Numerical Solution of Time-Dependent Physical systems by Means of Multi-dimensional Wave Digital Filtering network

1. 03/01/15 1
Numerical Solution of Time-
Dependent Physical systems by
Means of Multi-dimensional Wave
Digital Filters
Jason Tseng
School of Engineering
University of Warwick, UK

2. 2
Outline
 Physical systems modelling: Time-dependent PDEs
 Distinct advantages of the MD-WDF modelling
 MD-WDF modelling procedures
 Continuous mapping: lumped electrical networks
 Discrete mapping: bilinear transformation, wave digital filter
 Examples and computational results
 Sound wave propagation in a building (linear system).
 Mindlin plate (thick plate) vibration (linear system).
 Shallow water wave propagation (non-linear system).
 Future work and conclusions.

3. 3
Time-dependent differential equation models
 Original PDE models

Parabolic PDE:

Hyperbolic PDE:
 Models can represent:

Finite-element spatial- and time- discretization of PDEs

Finite-difference spatial- and time- discretization of PDEs

Lumped electrical circuits with linear and/ or non-linear
capacitors and inductors.
( ) fauuc
t
u
d =+∇⋅∇−
∂
∂
( ) fauuc
t
u
d =+∇⋅∇−
∂
∂
2
2
on time.dependcanand,,,where dfac

4. 4
Approaches for numerical modelling
of time-dependent PDEs
 Finite elements
 Advantages:

Easy inclusion of local grid refinement

Easy handling of complex geometries
 Disadvantages:

Computationally expensive

Hard to correctly set up the simulation plane
 Finite differences
 Advantages:

Computationally cheap

Easy to correctly set up the simulation plane
 Disadvantages:

Difficulties in handling irregular boundaries

A Need for local grid refinement to increase the accuracy

5. 5
Approaches for numerical modelling
of time-dependent PDEs (cont.)
 Multi-dimensional Wave digital Filters (MD-WDF)
 A member of finite difference family:
 Computationally cheap.
 Easy to correctly set up the simulation plane.
 Easy to handle complex geometries.
 Conservation of passivity:
 Achievement of full robustness due to positive port
resistances .
 Guarantee to all numerical stabilities required of an accurate
numerical integration method.

6. 6
Advantages of the MD-WDF Model (cont.)
 Fully local interconnectivity and massive parallelism
 Behaviour of the equivalent passive dynamical discrete
system at any point in space is directly influenced only by
the points in its nearest neighbourhood.
 Each point in the n-d grid can be updated simultaneously
when sufficient computing resources are available
 High accuracy:
 Low round-off noise characteristics of WDF structure
 Suppression of parasitic oscillations of WDF structure

7. 7
MD-WDF modelling procedures
Multi-dimensional
Kirchhoff circuit
Discrete mapping
Multi-dimensional
Wave digital filters algorithm
System behaviour description
by lumped electrical network
Discrete passive
dynamic system description
Time-dependent
PDEs
Generalized
Trapezoidal rule
Multi-dimensional
Wave quantities
Kirchhoff’s current and voltage laws
Original passive
Physical system
MD DSP
MDKC
MD WDF

8. 8
Lumped electrical networks
 Kirchhoff ‘s laws: n-port connection forming a loop.
 Passive circuit elements of electrical networks.
 Definition:
 Schematic representation:




=
===
∑=
(voltages)0
(currents)
1
21
n
k k
n
u
iii 
iRu 0=




≥=
≥
=
0)(),(
0),(
iLLiL
t
DL
Li
t
LD
u



=
−=
12
21
Riu
Riu
Resistor:
Inductor: Gyrator:




=
===
∑=
n
k k
n
i
uuu
1
21
0

Series connection Parallel connection
Ideal
transformer



−=
=
21
21
nii
unu

9. 9
Discrete mapping approach
 Generalized trapezoidal rule (bilinear transformation) for
inductor:
 Linear inductances:
 Non-linear inductances:
0),,,,(where)),()(()( 4321 ≥=+±±= kzyxt LtzyxiDLDLDLDLu xxx
[ ]
delaytime:shift;spatial:,,
,,,,
2222
where))()(()()(
4321
tzyx
tzyx
zyxt
TTTT
TTTT
T
L
T
L
T
L
T
L
R
iiRuu
±±±=====
−−=−+
T
TxxTxx
0)(),)(())(())(())(()( 44332211 ≥=±±±= iLLiLDLiLDLiLDLiLDLu kkzyxt xxxxx
approximated
[ ]
4321,
2222
where)))(())(()()(
LLLLL
TTTT
R
iLiLR
L
u
L
u
zyxt
===≡===≡
−−=−+ TxxTxx
approximated

10. 10
 MD wave quantities and adaptors.
 Wave quantities:

Voltage waves (linear circuit elements):

Power waves (non-linear circuit elements):
 Wave digital elements via bilinear transformation:






−
=
+
=
)power waveOutput(
2
)power wave(Input
2
R
Riu
b
R
Riu
a



−=
+=
wave)ltage(Output vo
wave)tage(Input vol
Riub
Riua
Resistor: Inductor: Gyrator:
)()( T−−= tatb



=
==



=
−=
es)(power wav
waves)(voltage
)()(
)()(
21
21
12
21
RRR
RRR
tatb
tatb
sourcevoltage:)(
0)(
)(2)(
te
tb
teta



=
=
Ideal
transformer




=
=
)(
1
)(
)()(
21
12
ta
n
tb
tnatb

11. 11
 MD wave quantities and adaptors (cont.)
 Relations of wave quantities in a n-port adaptor:

Voltage waves:
 Series connection:
 Parallel connection:

Power waves
 Series connection
 Parallel connection
∑
∑ =
=
=−=
n
j
jn
j j
k
kk nka
R
R
ab
1
1
,,1,
2

∑
∑ =
=
=+−=
n
j j
j
n
j
j
kk nk
R
a
R
ab
1
1
,,1,
1
2

nkaR
R
R
ab j
n
j
jn
j j
k
kk ,,1,
2
1
1
=−= ∑
∑ =
=
nk
R
a
R
R
ab
n
j j
j
n
j j
k
kk ,,1,
1
2
1
1
=+−= ∑
∑
=
=

12. 12
Stability conditions
 Linear system.
 CFL (courant-Friedrichs-levy) criterion to obtain the
maximum speed of wave propagation.
 Least restriction on the density of the sampling in time for a
given density of sampling in space.
 Non-linear system.

13. 13
Modelling example 1:
Sound wave propagation in a complex building
 Governing equations of motion and continuity
 System variables:












=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
0),(),(),(),(
1
0),(),(
0),(),(
0),(),(
2
0
0
0
0
tv
z
tv
y
tv
x
tp
tc
tp
z
tv
t
tp
y
tv
t
tp
x
tv
t
zyx
z
y
x
xxxx
xx
xx
xx
ρ
ρ
ρ
ρ
soundofspeed:
airtheofdensity:
//,//,//withsvelocitiefluidacoustic:,,
pressureacoustic:
time:
,,scoordinatespaceofvector:
0
c
zvyvxvvvv
p
t
zyx
zyxzyx
−
−
−
−
−
−
ρ


x

14. 14
Graphical network description of the system
 Equal physical dimension for system variables:
 Mesh equations representing MDKC















=++++−−++++
−−++++−−
=+++−−−
=+++−−−
=+++−−−
0)())((
2
1
))((
2
1
))((
2
1
))((
2
1
))((
2
1
))((
2
1
0))((
2
1
))((
2
1
)(
0))((
2
1
))((
2
1
)(
0))((
2
1
))((
2
1
)(
4343424
241414
34343
24242
14141
iDLiiztDiiztDiiytD
iiytDiixtDiixtD
iiztDiiztDiDL
iiytDiiytDiDL
iixtDiixtDiDL
tp
tz
ty
tx
resistancegraphical:0
voltagegraphical:
currentsgraphical:,,
where
),,,(),,,(
0
40321
≥−
−
−
=
r
p
vvv
iriiipvvv
zyx
zyx

15. 15
MDKC network description
 Partial derivative operators:
 Passivity of inductances





≤++<
≤<





≥−−−=
≥−=≥−=≥−=
2
0
2
0
0
2
0
2
0
000
0
,,0
0
0,0,0
c
r
c
r
L
LLL
zyx
zyx
zyxp
zzyyxx
ρ
δδδ
ρδδδ
δδδ
ρ
δρδρδρ
MDKC representation for 3D sound wave propagation in air
ztz
yty
xtx
DrDztD
DrDytD
DrDxtD
0
0
0
)(
)(
)(
−=±
±=±
±=±
δ
δ
δ

16. 16
Discrete mapping of MDKC
 Generalized trapezoidal rule for inductors-shift operators:
 MD voltage waves-port resistances:
sizesteptemporal:
,,insizesstepspatial:,,
where
],0,0,0[;
],,0,0[],,,0,0[
],0,,0[],,0,,0[
],0,0,[],,0,0,[
65
43
21
t
zyx
t
tztz
tyty
txtx
T
zyxTTT
T
TTTT
TTTT
TTTT
−
−
=





=−=
=−=
=−=

T
TT
TT
TT










≡′≡′≡′≡′
=≡=≡=≡
′=′=′=′=
======
t
p
t
z
t
y
t
x
t
z
zt
yx
yt
x
x
T
L
r
T
L
r
T
L
r
T
L
r
TT
r
r
TT
r
r
TT
r
r
rRrRrRrR
rRRrRRrRR
2
,
2
,
2
,
2
22
,
22
,
22
,,,,
,,
4321
0
3
0
2
0
1
414313212111
3107296185
δδδ

17. 17
MD WDF algorithm
 Relations of wave and state
quantities:
 Relations of state input-
output:







=−=
=−±=
=−±=
=−±=
==
10,,7),1,,,(),,,(
6;5),1,1,,(),,,(
4;3),1,,1,(),,,(
2;1),1,,,1(),,,(
,3,2,1,,,];,,,[],,,[


jknmlcknmld
jknmlcknmld
jknmlcknmld
jknmlcknmld
knmlkTnTmTlTtzyx
jj
jj
jj
jj
tzyxknml
MD WDF algorithm for 3D sound wave propagation in air










=−==−==−=
+=+
−
=−=−
−
=
+=+
−
=−=−
−
=
+=+
−
=−=−
−
=
139913128812117711
710656107105567
694349693346
582128581125
,;,;,
),(
2
1
;),(
2
1
),(
2
1
;),(
2
1
),(
2
1
;),(
2
1
bcdabcdabcda
bbcddabbcdda
bbcddabbcdda
bbcddabbcdda

18. 18
Numerical results 1:
Sound wave propagation in 2D of complex building
Floor plan of one story building with location of sound sources

19. 19
Numerical results 2:
Acoustic pressure propagation in true 3D of 2 storeys
complex building
Floor plan of two storeys building with location of sound sources

20. 20
Modelling example 2:
Mindlin plate (thick plate) vibration
 Governing equations of motion.
 System variables:









=−
∂
∂
−
∂
∂
=−
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−
∂
∂
0
1
0
1
0
2
2
y
y
x
x
yx
w
y
v
t
Q
Gh
w
x
v
t
Q
Gh
y
Q
x
Q
t
v
h
κ
κ
ρ
















=
∂
∂
−
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−
∂
∂
=+
∂
∂
−
∂
∂
−
∂
∂
=+
∂
∂
−
∂
∂
−
∂
∂
0
24
0
1
0
1
0
12
0
12
3
3
3
x
w
y
w
t
M
Gh
y
w
x
w
t
M
D
y
w
x
w
t
M
D
Q
y
M
x
M
t
wh
Q
y
M
x
M
t
wh
yxxy
yxy
yxx
x
yxyy
x
xyxx
ν
ν
ρ
ρ
shearinelasticityofmodulus:
)1(2
ncompressioandin tensionelasticityofdulusmodulus/mosYoung':
plateofrigidityflexural:
)1(12
ratiosPoisson'density,material,thicknessplate:),,(
platetheoflengthunitpermomentsbending:
platetheoflengthunitperforcesshearetransvers:),(
),(rotationsbendingtheofsvelocitie:),(
ntdisplacemeetransverstheofvelocity:
2
3
yx
ν
ν
νρ
ϕϕ
ϕϕ
+
=−
−
−
=−
−
−
∂
∂
=
∂
∂
=−
∂
∂
=−
E
G
E
Eh
D
h
),M,M-(M
QQ
t
w
t
w
t
w
v
xyyx
yx
y
y
x
x
0
12 2
2
2
2
2
2
2
23
2
=
∂
∂
+





∂
∂
−∇





∂
∂
−∇
t
w
hw
tGt
h
D ρ
κ
ρρ
+=
Sub-system 1
Sub-system 2

21. 21
Graphical network description of the system
 Equal physical dimension for system variables.
 Mesh equations representing MDKC.







=−+−+−+−
=−+−+−+−
=+−+++−
+−+++−
0)())(())((
0)())(())((
0)())(())((
))(())((
533312312
422211211
11312312
211211
iRiDLiiytDiiytD
iRiDLiixtDiixtD
iDLiiytDiiytD
iixtDiixtD
Glt
Gut
t













=+−+−+−
+−+−+−
=+−−−+−+−
=−++−+−+−
=+++−+−+
++−+−+
=++−+++−
+−+++−
0)())(())((
))(())((
0)()())(())((
0)()())(())((
0)())(())((
))(())((
0)())(())((
))(())((
88856856
845845
777667754754
766766643643
355856856
754754
244845845
643643
iDLiixtDiixtD
iiytDiiytD
iDLiiDLiiytDiiytD
iiDLiDLiixtDiixtD
iRiDLiixtDiixtD
iiytDiiytD
iRiDLiiytDiiytD
iixtDiixtD
t
tt
tt
Glt
Gut
sresistancegraphical:3,2,1,0
currentsgraphical:,,,,
voltagesgraphical:,,
where
),,,,(),,,,(
),,(),,(
8765342
3211
=≥−
−
−




=
=
jr
MMMQQ
wwv
iiiirirMMMww
iiirQQv
j
xyyxyx
yx
xyyxyx
yx
+
Sub-system 1
Sub-system 2

22. 22
MDKC network description
 Partial derivative operators:

Passivity of circuit elements:






==±=±
==±=±
2,3,1;5,4,2),(
2
1
)(
3,2,1;6,3,1),(
2
1
)(
ljDrDytD
ljDrDxtD
yltjj
xltjj
δ
δ
6,,1,0where
0
)1(24
;
12
7,6,0
)1(12
5,4,0
12
3,2,0
1
0
,
6538367
33
11
2
2
3
12
21
2
11
32
=≥















≥−−
+
==
=≥−
−
=
=≥−−=
=≥−=
≥−−=
==
−
+−
−
j
Eh
L
Eh
L
j
Eh
L
j
rh
L
j
Gh
L
hrL
rRrR
j
jj
jjj
jj
GlGu
δ
δδ
νν
δ
ν
δδ
ρ
δ
κ
δδρ
MDKC representation for Mindlin plate system
Sub-system 1 Sub-system 2

23. 23
Discrete mapping of MDKC
 Generalized trapezoidal rule for inductors-shift operators:
 MD voltage waves-port resistances:





=
=−=
=−=
],0,0[
],,0[],,,0[
],0,[],,0,[
43
21
t
tyty
txtx
T
TTTT
TTTT
T
TT
TT










==′







===
===
=


















−
−
=





=




==++
==+
=
′==′=



==
==
=
++
+
8,,1,
2
ˆ
2,3,1;5,4,2,
3,2,1;6,3,1,
ˆ
where
11
11
,
0
0
3,2;4,2,
3,2;3,1,
ˆ;8,5,4,3,2,1,ˆ
12,,5;6,,3
4,,1;2,1
ˆ2
67
12
6
0
3212
1
6767



j
T
L
r
lj
TT
r
lj
TT
r
r
R
R
R
kjRRR
kjRR
R
rRjrR
kj
kj
rR
t
j
j
t
j
y
l
t
j
x
l
j
c
skjj
skj
Gj
jsj
jk
δ
δ
RR

24. 24
MD WDF algorithm
 Relations of wave and state
quantities:
 Relations of state input-
output:
)1,,(),,(
8,,1),1,,(),,(
12,8,4/11,7,3),1,1,(
10,6,2/9,5,1),1,,1(
),,(
−=
=−=




=−±
=−±
=
knmknm
jknmcknmd
jknmc
jknmc
knmd
cc
sjsj
j
j
j
cd

MD WDF algorithm for numerical simulation of
Mindlin plate system



















=





=





=





=
=−=
==−=
=+
−
=−=
=−
−
=+=
++
++
14
13
14
13
14
13
14
13
11
11
,,,
where;
8,,1,,
12,,4,2),(
2
1
,
11,,3,1),(
2
1
,
d
d
c
c
b
b
a
a
jbcda
jbbcdda
jbbcdda
cccc
cccc
sjsjsjsj
jjjjjj
jjjjjj
dcba
bcda




25. 25
Numerical results 1:
Plane wave propagation of an isotropic square plate
 Plate material and size:
 Material: steel
 Size (volume):
1mx1mx0.1m
 Initial conditions:
 Boundary conditions:
( )
0
0
cos
)0,,(
22
22
=====
==
+
+
=
xyyxyx
yx
MMMww
QQ
yx
yx
yxv
0
edgesFree
=== xyyy MMQ

26. 26
Numerical results 2
Plate deformation of an isotropic square plate
 Plate material and size:
 Material: brass
 Size (volume):1m x 1m x
0.1m
 Initial conditions:
 Boundary conditions:
factorscalepositive:250
1such thatintegerssigned:44.0,9.0
modesofnumber:)2,2(),(
where
sinsin
sinsin
),,(
22
=−
=+==−
=−






















⋅





±








⋅





=
α
ππ
ππ
α
BABA
nm
l
ym
l
xn
B
l
yn
l
xm
A
oyxv
yx
yx
0
edgessupported-simplytype-Hard
=== yx Mwv

27. 27
Numerical results 3:
Combination of plate deformation and plane wave
propagation of an isotropic rectangular plate
 Plate material and size:
 Material: brass
 Size (volume):2m x 1m x
0.1m
 Initial conditions:
 Boundary conditions:









=====
==












=
0
0
2
cosarctan)0,,(
xyyxyx
yx
MMMww
QQ
xyxv
π
0:
SBandNBon(F)
edgesFree
0:
EBandon WB(SS)
edgessupported-simply
type-Soft
F-SS-F-SSedgesMixed
===
===
xyyy
yxy
MMQ
MMv

28. 28
Modelling example 3:
Non-linear water wave propagation
 Governing equations of motion and continuity.
 System variables:
( ) ( )








=
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
++
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+−
∂
∂
+
∂
∂
+
∂
∂
0
0
0
21
1
2
2
2
1
2
2
1
2
1
1
1
t
h
hv
y
hv
x
y
gfv
y
v
v
x
v
v
t
v
x
gfv
y
v
v
x
v
v
t
v
η
η
.(constant)onacceleratigravity:
(constant)parameterCoriolis:
nt.displacemesurfacefree:),,(
.(constant)depthmean:
depth.total:
.y//,//withocitieswater vel:, 2121
g
f
tyxz
H
Hh
vxvvv
η
η
=
+=


29. 29
Graphical network description
 Quantities normalization and equal physical dimension for
system variables:

Mesh equations representing MDKC.
parameterscaled:0
)(ithconstant w:0)(
where
ˆ,ˆ,ˆ,ˆ
3333
333
2
2
3
1
1
>−
=>=−
≡≡≡≡
ε
εεη
η
εε
ttttDv
v
h
h
vv
v
v
v
v
v
t
),,()ˆ,ˆ,ˆ( 32121 iiivv ≡η













=+++−−+
+−++
=++−−−−+
+−++
=−−−+++−
+−++
∑∑
∑ ∑
∑ ∑
∑ ∑
==
= = ++
= = ++
= =
0))(())((
)())(())((
0))(())((
)())(())((
0))(())((
)())(())((
13
2
1 313
2
1 3
2
1
2
1 3334334
232323231
2
1
2
1 2232232
131313132
2
1
2
1 11313
3
232
131
iittDiittD
iLDLittDittD
iittDiittDiR
iLDLittDittD
iittDiittDiR
iLDLittDittD
jjjj
j j tjjjj
g
j j vtvjjjj
g
j j vtvjjjj
ηη

30. 30
MDKC description
 Partial derivative operators:
 Passivity of non-linear
inductances:
MDKC representation for shallow water system
[ ]
02,2,2;,,;6,4,2
),)((
2
1
))((
01,1,1;,,;5,3,1
),)((
2
1
))((
2,1
,)()(
2
1
))((
21
23
21
13
443
23
13
3
===
•±±±=•±
===
•±±±=•±
=
•±•=•±
lvvkj
LDDLttD
lvvkj
LDDLttD
j
LDLDttD
lkttlkj
lkttlkj
tjtjj j
η
δδ
η
δδ
δδ









≥=
≥−−==≥−=
==≥−=
0ˆ
3
2
0;2,1,0
3,2,1;,,,02
3
0
21
2
3
3333
213
jj
jj
jkj
v
gv
L
v
LjLL
kvvjLL
ε
δδ
ε
δ
ηδ

31. 31
Discrete mapping of MDKC
 Generalized trapezoidal rule for non-linear inductors-shift
operators:
 MD power waves-port resistances:
 Stability criterion:
[ ] [ ]
[ ] [ ] [ ]
rTvTT
T
TTTT
TTTT
tyx
t
tyty
txtx
ˆwhere
00
0,0
0,0
3
43
21
≡==
=



−==
−==
T
TT
TT









=
=
±
=
±
=









=
=
±
=
±
=




=
=
15,
ˆ
2
14;13,
ˆ
12;11,
ˆ
;
10,5,
ˆ
2
9,4;8,3,
ˆ
7,2;6,1,
ˆ
;
4,,1
ˆ
02
01
2
1
4
j
r
L
j
r
L
j
r
L
R
j
r
L
j
r
L
j
r
L
R
j
r
L
R
sj
v
v
v
sj
j
η
η
η
δ
δ
δ
δ

)0,,(min),0,,(maxwhere
)
3
2
(2,
)3/2g(H
max
)(2
)3/2g(H
1if
)3/2g(H
)(2
)3/2g(H
1if)
3
2
(2
),(
min
),(
max
max
min
max
3
2
min
max
min
max
3
2
min
max
max3
yxyx
Hg
H
v
HH
v
H
Hgv
yxyx
ηηηη
η
η
η
η
η
ε
η
η
η
η
εηε
==

















+
+
+
≥
+
+
≤≠
+
+
≥
+
+
>≠+≥

32. 32
MD WDF algorithm
 Relations of wave and state
quantities:
 Relations of state input-
output:











==−=
−=−−=
+=+−=
+=+−=
−=−−=
15,,1,,
),(
2
1
;),(
2
1
),(
2
1
;),(
2
1
434434
433433
122122
121121
jbcda
bbcdda
bbcdda
bbcdda
bbcdda
sjsjsjsj









=−
=−+
=−−
=−+
=−−
=
15,10,5),1,,(
14,9,4,4),1,1,(
13,8,3,3),1,1,(
12,7,2,2),1,,1(
11,6,1,1),1,,1(
),,(
sssjknmc
sssjknmc
sssjknmc
sssjknmc
sssjknmc
knmd
j
j
j
j
j
j
MD WDF algorithm for numerical simulation of
shallow water system

33. 33
Numerical results 1:
Rectangular water basin
 Geometry:
 Initial conditions:
 Boundary conditions:
( )



+=
==
22
21
arctan)0,,(
0)0,,()0,,(
yxyx
yxvyxv
η
( ) 









=





•
boundaryclosedon the,pointeveryfor
,0
),(2
1
bb
yx
yx
v
v
n
bb

mhmrM
mH
mlml yx
5.1,15:mediumCircular
500:depthMean
150,200:lengthsSide
1 ==
=
==

34. 34
Numerical results 2:
Turnaround U-bend open channel
 Geometry:
 Initial conditions:
 Boundary conditions:
( )



+=
==
22
21
arctan)0,,(
0)0,,()0,,(
yxyx
yxvyxv
η
( ) 









=





•
boundaryclosedon the,pointeveryfor
,0
),(2
1
bb
yx
yx
v
v
n
bb

mlml
mH
mR
mR
yx 200,300:channeltheoflengthsSide
100:depthMean
72:circleExternal
5.22:circleInternal
2
1
==
=
=
=

35. 35
Conclusions
 An alternative approach to integrate physical systems described
by PDEs.