Mixing
Explicit and implicit scheme for third order PDE
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I have the following non-linear PDE: $frac{∂^3U}{∂t^2∂x}+ frac{∂U}{∂t}+ frac{∂^3U}{∂x^3}=f(x,t)$.
How is it possible to derive an explicit and implicit scheme for this equation? The problem is that there 2 layers in space for the first term, which means that on the last time step I will be having two points considering time. So, I derived such explicit scheme: $frac{U_{i,j+1}-2U_{i,j}+U_{i,j-1}-(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1})}{t^3x^2}+frac{U_{i+1,j}-3U_{i,j}+3U_{i-1,j}-U_{i-2,j}}{t^3}+frac{(U_{i,j+1}-U_{i,j-1})+(U_{i-1,j+1}-U_{i-1,j-1})}{4t}=f(x_{i-frac{1}{2}},t_j)$
That way my scheme could be considered explicit, only if I proceed the steps in space, not in time. Is it correct to do that?
pde partial-derivative finite-differences
asked Jan 13 at 16:31
P. YastrebovP. Yastrebov
32
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add a comment |
$begingroup$
I have the following non-linear PDE: $frac{∂^3U}{∂t^2∂x}+ frac{∂U}{∂t}+ frac{∂^3U}{∂x^3}=f(x,t)$.
How is it possible to derive an explicit and implicit scheme for this equation? The problem is that there 2 layers in space for the first term, which means that on the last time step I will be having two points considering time. So, I derived such explicit scheme: $frac{U_{i,j+1}-2U_{i,j}+U_{i,j-1}-(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1})}{t^3x^2}+frac{U_{i+1,j}-3U_{i,j}+3U_{i-1,j}-U_{i-2,j}}{t^3}+frac{(U_{i,j+1}-U_{i,j-1})+(U_{i-1,j+1}-U_{i-1,j-1})}{4t}=f(x_{i-frac{1}{2}},t_j)$
That way my scheme could be considered explicit, only if I proceed the steps in space, not in time. Is it correct to do that?
pde partial-derivative finite-differences
asked Jan 13 at 16:31
P. YastrebovP. Yastrebov
32
$endgroup$
1
$begingroup$
"Explicit/implicit" can only refer to the time stepping in my opinion. The values in the additional layers in space need to be determined from the boundary conditions on $U$. You did not say anything about the desired order of accuracy of your scheme (in space and time).
$endgroup$
– Christoph
Jan 14 at 3:49
add a comment |
$begingroup$
I have the following non-linear PDE: $frac{∂^3U}{∂t^2∂x}+ frac{∂U}{∂t}+ frac{∂^3U}{∂x^3}=f(x,t)$.
How is it possible to derive an explicit and implicit scheme for this equation? The problem is that there 2 layers in space for the first term, which means that on the last time step I will be having two points considering time. So, I derived such explicit scheme: $frac{U_{i,j+1}-2U_{i,j}+U_{i,j-1}-(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1})}{t^3x^2}+frac{U_{i+1,j}-3U_{i,j}+3U_{i-1,j}-U_{i-2,j}}{t^3}+frac{(U_{i,j+1}-U_{i,j-1})+(U_{i-1,j+1}-U_{i-1,j-1})}{4t}=f(x_{i-frac{1}{2}},t_j)$
That way my scheme could be considered explicit, only if I proceed the steps in space, not in time. Is it correct to do that?
pde partial-derivative finite-differences
asked Jan 13 at 16:31
P. YastrebovP. Yastrebov
32
$endgroup$
I have the following non-linear PDE: $frac{∂^3U}{∂t^2∂x}+ frac{∂U}{∂t}+ frac{∂^3U}{∂x^3}=f(x,t)$.
How is it possible to derive an explicit and implicit scheme for this equation? The problem is that there 2 layers in space for the first term, which means that on the last time step I will be having two points considering time. So, I derived such explicit scheme: $frac{U_{i,j+1}-2U_{i,j}+U_{i,j-1}-(U_{i-1,j+1}-2U_{i-1,j}+U_{i-1,j-1})}{t^3x^2}+frac{U_{i+1,j}-3U_{i,j}+3U_{i-1,j}-U_{i-2,j}}{t^3}+frac{(U_{i,j+1}-U_{i,j-1})+(U_{i-1,j+1}-U_{i-1,j-1})}{4t}=f(x_{i-frac{1}{2}},t_j)$
That way my scheme could be considered explicit, only if I proceed the steps in space, not in time. Is it correct to do that?
pde partial-derivative finite-differences
pde partial-derivative finite-differences
asked Jan 13 at 16:31
P. YastrebovP. Yastrebov
32
asked Jan 13 at 16:31
P. YastrebovP. Yastrebov
32
asked Jan 13 at 16:31
P. YastrebovP. Yastrebov
32
asked Jan 13 at 16:31
P. YastrebovP. Yastrebov
32
asked Jan 13 at 16:31
P. YastrebovP. Yastrebov
32
32
1
$begingroup$
"Explicit/implicit" can only refer to the time stepping in my opinion. The values in the additional layers in space need to be determined from the boundary conditions on $U$. You did not say anything about the desired order of accuracy of your scheme (in space and time).
$endgroup$
– Christoph
Jan 14 at 3:49
add a comment |
1
$begingroup$
"Explicit/implicit" can only refer to the time stepping in my opinion. The values in the additional layers in space need to be determined from the boundary conditions on $U$. You did not say anything about the desired order of accuracy of your scheme (in space and time).
$endgroup$
– Christoph
Jan 14 at 3:49
1
1
$begingroup$
"Explicit/implicit" can only refer to the time stepping in my opinion. The values in the additional layers in space need to be determined from the boundary conditions on $U$. You did not say anything about the desired order of accuracy of your scheme (in space and time).
$endgroup$
– Christoph
Jan 14 at 3:49
$begingroup$
"Explicit/implicit" can only refer to the time stepping in my opinion. The values in the additional layers in space need to be determined from the boundary conditions on $U$. You did not say anything about the desired order of accuracy of your scheme (in space and time).
$endgroup$
– Christoph
Jan 14 at 3:49
add a comment |
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$begingroup$
"Explicit/implicit" can only refer to the time stepping in my opinion. The values in the additional layers in space need to be determined from the boundary conditions on $U$. You did not say anything about the desired order of accuracy of your scheme (in space and time).
$endgroup$
– Christoph
Jan 14 at 3:49