the convergence of SG we introduce the semi-streamlinediffusion (SSD) method, through a modified form of the test functions.This add (automatically) a proper amount of viscosity resulting insmoothing effects on the equation.SSD method is performed only on the � � variable, whereas in the usualstreamline-diffusion (SD) method both � � and � discretizations areperformed is one, and the same, single variational formulation.In our approach, however, the penetration variable � is interpreted as atime variable and discretized by: discontinuous Galerkin (DG), backwardEuler (BE) and Crank-Nicolson (CN) methods.In fully discrete problem we combine SG or SSD schemes for
� � with atime discretization method for the penetration interval
�� .
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.15/20
The Standard Galerkin Method
This is a finite element approximation, in � � � � )� � � , based onquasi-uniform triangulation of
� � � ��� ��� , with a mesh size
�:
� � � � �
.To this approach we define the inflow boundary
� � � � �� � � �� � � �� � � � � � � � � � � � � �� � �
and a discrete, finite dimensional, function space��� � � �
� �� �
with
� � � ��� � � �� � � � � �� �� � � �
on
� �� �
such that,
�� � � �
� ��� � � �� � ��� �
,
� � �
� � ��� 7� � � 7�� � '� � � � 7� 7� � � � � � � and
� � � � � �
An example of such
� � is the set of sufficiently smooth piecewisepolynomials
� � � � �
of degree� �, satisfying the boundary conditions.
Now the objective is to find� � � ��� , such that� � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � ��� �
� � � ��� � � � � � � �� � �
�
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.16/20
A Smoothing Petrov Galerkin Method
Here we introduce the SSD approach which includes a diffusiongenerating test function in the ) direction over the usual SG procedure.
Using SSD we obtain a non-degenerate type equation with somewhatimproved regularity in the ) direction:
We let� � � � � with
� � �� � �
. Then the SSD test functions: � ��
automatically add the extra diffusion term,� �� � � �
, to the variationalformulation which, combined with
�� � � � � �� � � � � � � ��
, gives anon-degenerate equation with a diffusion term of order
� � � , for
�5 :
Multiplying the differential equation by� �� , integrating over
� � , andusing the boundary conditions yields,
� �� � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � �
� � � � �� � �� ��
� � � ��
The discrete version is now obtained by replacing
�
by a suitable
� � .M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.17/20
Numerical Implementation
We test convergence of SG and SSD through some numerical examples.Our implementations are performed over four different initial data:modified Dirac, hyperbolic, Maxwellian, and cone functions, approximatingour data: the
�
-function. The procedure is split into two steps:
Discretize the two dimensional domain
� � � �� ��� by means of
continuous piecewise linears: � � �� �
, and establish a mesh there inorder to obtain a semidiscrete solution.
Apply one of the time discretization methods (BE, CN or DG), to stepadvance in � direction.
The error � � � � � � � � , is measured in the weighted
� � norm
* *�� * * ��� � � ��
��* � *
�� � �
�� �� � � � � � � � � ��
with
� � � denoting the midpoints of the edges of the mesh triangles � .
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.18/20
Convergence results/tables
The reference domain is
� � � � ��� � � � � � � � � �
Parameters:
� � ��
� � � � �
�� �
(for discretization in �), and� � � ��
� � �
.SG in � � � � )� � � extrapolation error �� � � � �� � � �
discrete � Dirac Hyperbolic Maxwellian Cone
BE 13.63-1.806 .064-.013 .123-.042 .115-.047CN 13.73-1.814 .065-.014 .122-.041 .115-.047DG 13.40-2.065 .064-.012 .117-.043 .110-.051
SSD in � � � � )� � � , extrapolation error �� � � � �� � � �
discrete � Dirac Hyperbolic Maxwellian Cone
BE 13.33-1.801 .063-.014 .118-.041 .110-.045CN 13.44-1.806 .063-.015 .117-.040 .110-.045DG 13.28-2.068 .063-.014 .117-.042 .110-.049
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.19/20
Reliability of Asymptotic Expansions
Dose intensity (amount of deposited energy per unit volume, per unit time)radiating an elliptic target at the collision site � � �
��
and with � � � � ��
�
:
M. Asadzadeh, 23th RGD, Whistler, July 20-25, 2002 – p.20/20