Finite difference method for the third order partial differential equation
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I would like to use the finite difference method for the initial problem of the following form:
$$u_t(t,x) + u_{xxx} (t,x) = f(t,x), u(0,x) = u_0(x).$$
I have a compatible difference scheme for it, but I have no idea how to select the function $f$ and the initial condition $u_0$ (with an almost compact supp), so that zero boundary values can be given. The exercise has a hint to select $e^{-x ^ 2}$ as $u_0$. Below I give the scheme:
$$frac{u_{m}^{n+1} - u_{m}^n}{k} +
frac{u_{m+2}^{n} - 3 u_{m+1}{n} + 3 u_m^n - u_{m-1}^n}{h^3} = f^n_m.$$
In addition, the task is to estimate experimentally for which values of the constant $ nu = frac{k}{h ^ 3} $ the scheme is stable.
I can program the appropriate scheme in MATLAB, but I have a problem with choosing the appropriate function $ f $. I will be very grateful for your help with explanations.
Thank you in advance!
pde matlab finite-differences
asked Dec 5 '18 at 9:48
WawMathematicianWawMathematician
13
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add a comment |
$begingroup$
I would like to use the finite difference method for the initial problem of the following form:
$$u_t(t,x) + u_{xxx} (t,x) = f(t,x), u(0,x) = u_0(x).$$
I have a compatible difference scheme for it, but I have no idea how to select the function $f$ and the initial condition $u_0$ (with an almost compact supp), so that zero boundary values can be given. The exercise has a hint to select $e^{-x ^ 2}$ as $u_0$. Below I give the scheme:
$$frac{u_{m}^{n+1} - u_{m}^n}{k} +
frac{u_{m+2}^{n} - 3 u_{m+1}{n} + 3 u_m^n - u_{m-1}^n}{h^3} = f^n_m.$$
In addition, the task is to estimate experimentally for which values of the constant $ nu = frac{k}{h ^ 3} $ the scheme is stable.
I can program the appropriate scheme in MATLAB, but I have a problem with choosing the appropriate function $ f $. I will be very grateful for your help with explanations.
Thank you in advance!
pde matlab finite-differences
asked Dec 5 '18 at 9:48
WawMathematicianWawMathematician
13
$endgroup$
$begingroup$
Please edit your post. What requirements are given to chose $f$? Why not setting $f=0$? Which properties are wanted? Btw, you could have used a central difference to avoid introducing numerical anisotropy instead of your sided difference.
$endgroup$
– Harry49
Dec 5 '18 at 19:02
add a comment |
$begingroup$
I would like to use the finite difference method for the initial problem of the following form:
$$u_t(t,x) + u_{xxx} (t,x) = f(t,x), u(0,x) = u_0(x).$$
I have a compatible difference scheme for it, but I have no idea how to select the function $f$ and the initial condition $u_0$ (with an almost compact supp), so that zero boundary values can be given. The exercise has a hint to select $e^{-x ^ 2}$ as $u_0$. Below I give the scheme:
$$frac{u_{m}^{n+1} - u_{m}^n}{k} +
frac{u_{m+2}^{n} - 3 u_{m+1}{n} + 3 u_m^n - u_{m-1}^n}{h^3} = f^n_m.$$
In addition, the task is to estimate experimentally for which values of the constant $ nu = frac{k}{h ^ 3} $ the scheme is stable.
I can program the appropriate scheme in MATLAB, but I have a problem with choosing the appropriate function $ f $. I will be very grateful for your help with explanations.
Thank you in advance!
pde matlab finite-differences
asked Dec 5 '18 at 9:48
WawMathematicianWawMathematician
13
$endgroup$
I would like to use the finite difference method for the initial problem of the following form:
$$u_t(t,x) + u_{xxx} (t,x) = f(t,x), u(0,x) = u_0(x).$$
I have a compatible difference scheme for it, but I have no idea how to select the function $f$ and the initial condition $u_0$ (with an almost compact supp), so that zero boundary values can be given. The exercise has a hint to select $e^{-x ^ 2}$ as $u_0$. Below I give the scheme:
$$frac{u_{m}^{n+1} - u_{m}^n}{k} +
frac{u_{m+2}^{n} - 3 u_{m+1}{n} + 3 u_m^n - u_{m-1}^n}{h^3} = f^n_m.$$
In addition, the task is to estimate experimentally for which values of the constant $ nu = frac{k}{h ^ 3} $ the scheme is stable.
I can program the appropriate scheme in MATLAB, but I have a problem with choosing the appropriate function $ f $. I will be very grateful for your help with explanations.
Thank you in advance!
pde matlab finite-differences
pde matlab finite-differences
asked Dec 5 '18 at 9:48
WawMathematicianWawMathematician
13
asked Dec 5 '18 at 9:48
WawMathematicianWawMathematician
13
asked Dec 5 '18 at 9:48
WawMathematicianWawMathematician
13
asked Dec 5 '18 at 9:48
WawMathematicianWawMathematician
13
asked Dec 5 '18 at 9:48
WawMathematicianWawMathematician
13
13
$begingroup$
Please edit your post. What requirements are given to chose $f$? Why not setting $f=0$? Which properties are wanted? Btw, you could have used a central difference to avoid introducing numerical anisotropy instead of your sided difference.
$endgroup$
– Harry49
Dec 5 '18 at 19:02
add a comment |
$begingroup$
Please edit your post. What requirements are given to chose $f$? Why not setting $f=0$? Which properties are wanted? Btw, you could have used a central difference to avoid introducing numerical anisotropy instead of your sided difference.
$endgroup$
– Harry49
Dec 5 '18 at 19:02
$begingroup$
Please edit your post. What requirements are given to chose $f$? Why not setting $f=0$? Which properties are wanted? Btw, you could have used a central difference to avoid introducing numerical anisotropy instead of your sided difference.
$endgroup$
– Harry49
Dec 5 '18 at 19:02
$begingroup$
Please edit your post. What requirements are given to chose $f$? Why not setting $f=0$? Which properties are wanted? Btw, you could have used a central difference to avoid introducing numerical anisotropy instead of your sided difference.
$endgroup$
– Harry49
Dec 5 '18 at 19:02
add a comment |
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$begingroup$
Please edit your post. What requirements are given to chose $f$? Why not setting $f=0$? Which properties are wanted? Btw, you could have used a central difference to avoid introducing numerical anisotropy instead of your sided difference.
$endgroup$
– Harry49
Dec 5 '18 at 19:02