Calculating the constants for Runge-Kutta order 4 in other form
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I know why Runge-Kutta order 4 can be written in the below form I guess. But I don't know how I should go about to calculate the constants required.
Runge-Kutta order 4 can also be written in the below form:
$$w_0 = alpha_{0}$$
$$
w_{i+1} = w_i + frac{h}{6}f(t_i, w_i)+frac{h}{3}f(t_i+alpha_1h,w_i+delta_1f(t_i,w_i))+frac{h}{3}f(t_i+alpha_2h,w_i+delta_2hf(t_i+gamma_2h,w_i+gamma_3hf(t_i,w_i)))+frac{h}{6}f(t_i+alpha_3h,w_i+delta_3hf(t_i+gamma_4h,w_i+gamma_5hf(t_i+gamma_6h,w_i+gamma_7hf(t_i,w_i))))
$$
How should I find out the constants $alpha_1$, $alpha_2$, $alpha_3$, $delta_1$, $delta_2$, $delta_3$, $gamma_2$, $gamma_3$, $gamma_4$, $gamma_5$, $gamma_6$, $gamma_7$ ?
I tried to somehow write the taylor expansion for the function and compare the coefficients but my solution leaded some unsolvable equations.
ordinary-differential-equations numerical-methods taylor-expansion runge-kutta-methods
asked Dec 29 '18 at 23:42
FreeMindFreeMind
9601133
$endgroup$
|
show 2 more comments
$begingroup$
I know why Runge-Kutta order 4 can be written in the below form I guess. But I don't know how I should go about to calculate the constants required.
Runge-Kutta order 4 can also be written in the below form:
$$w_0 = alpha_{0}$$
$$
w_{i+1} = w_i + frac{h}{6}f(t_i, w_i)+frac{h}{3}f(t_i+alpha_1h,w_i+delta_1f(t_i,w_i))+frac{h}{3}f(t_i+alpha_2h,w_i+delta_2hf(t_i+gamma_2h,w_i+gamma_3hf(t_i,w_i)))+frac{h}{6}f(t_i+alpha_3h,w_i+delta_3hf(t_i+gamma_4h,w_i+gamma_5hf(t_i+gamma_6h,w_i+gamma_7hf(t_i,w_i))))
$$
How should I find out the constants $alpha_1$, $alpha_2$, $alpha_3$, $delta_1$, $delta_2$, $delta_3$, $gamma_2$, $gamma_3$, $gamma_4$, $gamma_5$, $gamma_6$, $gamma_7$ ?
I tried to somehow write the taylor expansion for the function and compare the coefficients but my solution leaded some unsolvable equations.
ordinary-differential-equations numerical-methods taylor-expansion runge-kutta-methods
asked Dec 29 '18 at 23:42
FreeMindFreeMind
9601133
$endgroup$
$begingroup$
en.wikipedia.org/wiki/…
$endgroup$
– caverac
Dec 29 '18 at 23:45
$begingroup$
@caverac But the one in the wikipedia is not in the form I have posed the question.
$endgroup$
– FreeMind
Dec 29 '18 at 23:46
$begingroup$
This does not make sense. Where does this problem come from? RK4 requires 4 evaluations of $f$ per step. The form you wrote requires up to $7$ evaluations of $f$, why make this over-general formula?
$endgroup$
– LutzL
Dec 30 '18 at 11:47
$begingroup$
Essentially, you are asking about the order equations for the Butcher tableau begin{array}{l|lllllll} 0&\ α_1&δ_1 \ γ_2&γ_3 \ α_2&0&0&δ_1 \ γ_6&γ_7 \ γ_4&0&0&0&0&γ_5 \ α_2&0&0&0&0&0&δ_1 \ hline &frac16&frac13&0&frac13&0&0&frac16 end{array}
$endgroup$
– LutzL
Dec 30 '18 at 11:54
1
$begingroup$
@FreeMind: I think you misunderstood the exercise. The task is not to derive these constants from some order conditions, but rather to start with the classical fourth-order Runge-Kutta method for the initial-value problem $w' = f(t,w)$, $w(t_0)=alpha_0$, and to rewrite it in the form above, which gives you the values of the constants.
$endgroup$
– Christoph
Jan 1 at 9:15
|
show 2 more comments
$begingroup$
I know why Runge-Kutta order 4 can be written in the below form I guess. But I don't know how I should go about to calculate the constants required.
Runge-Kutta order 4 can also be written in the below form:
$$w_0 = alpha_{0}$$
$$
w_{i+1} = w_i + frac{h}{6}f(t_i, w_i)+frac{h}{3}f(t_i+alpha_1h,w_i+delta_1f(t_i,w_i))+frac{h}{3}f(t_i+alpha_2h,w_i+delta_2hf(t_i+gamma_2h,w_i+gamma_3hf(t_i,w_i)))+frac{h}{6}f(t_i+alpha_3h,w_i+delta_3hf(t_i+gamma_4h,w_i+gamma_5hf(t_i+gamma_6h,w_i+gamma_7hf(t_i,w_i))))
$$
How should I find out the constants $alpha_1$, $alpha_2$, $alpha_3$, $delta_1$, $delta_2$, $delta_3$, $gamma_2$, $gamma_3$, $gamma_4$, $gamma_5$, $gamma_6$, $gamma_7$ ?
I tried to somehow write the taylor expansion for the function and compare the coefficients but my solution leaded some unsolvable equations.
ordinary-differential-equations numerical-methods taylor-expansion runge-kutta-methods
asked Dec 29 '18 at 23:42
FreeMindFreeMind
9601133
$endgroup$
I know why Runge-Kutta order 4 can be written in the below form I guess. But I don't know how I should go about to calculate the constants required.
Runge-Kutta order 4 can also be written in the below form:
$$w_0 = alpha_{0}$$
$$
w_{i+1} = w_i + frac{h}{6}f(t_i, w_i)+frac{h}{3}f(t_i+alpha_1h,w_i+delta_1f(t_i,w_i))+frac{h}{3}f(t_i+alpha_2h,w_i+delta_2hf(t_i+gamma_2h,w_i+gamma_3hf(t_i,w_i)))+frac{h}{6}f(t_i+alpha_3h,w_i+delta_3hf(t_i+gamma_4h,w_i+gamma_5hf(t_i+gamma_6h,w_i+gamma_7hf(t_i,w_i))))
$$
How should I find out the constants $alpha_1$, $alpha_2$, $alpha_3$, $delta_1$, $delta_2$, $delta_3$, $gamma_2$, $gamma_3$, $gamma_4$, $gamma_5$, $gamma_6$, $gamma_7$ ?
I tried to somehow write the taylor expansion for the function and compare the coefficients but my solution leaded some unsolvable equations.
ordinary-differential-equations numerical-methods taylor-expansion runge-kutta-methods
ordinary-differential-equations numerical-methods taylor-expansion runge-kutta-methods
asked Dec 29 '18 at 23:42
FreeMindFreeMind
9601133
asked Dec 29 '18 at 23:42
FreeMindFreeMind
9601133
edited Dec 30 '18 at 1:06
FreeMind
asked Dec 29 '18 at 23:42
FreeMindFreeMind
9601133
asked Dec 29 '18 at 23:42
FreeMindFreeMind
9601133
asked Dec 29 '18 at 23:42
FreeMindFreeMind
9601133
9601133
$begingroup$
en.wikipedia.org/wiki/…
$endgroup$
– caverac
Dec 29 '18 at 23:45
$begingroup$
@caverac But the one in the wikipedia is not in the form I have posed the question.
$endgroup$
– FreeMind
Dec 29 '18 at 23:46
$begingroup$
This does not make sense. Where does this problem come from? RK4 requires 4 evaluations of $f$ per step. The form you wrote requires up to $7$ evaluations of $f$, why make this over-general formula?
$endgroup$
– LutzL
Dec 30 '18 at 11:47
$begingroup$
Essentially, you are asking about the order equations for the Butcher tableau begin{array}{l|lllllll} 0&\ α_1&δ_1 \ γ_2&γ_3 \ α_2&0&0&δ_1 \ γ_6&γ_7 \ γ_4&0&0&0&0&γ_5 \ α_2&0&0&0&0&0&δ_1 \ hline &frac16&frac13&0&frac13&0&0&frac16 end{array}
$endgroup$
– LutzL
Dec 30 '18 at 11:54
1
$begingroup$
@FreeMind: I think you misunderstood the exercise. The task is not to derive these constants from some order conditions, but rather to start with the classical fourth-order Runge-Kutta method for the initial-value problem $w' = f(t,w)$, $w(t_0)=alpha_0$, and to rewrite it in the form above, which gives you the values of the constants.
$endgroup$
– Christoph
Jan 1 at 9:15
|
show 2 more comments
$begingroup$
en.wikipedia.org/wiki/…
$endgroup$
– caverac
Dec 29 '18 at 23:45
$begingroup$
@caverac But the one in the wikipedia is not in the form I have posed the question.
$endgroup$
– FreeMind
Dec 29 '18 at 23:46
$begingroup$
This does not make sense. Where does this problem come from? RK4 requires 4 evaluations of $f$ per step. The form you wrote requires up to $7$ evaluations of $f$, why make this over-general formula?
$endgroup$
– LutzL
Dec 30 '18 at 11:47
$begingroup$
Essentially, you are asking about the order equations for the Butcher tableau begin{array}{l|lllllll} 0&\ α_1&δ_1 \ γ_2&γ_3 \ α_2&0&0&δ_1 \ γ_6&γ_7 \ γ_4&0&0&0&0&γ_5 \ α_2&0&0&0&0&0&δ_1 \ hline &frac16&frac13&0&frac13&0&0&frac16 end{array}
$endgroup$
– LutzL
Dec 30 '18 at 11:54
1
$begingroup$
@FreeMind: I think you misunderstood the exercise. The task is not to derive these constants from some order conditions, but rather to start with the classical fourth-order Runge-Kutta method for the initial-value problem $w' = f(t,w)$, $w(t_0)=alpha_0$, and to rewrite it in the form above, which gives you the values of the constants.
$endgroup$
– Christoph
Jan 1 at 9:15
$begingroup$
en.wikipedia.org/wiki/…
$endgroup$
– caverac
Dec 29 '18 at 23:45
$begingroup$
en.wikipedia.org/wiki/…
$endgroup$
– caverac
Dec 29 '18 at 23:45
$begingroup$
@caverac But the one in the wikipedia is not in the form I have posed the question.
$endgroup$
– FreeMind
Dec 29 '18 at 23:46
$begingroup$
@caverac But the one in the wikipedia is not in the form I have posed the question.
$endgroup$
– FreeMind
Dec 29 '18 at 23:46
$begingroup$
This does not make sense. Where does this problem come from? RK4 requires 4 evaluations of $f$ per step. The form you wrote requires up to $7$ evaluations of $f$, why make this over-general formula?
$endgroup$
– LutzL
Dec 30 '18 at 11:47
$begingroup$
This does not make sense. Where does this problem come from? RK4 requires 4 evaluations of $f$ per step. The form you wrote requires up to $7$ evaluations of $f$, why make this over-general formula?
$endgroup$
– LutzL
Dec 30 '18 at 11:47
$begingroup$
Essentially, you are asking about the order equations for the Butcher tableau begin{array}{l|lllllll} 0&\ α_1&δ_1 \ γ_2&γ_3 \ α_2&0&0&δ_1 \ γ_6&γ_7 \ γ_4&0&0&0&0&γ_5 \ α_2&0&0&0&0&0&δ_1 \ hline &frac16&frac13&0&frac13&0&0&frac16 end{array}
$endgroup$
– LutzL
Dec 30 '18 at 11:54
$begingroup$
Essentially, you are asking about the order equations for the Butcher tableau begin{array}{l|lllllll} 0&\ α_1&δ_1 \ γ_2&γ_3 \ α_2&0&0&δ_1 \ γ_6&γ_7 \ γ_4&0&0&0&0&γ_5 \ α_2&0&0&0&0&0&δ_1 \ hline &frac16&frac13&0&frac13&0&0&frac16 end{array}
$endgroup$
– LutzL
Dec 30 '18 at 11:54
1
1
$begingroup$
@FreeMind: I think you misunderstood the exercise. The task is not to derive these constants from some order conditions, but rather to start with the classical fourth-order Runge-Kutta method for the initial-value problem $w' = f(t,w)$, $w(t_0)=alpha_0$, and to rewrite it in the form above, which gives you the values of the constants.
$endgroup$
– Christoph
Jan 1 at 9:15
$begingroup$
@FreeMind: I think you misunderstood the exercise. The task is not to derive these constants from some order conditions, but rather to start with the classical fourth-order Runge-Kutta method for the initial-value problem $w' = f(t,w)$, $w(t_0)=alpha_0$, and to rewrite it in the form above, which gives you the values of the constants.
$endgroup$
– Christoph
Jan 1 at 9:15
|
show 2 more comments
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$begingroup$
en.wikipedia.org/wiki/…
$endgroup$
– caverac
Dec 29 '18 at 23:45
$begingroup$
@caverac But the one in the wikipedia is not in the form I have posed the question.
$endgroup$
– FreeMind
Dec 29 '18 at 23:46
$begingroup$
This does not make sense. Where does this problem come from? RK4 requires 4 evaluations of $f$ per step. The form you wrote requires up to $7$ evaluations of $f$, why make this over-general formula?
$endgroup$
– LutzL
Dec 30 '18 at 11:47
$begingroup$
Essentially, you are asking about the order equations for the Butcher tableau begin{array}{l|lllllll} 0&\ α_1&δ_1 \ γ_2&γ_3 \ α_2&0&0&δ_1 \ γ_6&γ_7 \ γ_4&0&0&0&0&γ_5 \ α_2&0&0&0&0&0&δ_1 \ hline &frac16&frac13&0&frac13&0&0&frac16 end{array}
$endgroup$
– LutzL
Dec 30 '18 at 11:54
1
$begingroup$
@FreeMind: I think you misunderstood the exercise. The task is not to derive these constants from some order conditions, but rather to start with the classical fourth-order Runge-Kutta method for the initial-value problem $w' = f(t,w)$, $w(t_0)=alpha_0$, and to rewrite it in the form above, which gives you the values of the constants.
$endgroup$
– Christoph
Jan 1 at 9:15