Use of the numerical solution on finite interval to describe the behavior of PDE on $Bbb R$
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When we want to simulate the solution of some one dimensional PDE
$$ mathcalL(u) = f,quad texton mathbbR $$
on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $Bbb R$?
pde numerical-methods
edited Sep 4 at 20:41
Harry49
5,0052825
asked Sep 4 at 10:18
Panasun
33
add a comment |Â
up vote
0
down vote
favorite
When we want to simulate the solution of some one dimensional PDE
$$ mathcalL(u) = f,quad texton mathbbR $$
on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $Bbb R$?
pde numerical-methods
edited Sep 4 at 20:41
Harry49
5,0052825
asked Sep 4 at 10:18
Panasun
33
In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
– Ian
Sep 4 at 20:53
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
When we want to simulate the solution of some one dimensional PDE
$$ mathcalL(u) = f,quad texton mathbbR $$
on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $Bbb R$?
pde numerical-methods
edited Sep 4 at 20:41
Harry49
5,0052825
asked Sep 4 at 10:18
Panasun
33
When we want to simulate the solution of some one dimensional PDE
$$ mathcalL(u) = f,quad texton mathbbR $$
on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $Bbb R$?
pde numerical-methods
pde numerical-methods
edited Sep 4 at 20:41
Harry49
5,0052825
asked Sep 4 at 10:18
Panasun
33
edited Sep 4 at 20:41
Harry49
5,0052825
asked Sep 4 at 10:18
Panasun
33
edited Sep 4 at 20:41
Harry49
5,0052825
edited Sep 4 at 20:41
Harry49
5,0052825
edited Sep 4 at 20:41
Harry49
5,0052825
5,0052825
asked Sep 4 at 10:18
Panasun
33
asked Sep 4 at 10:18
Panasun
33
asked Sep 4 at 10:18
Panasun
33
33
In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
– Ian
Sep 4 at 20:53
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
add a comment |Â
In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
– Ian
Sep 4 at 20:53
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
– Ian
Sep 4 at 20:53
In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
– Ian
Sep 4 at 20:53
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
add a comment |Â
1 Answer
1
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0
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accepted
One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.
answered Sep 4 at 20:49
Harry49
5,0052825
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.
answered Sep 4 at 20:49
Harry49
5,0052825
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
add a comment |Â
up vote
0
down vote
accepted
One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.
answered Sep 4 at 20:49
Harry49
5,0052825
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.
answered Sep 4 at 20:49
Harry49
5,0052825
One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.
answered Sep 4 at 20:49
Harry49
5,0052825
edited Sep 4 at 20:54
answered Sep 4 at 20:49
Harry49
5,0052825
answered Sep 4 at 20:49
Harry49
5,0052825
answered Sep 4 at 20:49
Harry49
5,0052825
5,0052825
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
add a comment |Â
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11
add a comment |Â
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In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
– Ian
Sep 4 at 20:53
Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11