University of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Euler’s Method
For t
n
= t
0
+ n
×
h,
0
≤
n
≤
(t
last
–t
0
)/h
an approximate solution is given by
u
n+1
= u
n
+ h*f(u
n
, t
n
)
Inaccurate and can be unstable: should not be used!
Linear approximation is used
to get the next point
t
n
t
n+1
t
n+2
u
Poor guess
Real value
u
n
h
The main problem with Euler method is that only information from the
beginning of the interval is used to extrapolate the value at the other side of the
interval.
In other words,
only slope of the function is taken into account, the
curvature is ignored
.
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View Full DocumentUniversity of Virginia, MSE 4270/6270: Introduction to Atomistic Simulations, Leonid Zhigilei
Error in Euler’s method
()
( )
( )
...
dt
t
u
d
p!
h
...
dt
t
u
d
2!
h
dt
t
du
h
u
u
p
n
p
p
2
n
2
2
n
n
1
n
+
+
+
+
+
=
+
Let’s use Taylor expansion to get the exact solution:
Assuming that u
n
is exact, the estimation of local error due to the truncation is
)
h
(
O
t
,
u
hf
u
u
2
n
n
1
n
+
+
=
+
This is a local error.
As it accumulates over the h
1
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 Fall '11
 Zhigilei
 Numerical Analysis, Derivative, Midpoint method, Runge–Kutta methods, Atomistic Simulations, Leonid Zhigilei

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