2 Layer Finite Difference Scheme PDE
I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.
$frac{∂^2}{∂x^2}(k(x,t) frac{∂^2U(x,t)}{∂t^2})=0$
k is just a parameter, which is dependent on x and t. The problem is there are second order derivatives, and I don't know how to couple them together.
pde partial-derivative boundary-value-problem finite-difference-methods
asked Dec 26 at 0:49
P. Yastrebov
32
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P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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add a comment |
I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.
$frac{∂^2}{∂x^2}(k(x,t) frac{∂^2U(x,t)}{∂t^2})=0$
k is just a parameter, which is dependent on x and t. The problem is there are second order derivatives, and I don't know how to couple them together.
pde partial-derivative boundary-value-problem finite-difference-methods
asked Dec 26 at 0:49
P. Yastrebov
32
New contributor
P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.
$frac{∂^2}{∂x^2}(k(x,t) frac{∂^2U(x,t)}{∂t^2})=0$
k is just a parameter, which is dependent on x and t. The problem is there are second order derivatives, and I don't know how to couple them together.
pde partial-derivative boundary-value-problem finite-difference-methods
asked Dec 26 at 0:49
P. Yastrebov
32
New contributor
P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I have this PDE, and I need to build a 2 layer Finite Difference scheme for it.
$frac{∂^2}{∂x^2}(k(x,t) frac{∂^2U(x,t)}{∂t^2})=0$
k is just a parameter, which is dependent on x and t. The problem is there are second order derivatives, and I don't know how to couple them together.
pde partial-derivative boundary-value-problem finite-difference-methods
pde partial-derivative boundary-value-problem finite-difference-methods
asked Dec 26 at 0:49
P. Yastrebov
32
New contributor
P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 at 0:49
P. Yastrebov
32
New contributor
P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Dec 26 at 0:54
asked Dec 26 at 0:49
P. Yastrebov
32
New contributor
P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 26 at 0:49
P. Yastrebov
32
asked Dec 26 at 0:49
P. Yastrebov
32
32
New contributor
P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
P. Yastrebov is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
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1 Answer
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First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
$$
x_i = x_0 + icdotDelta x
quadquad
t_j = t_0 + jcdotDelta t
$$
and for a general funciton $f(x,t)$, define
$$
f_{i,j} = f(x_i,t_j)
$$
which, on a uniform grid, has approximations to its (unmixed) second derivatives
$$
left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
$$
$$
left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
$$
For your given equation, we first define
$$
F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
$$
thus
$$
0 = frac{partial^2 F}{partial x^2}(x,t).
$$
Discretising:
begin{align}
0
&= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
\
&= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
\
&= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
end{align}
To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.
answered 2 days ago
Eddy
774412
add a comment |
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1 Answer
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1 Answer
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First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
$$
x_i = x_0 + icdotDelta x
quadquad
t_j = t_0 + jcdotDelta t
$$
and for a general funciton $f(x,t)$, define
$$
f_{i,j} = f(x_i,t_j)
$$
which, on a uniform grid, has approximations to its (unmixed) second derivatives
$$
left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
$$
$$
left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
$$
For your given equation, we first define
$$
F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
$$
thus
$$
0 = frac{partial^2 F}{partial x^2}(x,t).
$$
Discretising:
begin{align}
0
&= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
\
&= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
\
&= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
end{align}
To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.
answered 2 days ago
Eddy
774412
add a comment |
First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
$$
x_i = x_0 + icdotDelta x
quadquad
t_j = t_0 + jcdotDelta t
$$
and for a general funciton $f(x,t)$, define
$$
f_{i,j} = f(x_i,t_j)
$$
which, on a uniform grid, has approximations to its (unmixed) second derivatives
$$
left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
$$
$$
left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
$$
For your given equation, we first define
$$
F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
$$
thus
$$
0 = frac{partial^2 F}{partial x^2}(x,t).
$$
Discretising:
begin{align}
0
&= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
\
&= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
\
&= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
end{align}
To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.
answered 2 days ago
Eddy
774412
add a comment |
First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
$$
x_i = x_0 + icdotDelta x
quadquad
t_j = t_0 + jcdotDelta t
$$
and for a general funciton $f(x,t)$, define
$$
f_{i,j} = f(x_i,t_j)
$$
which, on a uniform grid, has approximations to its (unmixed) second derivatives
$$
left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
$$
$$
left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
$$
For your given equation, we first define
$$
F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
$$
thus
$$
0 = frac{partial^2 F}{partial x^2}(x,t).
$$
Discretising:
begin{align}
0
&= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
\
&= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
\
&= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
end{align}
To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.
answered 2 days ago
Eddy
774412
First we need to define a grid for the finite difference scheme. I will be using a regular gird here, but the same approach can be used for an irregular grid if the appropriate approximations to derivatives are used. The grid is
$$
x_i = x_0 + icdotDelta x
quadquad
t_j = t_0 + jcdotDelta t
$$
and for a general funciton $f(x,t)$, define
$$
f_{i,j} = f(x_i,t_j)
$$
which, on a uniform grid, has approximations to its (unmixed) second derivatives
$$
left(frac{partial^2 f}{partial x^2}right)_{i,j} = frac{f_{i+1,j} - 2f_{i,j} + f_{i-1,j}}{Delta x^2}
$$
$$
left(frac{partial^2 f}{partial t^2}right)_{i,j} = frac{f_{i,j+1} - 2f_{i,j} + f_{i,j-1}}{Delta t^2}.
$$
For your given equation, we first define
$$
F(x,t) = k(x,t) cdot frac{partial^2 U}{partial t^2}(x,t)
$$
thus
$$
0 = frac{partial^2 F}{partial x^2}(x,t).
$$
Discretising:
begin{align}
0
&= frac{1}{Delta x^2}left[F_{i+1,j} - 2F_{i,j} + F_{i-1,j}right]
\
&= frac{1}{Delta x^2}left[k_{i+1,j}left(frac{partial^2 U}{partial t^2}right)_{i+1,j} - 2k_{i,j}left(frac{partial^2 U}{partial t^2}right)_{i,j} + k_{i-1,j}left(frac{partial^2 U}{partial t^2}right)_{i-1,j}right]
\
&= frac{1}{Delta x^2Delta t^2}left[k_{i+1,j}left(U_{i+1,j+1}-2U_{i+1,j}+U_{i+1,j-1}right) - 2k_{i,j}left(U_{i,j+1}-2U_{i,j}+U_{i,j-1}right) +
k_{i-1,j}left(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1}right)right].
end{align}
To use this in a finite difference scheme you will need to specify $U_{i,j}$ at the first two time instances, as well as the left and right endpoints.
answered 2 days ago
Eddy
774412
answered 2 days ago
Eddy
774412
answered 2 days ago
Eddy
774412
answered 2 days ago
Eddy
774412
774412
add a comment |
add a comment |
P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.
P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.
P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.
P. Yastrebov is a new contributor. Be nice, and check out our Code of Conduct.
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