Novel techniques for Numerical Relativity
Manuel Tiglio, LSU (KITP Gravitation Conf 5/14/03)
2
Overview, main ideas
l
Method of lines
: from semidiscrete to fully discrete stability.
l
Numerical stability as a discrete version of
well posedness
.
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Numerical stability through
discrete energy estimates
:
Summation by parts
Boundary conditions
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Going beyond numerical stability:
preventing errors from growing
.
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Application: black hole
excision
.
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Dynamical control of
discrete constraint violations
.
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Multi-patch
evolution.
The method of lines
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Given a set of differential equations,
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Semidiscrete problem: first discretize space but not time:
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Prove numerical stability for the semidiscrete problem (system of ODE’s).
u
t
x
u
B
u
D
t
x
u
A
u
i
i
i
t
)
,
,
(
)
,
,
(
r
r
+
=
∂
k
j
i
k
j
i
i
k
j
i
i
k
j
i
i
k
j
i
u
t
x
u
B
u
D
t
x
u
A
u
dt
d
,
,
,
,
,
,
,
,
,
,
)
,
,
(
)
,
,
(
+
=
l
If now you use appropriate time integrators, stability for the fully discrete
problem follows. The details of what you did in the semidiscrete case do not
matter, provided that the semidiscrete problem was stable.
l
Get spatial discretizations that give semidiscrete stability, and time integrators
that are “locally stable”.
Combine them at will.