18 the Lax Wendroff Technique
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Transcript of 18 the Lax Wendroff Technique
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8/6/2019 18 the Lax Wendroff Technique
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The Lax-Wendroff Technique
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Introduction What a kind of techniques is it ?
The Lax-Wendroff techniques is an explicit,
finite-differencemethod particularly suited to
marchingsolutions. A brief review on the marching problems
Hyperbolic and parabolic partial differentialequations
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Introduction An example of hyperbolic equations
The time-marching solution of an invicid flow using theunsteady Euler equations (2D).
Continuity
+
+
+
=
y
v
y
v
x
u
x
u
t
+
+
=
x
p
y
vv
x
uu
t
u
1x momentum
y momentum
+
+
=
y
p
y
vv
x
vu
t
v
1
Energy
+
+
+
=
y
vp
x
up
y
eu
x
e
t
e
(6.1)
(6.2)
(6.3)
(6.4)
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Procedure Predict the the value of the variables at next time step
based on a Taylor series expansion.
Choose density for purposes of illustration
+
+
+=+
2
)(2
,2
2
,
,,
t
tt
t
t
ji
t
ji
t
ji
tt
ji
(6.5)
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Procedure Analogous Taylor series are written for all the other
dependent variables.
+
+
+=+2
)(2
,
2
2
,
,,
t
t
ut
t
uuu
t
ji
t
ji
t
ji
tt
ji
+
+
+=+2
)(2
,
2
2
,
,,
t
t
vt
t
vvv
t
ji
t
ji
t
ji
tt
ji
+
+
+=+2
)(2
,
2
2
,
,,
t
t
et
t
eee
t
ji
t
ji
t
ji
tt
ji
(6.6)
(6.7)
(6.8)
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Procedure
A number for (/t) is obtained from the continuityequation, Eq. (6.1)
The spatial derivative are given by second-order
central differences
+
+
+=+2
)(2
,
2
2
,
,,
t
tt
t
t
ji
t
ji
t
ji
tt
ji
(
)y
u
y
vv
xu
x
uu
t
t
ji
t
jit
ji
t
ji
t
jit
ji
t
ji
t
jitji
t
ji
t
jitji
t
ji
+
+
+
=
++
++
22
22
,1,1
,
,1,1
,
,1,1,
,1,1,
,
(6.9)
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Procedure+
+
+=+2
)(2
,
2
2
,
,,
t
tt
t
t
ji
t
ji
t
ji
tt
ji
t
v
ytyvty
v
ty
v
t
u
xtxutx
u
tx
u
t
+
+
+
+
+
+
+
=
2222
2
2
The mixed derivative is found by differentiating Eq.(6.2) with
respect to x
xx
p
x
p
x
v
y
u
yx
uv
x
u
x
uu
tx
u
+
+
+
+
=
22
222
2
2211
Differentiate Eq. (6.1) with respect to time
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Procedure
Second-order, centered finite-difference quotients at time t
xx
pp
x
ppp
x
vv
y
uu
yx
uuuu
v
x
uu
x
uuuu
tx
u
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
ji
t
jit
ji
tji
tji
tji
tji
tjit
ji
t
ji
++
+
+++
+
+=
+++
++++++
++
22)(
1
)(
21
22))((4
)2
()(
2
,1,1,1,1
2
,
2
,1,,1
,
,1,11,1,1,11,11,11,1
,
2,1,1
2
,1,,1
,
,
2
xx
p
x
p
x
v
y
u
yx
uv
x
u
x
uu
tx
u
+
+
+
+
=
22
222
2
2211
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Procedure
The remaining derivatives are first spatial derivatives, namely, u/x,v/y,,/x,and /y,replaced by second-order central differences
t
v
ytyv
ty
v
ty
v
t
u
xtxu
tx
u
tx
u
t
+
+
+
+
+
+
+
=
2222
2
2
x
uu
x
u jijit
ji
+=
+
2
,1,1
,
The other remaining derivatives are first time derivatives, /t, u/t,and v/t.
/t has already been obtained from (6.9)
u/t, and v/t can be obtained, analogous to the form of /t
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Procedure
We now have known values at time t for all three terms
on the right side of the above equation.
This allows the calculation of density at time t+t.
Repeat the above procedures to find the remainingflow-field variables at grid point (i,j) at time t+t.
+
+
+=+
2
)(2
,
2
2
,,,
t
ttt
t
ji
t
ji
t
ji
tt
ji
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The essence of the Lax-Wendroff methodObtain explicitly the flow-field variables at grid point (i,j) at time t+tfrom the known flow-field variables at grid points (i,j), (I+1,j), (I-1,j),
and (i,j+1) at time t.
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Discussion on the technique
It has second order accuracy in both space and time.
The idea is straightforward, but algebra is lengthy.
Most of the lengthy algebra is associated with the second timederivative in Eqs. (6.6) to (6.8).
+
+
+=+2
)(2
,
2
2
,
,,
t
t
vt
t
vvv
t
ji
t
ji
t
ji
tt
ji
+
+
+=+
2
)( 2
,
2
2
,
,,
t
t
et
t
eee
t
ji
t
ji
t
ji
tt
ji
(6.7)
(6.8)
+
+
+=+2
)(2
,
2
2
,
,,
t
t
ut
t
uuu
t
ji
t
ji
t
ji
tt
ji (6.6)
Much of this algebra can be cut by MacCormacks techniques
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