=Paper=
{{Paper
|id=Vol-2543/spaper07
|storemode=property
|title=Numerical Solution Tasks with Mixed Operator in Unlimited Area
|pdfUrl=https://ceur-ws.org/Vol-2543/spaper07.pdf
|volume=Vol-2543
|authors=Mikhail Galanin,Dmitry Sorokin
|dblpUrl=https://dblp.org/rec/conf/ssi/GalaninS19
}}
==Numerical Solution Tasks with Mixed Operator in Unlimited Area==
https://ceur-ws.org/Vol-2543/spaper07.pdf
Numerical Solution Tasks with Mixed Operator
in Unlimited Area
Mikhail Galanin1,2[0000-0003-0264-6270] and Dmitry Sorokin1,2[0000-0002-6273-3985]
1 Keldysh Institute of Applied Mathematics (KIAM), Moscow
2 Bauman Moscow State Technical University (BMSTU), Moscow
sorokin.dmitr@yandex.ru
Abstract. Methods for solving problems of elliptic equations, based on the in-
tegrated Green formula, were analyzed. New methods for solving a problem
with a mixed-type operator in an unbounded domain are proposed. On the basis
of the proposed methods, programs for solving problems with a mixed type op-
erator have been created. The results of computational experiments, showing
the correctness of the application of methods, are presented. Moreover, the con-
vergence of the developed iterative methods has been studied; protection of the
order of the original difference scheme using the developed methods has been
shown.
Keywords: Unlimited Area, Mixed Operator, Electrodynamics Accelerator,
Railgun, Electromagnetic Field, Maxwell’s Equations.
1 Introduction
In the study of physical phenomena, it is often necessary to perform modeling in an
unlimited area, for example, in the model-roving of the electrostatic field of charges,
solving the problem of thermal conductivity, etc. In the case where the phenomenon
can be described by means of the simplest linear elliptic operators, the problem is
solved and described, for example, in [1, 2]. The setting of non-reflective boundary
conditions for wave equations is discussed in detail in [3]. However, for a wider class
of problems for which the problem operator may have a mixed type but outside some
finite domain, the fundamental solution of the operator is known and easily calculat-
ed, the above methods are not applicable.
In particular, problems with operators of mixed type arise when modulating the
electromagnetic field in electrodynamic accelerators of the rail type. Process of cur-
rent flow in conductors in this case can be described by parabolic equation and elec-
tromagnetic field in dielectric – elliptical [4, 5].
Copyright © 2020 for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
371
2 Methods of Solving the Problem in an Unlimited Area
In order to solve the problem, three methods of solving are proposed: the method of
expansion of the area, the method of setting integral boundary conditions, three-stage
method [6].
When solving problems using grid numerical methods, accuracy is determined by
grid spacing and accuracy of setting boundary conditions. If you set the same bounda-
ry conditions as infinity in the original task when you solve a problem in the target
area on the boundary, the error defined by the boundary condition setting will gener-
ally decrease as the calculation area increases in size. Therefore, you can use the
scope extension method to solve tasks in an unlimited scope, that is, you can find the
size of the calculation scope so that the effect of the artificial boundary is less than the
application of the numerical method.
The method of expansion of the design area is easy to implement, but has a high
computational complexity [6]. Alternative methods are methods based on the use of
Green formulas.
The first is the method of defining the integral boundary condition [6]. It is based
on the fact that outside some finite region there is an operator with a known funda-
mental solution and a trivial right part, then the solution on the boundary of the calcu-
lation region is determined by means of the basic integral Green formula. Thus, solv-
ing the problem using the method of defining integral boundary conditions requires
solving a system of equations with filled lines corresponding to the boundary.
Three-stage method of solution is built in [6]. When using the three-step method, it
is necessary to solve two systems, but with sparse lines related to boundary condi-
tions. The results of calculations show that when solving problems reduced to solving
linear equations with strongly filled matrices, the method is ineffective, but for prob-
lems with sparse matrix of the system, the three-step method allows to obtain the
result significantly faster than the method of setting the integral boundary condition.
3 Modification of Three-Stage Method. Iterative Method
of Problem Solution in Unlimited Area
Consider in more detail the three-step algorithm [6] on the example of solving the
problem with an operator of mixed type:
ut a u f , r DS ,
2
(1)
u 0, r \DS ,
2
where DS is the boundary of the region DS (Fig. 1).
After applying the finite difference method to approximate the production time in-
stead of the system (1), it is necessary to solve the problem:
1 1 u
uˆ 2 uˆ 2 fˆ 2 , r DS ,
a a a (2)
uˆ 0, r \DS .
2
372
Fig. 1. Structure of area.
In subsequent records of solved tasks, temporary derivatives use approxima-
tion (2).
At the first stage of the three-stage method it is proposed to solve the system with
time-consuming boundary conditions:
vt a 2 vˆ f , r DS ;
vˆ 0, r D \DS ,
vˆ u ,
v(t ) u.
i 1
At the second stage it is assumed that the solution v̂ in the area DS is slightly dif-
ferent from the solution û of the initial problem (3), and the difference is related to
the error in the boundary condition setting. Based on these considerations and Green's
basic integral formula, the objective is:
ut a 2 uˆ f , r DS ;
uˆ 0, r D \DS ,
vˆi P
uˆ vˆi P n vˆi n dS ,
u (ti 1 ) u.
At the third stage, according to the main integrated Green formula, the value û on
the boundary by the value û in the area D :
uˆ ФPL
uˆ ( P) ФPL uˆ dS .
S
n n
The main advantage of the three-stage algorithm is that without the use of an itera-
tive process it allows to obtain results of high accuracy [6, 7]. However, the presence
of the second stage deliberately creates an error that is not eliminated by iteratively
repeating steps 1–3 of the algorithm and can grow with a reduction in the time inte-
gration step . In order to overcome this disadvantage of the method, it is proposed
to delete the second step of the method, and to perform the first and third steps several
times on each time layer.
373
The first stage of the new iterative algorithm
utk a 2 uˆ k f , r DS ;
k
uˆ 0, r D \DS ,
k k 1
uˆ gˆ ,
u (ti 1 ) u , k 1..N .
On the first iteration, you are prompted to select a lagging boundary condition as
the boundary condition
ĝ 0 g .
The second stage of the algorithm
uˆ k ФPL
gˆ k ( P) ФPL uˆ k dS .
S
n n
4 Application of Iterative Algorithm to Solving Problems
in Unlimited Area
The built algorithm is applicable to solving the following problem:
ut a u f , r R0 ,
2
(3)
u 0, r R0 ,
2 m 22
f (r , , t ) a 2 1 r 2 n2 k 2 r k 2 yk C1r n2 e n a t cos .
k 2
Initial condition ( r R0 ):
m2
u (r , , 0) yk r k C1r
2c
J1 (r n ) cos .
k 2 R
n 0
2
( J 0 ( R
0 n ) J 2 ( R
0 n ))
Where m and n — parameters; J j — j order Bessel function ( j 0,1, 2 );
1 Cmk 2
k
n
n , n — root of equation J1 (n ) 0; yk ; 2;
R0 R0k 2 k 2 1
1 m2 m2
1
C1 k 1 yk R0k 1 ; C2 yk R0k 1 C1 R02 ; c C2 1 .
1 k 2 k 2
The analytical solution is as follows:
m 2 22
yk r C1r
2c
k
J1 (r n ) e n a t cos , r R0 ;
k 2 n R02 ( J 0 ( R0 n ) J 2 ( R0 n ))
u (r , , t )
2 n a t
C
cos , r R0 .
2 2
r e
374
Let's set m 1, n 2, a 0.1, R0 1.0. Solve the problem with a new iterative
method and a three-step algorithm. Table 1 shows the calculation results. We see that
all the methods considered lead to the correct result.
Table 1. Task (3) results. R0 1.0, R 2.0, 0.001, h 0.2, T 0.008 .
uh u C uh u C uh u C
uh u C
uC uC uC
uC
h analytical p p application of p application of p
N
definition of
application of
iterative iterative
a three-stage
boundary algorithm algorithm
algorithm
conditions (1 iter) (2 iter)
h 3.15114 103 3.90629 103 3.20485 103 3.25578 103
8
(t = 0.211 s.) (t = 0.489 s.) (t = 0.267 s.) (t = 0.555 s.)
h/2 4.79565 104 5.05743 104 4.50521 104 4.57497 104
32 2.7 3.0 2.8 2.8
/4 ( t = 1.153 s.) (t = 2.322 s.) (t = 1.367 s.) (t = 2.660 s.)
5 5 5 5
h/4 6.82737 10 6.8335 10 (t 8.60114 10 7.39768 10
128 2.8 2.9 2.4 2.6
/16 ( t = 8.141 s.) = 15.62 s.) (t = 9.013 s.) (t = 17.71 s.)
h/8 1.75524 105 ( 1.75524 105 2.65021 105 2.32554 105
512 2.0 2.0 1.7 1.7
/64 t = 114.5 s.) (t = 233.4 s.) (t = 133.1 s.) (t = 260.5 s.)
Here p log 2 uh u C / u C
h
u u / u
h C C h 2
— value characterizing
the order of the method, h — value calculated on a grid with step h .
Table 2. The convergence of iterative methods in the solution of task (3) in case of selection at
each step as the initial approximation of the zero vector 1.5625 10 , h 0.025,
5
T0 0.000125. Error in analytical method of setting boundary conditions is equal to
8.45 106
uh u C
uh u C
uC
Iteration number uC
(iterative process based
(two-step iterative method)
on a three-step method)
1 3.82473 101 1.39064 101
2 1.25412 101 1.65792 102
4 1.34993 102 2.42915 104
8 1.80879 104 1.07342 105
16 2.53095 105 1.07263 105
32 2.53009 105 —
64 2.53275 105 —
375
Note that the iterative process could also be built on the basis of a three-stage algo-
rithm [6]. However, according to the results in Table 2, this will not help to get the
answer with the required accuracy in less time, since one iteration of the three-step
method contains almost twice as many deductive operations as the number of opera-
tions of the new iterative method.
By studying Tables 1 and 2, it can be observed that the iterative methods con-
structed have a high convergence rate as long as the distance from the approximation
to the true solution is large. In the vicinity of the solution, iterative projects do not
converge, which is confirmed by the results of numerical calculations (Table 3).
Table 3. Convergence of two-step iterative method.
0.001 0.2
R0 1.0, R 2.0, , h , T 0.008
64 8
uh u C uh u C uh u C uh u C uh u C
uC uC uC uC uC
analytical application of application of application of application of
definition of iterative algo- iterative algo- iterative algo- iterative algo-
boundary rithm rithm rithm rithm
conditions (1 iter) (2 iter) (3 iter) (4 iter)
Error 1.75524 105 2.65021 105 2.32554 105 2.45878 105 4.61259 105
Time (s.) 114.472 133.145 260.535 359.556 486.5
One way to improve convergence rate in the vicinity of the solution (Table 4) is to
change the second step of the iterative method, for example:
1 uˆ k ФPL
gˆ k ( P) gˆ k 1 ( P) ФPL uˆ k dS .
2 S
n n
It is also worth noting that the newly built method is less demanding to the size of
the auxiliary sub area D\DS , which allows reducing its time-measures, and thus reduc-
ing the computational complexity of the task.
Table 4. Convergence of modified two-step iterative method.
0.001 0.2
R0 1.0, R 2.0, , h , T 0.008
64 8
uh u C uh u C uh u C uh u C uh u C
uC uC uC uC uC
analytical application of application of application of application of
definition of iterative algo- iterative algo- iterative algo- iterative algo-
boundary rithm rithm rithm rithm
conditions (1 iter) (2 iter) (4 iter) (8 iter)
Error 1.75524 105 2.65021 105 2.51139 105 2.4427 105 2.42558 105
Time (s.) 114.472 133.145 236.651 468.127 925.113
376
5 Conclusion
On the basis of the method of setting integral boundary conditions and the three-stage
method, computational algorithms are built and software implemented to solve a
number of problems in an unlimited area. The results of computational experiments
confirm the correctness of the methods. They also show that if the order of accuracy
of quadrature formulas used in the implementation of methods is consistent with the
order of the difference scheme, the order of the scheme is preserved.
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