Discretization of second order nonlinear ODE using finite difference approximation not correct
I have the differential equation
$$y'' + x(y^2)' - 2y^2 = g(x) Longleftrightarrow y'' + x2yy'-2y^2 = g(x).$$
Using finite-difference approximations
$$y''(x_m) approx frac{Y_{m-1} - 2Y_m + Y_{m+1}}{Delta x^2},$$
$$y'(x_m) approx frac{Y_{m+1} - Y_{m-1}}{2Delta x},$$
$$y(x_m) approx Y_m,$$
I get
$$frac{Y_{m-1} - 2Y_m + Y_{m+1}}{Delta x^2} + frac{x_m}{Delta x}Y_m(Y_{m+1}-Y_{m-1}) - 2Y_m^2 = g(x_m).$$
However, the answer is supposed to be
Why is the second term in my answer wrong?
differential-equations numerical-methods nonlinear-system finite-differences finite-difference-methods
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I have the differential equation
$$y'' + x(y^2)' - 2y^2 = g(x) Longleftrightarrow y'' + x2yy'-2y^2 = g(x).$$
Using finite-difference approximations
$$y''(x_m) approx frac{Y_{m-1} - 2Y_m + Y_{m+1}}{Delta x^2},$$
$$y'(x_m) approx frac{Y_{m+1} - Y_{m-1}}{2Delta x},$$
$$y(x_m) approx Y_m,$$
I get
$$frac{Y_{m-1} - 2Y_m + Y_{m+1}}{Delta x^2} + frac{x_m}{Delta x}Y_m(Y_{m+1}-Y_{m-1}) - 2Y_m^2 = g(x_m).$$
However, the answer is supposed to be
Why is the second term in my answer wrong?
differential-equations numerical-methods nonlinear-system finite-differences finite-difference-methods
add a comment |
I have the differential equation
$$y'' + x(y^2)' - 2y^2 = g(x) Longleftrightarrow y'' + x2yy'-2y^2 = g(x).$$
Using finite-difference approximations
$$y''(x_m) approx frac{Y_{m-1} - 2Y_m + Y_{m+1}}{Delta x^2},$$
$$y'(x_m) approx frac{Y_{m+1} - Y_{m-1}}{2Delta x},$$
$$y(x_m) approx Y_m,$$
I get
$$frac{Y_{m-1} - 2Y_m + Y_{m+1}}{Delta x^2} + frac{x_m}{Delta x}Y_m(Y_{m+1}-Y_{m-1}) - 2Y_m^2 = g(x_m).$$
However, the answer is supposed to be
Why is the second term in my answer wrong?
differential-equations numerical-methods nonlinear-system finite-differences finite-difference-methods
I have the differential equation
$$y'' + x(y^2)' - 2y^2 = g(x) Longleftrightarrow y'' + x2yy'-2y^2 = g(x).$$
Using finite-difference approximations
$$y''(x_m) approx frac{Y_{m-1} - 2Y_m + Y_{m+1}}{Delta x^2},$$
$$y'(x_m) approx frac{Y_{m+1} - Y_{m-1}}{2Delta x},$$
$$y(x_m) approx Y_m,$$
I get
$$frac{Y_{m-1} - 2Y_m + Y_{m+1}}{Delta x^2} + frac{x_m}{Delta x}Y_m(Y_{m+1}-Y_{m-1}) - 2Y_m^2 = g(x_m).$$
However, the answer is supposed to be
Why is the second term in my answer wrong?
differential-equations numerical-methods nonlinear-system finite-differences finite-difference-methods
differential-equations numerical-methods nonlinear-system finite-differences finite-difference-methods
edited yesterday
LutzL
56.3k42054
56.3k42054
asked yesterday
Heuristics
470211
470211
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add a comment |
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In the second version they are using a finite difference representation of the term $y^2$:
$$
(y^2)' approx frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} tag{1}
$$
While you are using a finite representation of $2y y'$:
$$
2y y' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} tag{2}
$$
But they are equivalent, if you take (if the solution $y$ is smooth enough you can do approximate the value at a node with the average of its neighbor nodes)
$$
y_m approx frac{y_{m + 1} + y_{m - 1}}{2} tag{3}
$$
Then Eq. (2) becomes
$$
2yy' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} approx (y_{m + 1} + y_{m - 1})frac{y_{m + 1} - y_{m - 1}}{2Delta x} = frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} approx (y^2)'
$$
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
In the second version they are using a finite difference representation of the term $y^2$:
$$
(y^2)' approx frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} tag{1}
$$
While you are using a finite representation of $2y y'$:
$$
2y y' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} tag{2}
$$
But they are equivalent, if you take (if the solution $y$ is smooth enough you can do approximate the value at a node with the average of its neighbor nodes)
$$
y_m approx frac{y_{m + 1} + y_{m - 1}}{2} tag{3}
$$
Then Eq. (2) becomes
$$
2yy' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} approx (y_{m + 1} + y_{m - 1})frac{y_{m + 1} - y_{m - 1}}{2Delta x} = frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} approx (y^2)'
$$
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In the second version they are using a finite difference representation of the term $y^2$:
$$
(y^2)' approx frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} tag{1}
$$
While you are using a finite representation of $2y y'$:
$$
2y y' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} tag{2}
$$
But they are equivalent, if you take (if the solution $y$ is smooth enough you can do approximate the value at a node with the average of its neighbor nodes)
$$
y_m approx frac{y_{m + 1} + y_{m - 1}}{2} tag{3}
$$
Then Eq. (2) becomes
$$
2yy' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} approx (y_{m + 1} + y_{m - 1})frac{y_{m + 1} - y_{m - 1}}{2Delta x} = frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} approx (y^2)'
$$
add a comment |
In the second version they are using a finite difference representation of the term $y^2$:
$$
(y^2)' approx frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} tag{1}
$$
While you are using a finite representation of $2y y'$:
$$
2y y' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} tag{2}
$$
But they are equivalent, if you take (if the solution $y$ is smooth enough you can do approximate the value at a node with the average of its neighbor nodes)
$$
y_m approx frac{y_{m + 1} + y_{m - 1}}{2} tag{3}
$$
Then Eq. (2) becomes
$$
2yy' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} approx (y_{m + 1} + y_{m - 1})frac{y_{m + 1} - y_{m - 1}}{2Delta x} = frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} approx (y^2)'
$$
In the second version they are using a finite difference representation of the term $y^2$:
$$
(y^2)' approx frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} tag{1}
$$
While you are using a finite representation of $2y y'$:
$$
2y y' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} tag{2}
$$
But they are equivalent, if you take (if the solution $y$ is smooth enough you can do approximate the value at a node with the average of its neighbor nodes)
$$
y_m approx frac{y_{m + 1} + y_{m - 1}}{2} tag{3}
$$
Then Eq. (2) becomes
$$
2yy' approx 2y_mfrac{y_{m + 1} - y_{m - 1}}{2Delta x} approx (y_{m + 1} + y_{m - 1})frac{y_{m + 1} - y_{m - 1}}{2Delta x} = frac{y^2_{m + 1} - y^2_{m - 1}}{2Delta x} approx (y^2)'
$$
edited yesterday
answered yesterday
caverac
13.8k21030
13.8k21030
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