$4^textth$ order Runge-Kutta method
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
I would like to know the motivation behind the choice of numbers or coefficients in front of $k_1$, $k_2$, $k_3$ and $k_4$ in $4^textth$ order Runge-Kutta method. There are many choices of the coefficients one can make. However, $frac16$, $frac13$, $frac13$, $frac16$ are the most popular set. Can anyone explain this point?
numerical-methods motivation runge-kutta-methods
edited Aug 22 at 10:46
Daniel Buck
2,5151625
asked Aug 20 at 16:11
Soumen Basak
161
add a comment |Â
up vote
3
down vote
favorite
I would like to know the motivation behind the choice of numbers or coefficients in front of $k_1$, $k_2$, $k_3$ and $k_4$ in $4^textth$ order Runge-Kutta method. There are many choices of the coefficients one can make. However, $frac16$, $frac13$, $frac13$, $frac16$ are the most popular set. Can anyone explain this point?
numerical-methods motivation runge-kutta-methods
edited Aug 22 at 10:46
Daniel Buck
2,5151625
asked Aug 20 at 16:11
Soumen Basak
161
Relevant: What's the motivation of Runge-Kutta method?, What is the motivation behind the Runge-Kutta method?.
– Frenzy Li
Aug 20 at 16:21
The old posts on similar topic do not have the answer of my question.
– Soumen Basak
Aug 20 at 16:24
Well, I did not start my comment with "Duplicate:", just "Relevant." In particular, the second post probably points in the correct direction but did not lay everything out explicitly.
– Frenzy Li
Aug 20 at 16:26
I suppose this particular choice of the coefficients has some advantages over other choices. I just wanted know those advantages with proper explanation.
– Soumen Basak
Aug 20 at 16:30
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I would like to know the motivation behind the choice of numbers or coefficients in front of $k_1$, $k_2$, $k_3$ and $k_4$ in $4^textth$ order Runge-Kutta method. There are many choices of the coefficients one can make. However, $frac16$, $frac13$, $frac13$, $frac16$ are the most popular set. Can anyone explain this point?
numerical-methods motivation runge-kutta-methods
edited Aug 22 at 10:46
Daniel Buck
2,5151625
asked Aug 20 at 16:11
Soumen Basak
161
I would like to know the motivation behind the choice of numbers or coefficients in front of $k_1$, $k_2$, $k_3$ and $k_4$ in $4^textth$ order Runge-Kutta method. There are many choices of the coefficients one can make. However, $frac16$, $frac13$, $frac13$, $frac16$ are the most popular set. Can anyone explain this point?
numerical-methods motivation runge-kutta-methods
edited Aug 22 at 10:46
Daniel Buck
2,5151625
asked Aug 20 at 16:11
Soumen Basak
161
edited Aug 22 at 10:46
Daniel Buck
2,5151625
edited Aug 22 at 10:46
Daniel Buck
2,5151625
edited Aug 22 at 10:46
Daniel Buck
2,5151625
2,5151625
asked Aug 20 at 16:11
Soumen Basak
161
asked Aug 20 at 16:11
Soumen Basak
161
asked Aug 20 at 16:11
Soumen Basak
161
161
Relevant: What's the motivation of Runge-Kutta method?, What is the motivation behind the Runge-Kutta method?.
– Frenzy Li
Aug 20 at 16:21
The old posts on similar topic do not have the answer of my question.
– Soumen Basak
Aug 20 at 16:24
Well, I did not start my comment with "Duplicate:", just "Relevant." In particular, the second post probably points in the correct direction but did not lay everything out explicitly.
– Frenzy Li
Aug 20 at 16:26
I suppose this particular choice of the coefficients has some advantages over other choices. I just wanted know those advantages with proper explanation.
– Soumen Basak
Aug 20 at 16:30
add a comment |Â
Relevant: What's the motivation of Runge-Kutta method?, What is the motivation behind the Runge-Kutta method?.
– Frenzy Li
Aug 20 at 16:21
The old posts on similar topic do not have the answer of my question.
– Soumen Basak
Aug 20 at 16:24
Well, I did not start my comment with "Duplicate:", just "Relevant." In particular, the second post probably points in the correct direction but did not lay everything out explicitly.
– Frenzy Li
Aug 20 at 16:26
I suppose this particular choice of the coefficients has some advantages over other choices. I just wanted know those advantages with proper explanation.
– Soumen Basak
Aug 20 at 16:30
Relevant: What's the motivation of Runge-Kutta method?, What is the motivation behind the Runge-Kutta method?.
– Frenzy Li
Aug 20 at 16:21
Relevant: What's the motivation of Runge-Kutta method?, What is the motivation behind the Runge-Kutta method?.
– Frenzy Li
Aug 20 at 16:21
The old posts on similar topic do not have the answer of my question.
– Soumen Basak
Aug 20 at 16:24
The old posts on similar topic do not have the answer of my question.
– Soumen Basak
Aug 20 at 16:24
Well, I did not start my comment with "Duplicate:", just "Relevant." In particular, the second post probably points in the correct direction but did not lay everything out explicitly.
– Frenzy Li
Aug 20 at 16:26
Well, I did not start my comment with "Duplicate:", just "Relevant." In particular, the second post probably points in the correct direction but did not lay everything out explicitly.
– Frenzy Li
Aug 20 at 16:26
I suppose this particular choice of the coefficients has some advantages over other choices. I just wanted know those advantages with proper explanation.
– Soumen Basak
Aug 20 at 16:30
I suppose this particular choice of the coefficients has some advantages over other choices. I just wanted know those advantages with proper explanation.
– Soumen Basak
Aug 20 at 16:30
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
Runge Kutte forms an infinite family of ODE solvers. The coefficients come from something called a Butcher Tableau. The most basic form of Runge Kutte is Eulers method. Later there were better methods made with the mid-point method then Heun's method. Then Kutta gave an explanation of $4th$ order methods. The evaluation of the stages give the tableau. There is a lecture here on derivation for RK4 which is typically done in numerical analysis.
answered Aug 20 at 20:49
RHowe
1,069815
Thanks for sharing two useful references on numerical analyis. In the 2nd reference on numerical analysis, it is mentioned that alpha2=1/2 and beta31=0 is the most useful (see the section above table 2) choice. Why this particular choice is most useful ? Dow you know the reason ? This can not be just human preference.
– Soumen Basak
Aug 21 at 16:34
I would imagine so then you can solve it easier if $beta_31 =0$ then $ beta_32 = alpha_3$
– RHowe
Aug 21 at 17:28
add a comment |Â
up vote
0
down vote
Consider a differential equation $y' = f(x,y)$ with initial condition $y(0)=0$, and write out the series solution to order $x^5$ (in terms of the coefficients of the bivariate Taylor series of $f$ at $(0,0))$. Then compare with what you get from the Runge-Kutta scheme with coefficients $k_1, ldots, k_4$. In order for the Runge-Kutta to agree with the series solution to order $x^5$, there will be a set of equations to solve. It will turn out that $k_1 = 1/6$, $k_2 = 1/3$, $k_3 = 1/3$, $k_4 = 1/6$ are the solution.
answered Aug 20 at 16:35
Robert Israel
306k22201443
Thats not true. In case of 4th order "The system involves 11 equations in 13 unknowns, so two of them could be chosen arbitrary." cfm.brown.edu/people/dobrush/am33/Matlab/RK4.html
– Soumen Basak
Aug 20 at 16:40
That's if you want to choose not just $k_1, ldots, k_4$, but the points at which to evaluate the $f$'s. Once you make the classical choices for those, the $k_1,...k_4$ are determined.
– Robert Israel
Aug 20 at 16:49
For the specific sequence of Euler predictor, midpoint corrector twice and evaluation at the end-point, it is sufficient to compute the Taylor expansions for the special case $y'=f(x,y)=y$ to get the uniqueness of the coefficients.
– LutzL
Aug 20 at 17:00
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Runge Kutte forms an infinite family of ODE solvers. The coefficients come from something called a Butcher Tableau. The most basic form of Runge Kutte is Eulers method. Later there were better methods made with the mid-point method then Heun's method. Then Kutta gave an explanation of $4th$ order methods. The evaluation of the stages give the tableau. There is a lecture here on derivation for RK4 which is typically done in numerical analysis.
answered Aug 20 at 20:49
RHowe
1,069815
Thanks for sharing two useful references on numerical analyis. In the 2nd reference on numerical analysis, it is mentioned that alpha2=1/2 and beta31=0 is the most useful (see the section above table 2) choice. Why this particular choice is most useful ? Dow you know the reason ? This can not be just human preference.
– Soumen Basak
Aug 21 at 16:34
I would imagine so then you can solve it easier if $beta_31 =0$ then $ beta_32 = alpha_3$
– RHowe
Aug 21 at 17:28
add a comment |Â
up vote
1
down vote
Runge Kutte forms an infinite family of ODE solvers. The coefficients come from something called a Butcher Tableau. The most basic form of Runge Kutte is Eulers method. Later there were better methods made with the mid-point method then Heun's method. Then Kutta gave an explanation of $4th$ order methods. The evaluation of the stages give the tableau. There is a lecture here on derivation for RK4 which is typically done in numerical analysis.
answered Aug 20 at 20:49
RHowe
1,069815
Thanks for sharing two useful references on numerical analyis. In the 2nd reference on numerical analysis, it is mentioned that alpha2=1/2 and beta31=0 is the most useful (see the section above table 2) choice. Why this particular choice is most useful ? Dow you know the reason ? This can not be just human preference.
– Soumen Basak
Aug 21 at 16:34
I would imagine so then you can solve it easier if $beta_31 =0$ then $ beta_32 = alpha_3$
– RHowe
Aug 21 at 17:28
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Runge Kutte forms an infinite family of ODE solvers. The coefficients come from something called a Butcher Tableau. The most basic form of Runge Kutte is Eulers method. Later there were better methods made with the mid-point method then Heun's method. Then Kutta gave an explanation of $4th$ order methods. The evaluation of the stages give the tableau. There is a lecture here on derivation for RK4 which is typically done in numerical analysis.
answered Aug 20 at 20:49
RHowe
1,069815
Runge Kutte forms an infinite family of ODE solvers. The coefficients come from something called a Butcher Tableau. The most basic form of Runge Kutte is Eulers method. Later there were better methods made with the mid-point method then Heun's method. Then Kutta gave an explanation of $4th$ order methods. The evaluation of the stages give the tableau. There is a lecture here on derivation for RK4 which is typically done in numerical analysis.
answered Aug 20 at 20:49
RHowe
1,069815
answered Aug 20 at 20:49
RHowe
1,069815
answered Aug 20 at 20:49
RHowe
1,069815
answered Aug 20 at 20:49
RHowe
1,069815
1,069815
Thanks for sharing two useful references on numerical analyis. In the 2nd reference on numerical analysis, it is mentioned that alpha2=1/2 and beta31=0 is the most useful (see the section above table 2) choice. Why this particular choice is most useful ? Dow you know the reason ? This can not be just human preference.
– Soumen Basak
Aug 21 at 16:34
I would imagine so then you can solve it easier if $beta_31 =0$ then $ beta_32 = alpha_3$
– RHowe
Aug 21 at 17:28
add a comment |Â
Thanks for sharing two useful references on numerical analyis. In the 2nd reference on numerical analysis, it is mentioned that alpha2=1/2 and beta31=0 is the most useful (see the section above table 2) choice. Why this particular choice is most useful ? Dow you know the reason ? This can not be just human preference.
– Soumen Basak
Aug 21 at 16:34
I would imagine so then you can solve it easier if $beta_31 =0$ then $ beta_32 = alpha_3$
– RHowe
Aug 21 at 17:28
Thanks for sharing two useful references on numerical analyis. In the 2nd reference on numerical analysis, it is mentioned that alpha2=1/2 and beta31=0 is the most useful (see the section above table 2) choice. Why this particular choice is most useful ? Dow you know the reason ? This can not be just human preference.
– Soumen Basak
Aug 21 at 16:34
Thanks for sharing two useful references on numerical analyis. In the 2nd reference on numerical analysis, it is mentioned that alpha2=1/2 and beta31=0 is the most useful (see the section above table 2) choice. Why this particular choice is most useful ? Dow you know the reason ? This can not be just human preference.
– Soumen Basak
Aug 21 at 16:34
I would imagine so then you can solve it easier if $beta_31 =0$ then $ beta_32 = alpha_3$
– RHowe
Aug 21 at 17:28
I would imagine so then you can solve it easier if $beta_31 =0$ then $ beta_32 = alpha_3$
– RHowe
Aug 21 at 17:28
add a comment |Â
up vote
0
down vote
Consider a differential equation $y' = f(x,y)$ with initial condition $y(0)=0$, and write out the series solution to order $x^5$ (in terms of the coefficients of the bivariate Taylor series of $f$ at $(0,0))$. Then compare with what you get from the Runge-Kutta scheme with coefficients $k_1, ldots, k_4$. In order for the Runge-Kutta to agree with the series solution to order $x^5$, there will be a set of equations to solve. It will turn out that $k_1 = 1/6$, $k_2 = 1/3$, $k_3 = 1/3$, $k_4 = 1/6$ are the solution.
answered Aug 20 at 16:35
Robert Israel
306k22201443
Thats not true. In case of 4th order "The system involves 11 equations in 13 unknowns, so two of them could be chosen arbitrary." cfm.brown.edu/people/dobrush/am33/Matlab/RK4.html
– Soumen Basak
Aug 20 at 16:40
That's if you want to choose not just $k_1, ldots, k_4$, but the points at which to evaluate the $f$'s. Once you make the classical choices for those, the $k_1,...k_4$ are determined.
– Robert Israel
Aug 20 at 16:49
For the specific sequence of Euler predictor, midpoint corrector twice and evaluation at the end-point, it is sufficient to compute the Taylor expansions for the special case $y'=f(x,y)=y$ to get the uniqueness of the coefficients.
– LutzL
Aug 20 at 17:00
add a comment |Â
up vote
0
down vote
Consider a differential equation $y' = f(x,y)$ with initial condition $y(0)=0$, and write out the series solution to order $x^5$ (in terms of the coefficients of the bivariate Taylor series of $f$ at $(0,0))$. Then compare with what you get from the Runge-Kutta scheme with coefficients $k_1, ldots, k_4$. In order for the Runge-Kutta to agree with the series solution to order $x^5$, there will be a set of equations to solve. It will turn out that $k_1 = 1/6$, $k_2 = 1/3$, $k_3 = 1/3$, $k_4 = 1/6$ are the solution.
answered Aug 20 at 16:35
Robert Israel
306k22201443
Thats not true. In case of 4th order "The system involves 11 equations in 13 unknowns, so two of them could be chosen arbitrary." cfm.brown.edu/people/dobrush/am33/Matlab/RK4.html
– Soumen Basak
Aug 20 at 16:40
That's if you want to choose not just $k_1, ldots, k_4$, but the points at which to evaluate the $f$'s. Once you make the classical choices for those, the $k_1,...k_4$ are determined.
– Robert Israel
Aug 20 at 16:49
For the specific sequence of Euler predictor, midpoint corrector twice and evaluation at the end-point, it is sufficient to compute the Taylor expansions for the special case $y'=f(x,y)=y$ to get the uniqueness of the coefficients.
– LutzL
Aug 20 at 17:00
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Consider a differential equation $y' = f(x,y)$ with initial condition $y(0)=0$, and write out the series solution to order $x^5$ (in terms of the coefficients of the bivariate Taylor series of $f$ at $(0,0))$. Then compare with what you get from the Runge-Kutta scheme with coefficients $k_1, ldots, k_4$. In order for the Runge-Kutta to agree with the series solution to order $x^5$, there will be a set of equations to solve. It will turn out that $k_1 = 1/6$, $k_2 = 1/3$, $k_3 = 1/3$, $k_4 = 1/6$ are the solution.
answered Aug 20 at 16:35
Robert Israel
306k22201443
Consider a differential equation $y' = f(x,y)$ with initial condition $y(0)=0$, and write out the series solution to order $x^5$ (in terms of the coefficients of the bivariate Taylor series of $f$ at $(0,0))$. Then compare with what you get from the Runge-Kutta scheme with coefficients $k_1, ldots, k_4$. In order for the Runge-Kutta to agree with the series solution to order $x^5$, there will be a set of equations to solve. It will turn out that $k_1 = 1/6$, $k_2 = 1/3$, $k_3 = 1/3$, $k_4 = 1/6$ are the solution.
answered Aug 20 at 16:35
Robert Israel
306k22201443
answered Aug 20 at 16:35
Robert Israel
306k22201443
answered Aug 20 at 16:35
Robert Israel
306k22201443
answered Aug 20 at 16:35
Robert Israel
306k22201443
306k22201443
Thats not true. In case of 4th order "The system involves 11 equations in 13 unknowns, so two of them could be chosen arbitrary." cfm.brown.edu/people/dobrush/am33/Matlab/RK4.html
– Soumen Basak
Aug 20 at 16:40
That's if you want to choose not just $k_1, ldots, k_4$, but the points at which to evaluate the $f$'s. Once you make the classical choices for those, the $k_1,...k_4$ are determined.
– Robert Israel
Aug 20 at 16:49
For the specific sequence of Euler predictor, midpoint corrector twice and evaluation at the end-point, it is sufficient to compute the Taylor expansions for the special case $y'=f(x,y)=y$ to get the uniqueness of the coefficients.
– LutzL
Aug 20 at 17:00
add a comment |Â
Thats not true. In case of 4th order "The system involves 11 equations in 13 unknowns, so two of them could be chosen arbitrary." cfm.brown.edu/people/dobrush/am33/Matlab/RK4.html
– Soumen Basak
Aug 20 at 16:40
That's if you want to choose not just $k_1, ldots, k_4$, but the points at which to evaluate the $f$'s. Once you make the classical choices for those, the $k_1,...k_4$ are determined.
– Robert Israel
Aug 20 at 16:49
For the specific sequence of Euler predictor, midpoint corrector twice and evaluation at the end-point, it is sufficient to compute the Taylor expansions for the special case $y'=f(x,y)=y$ to get the uniqueness of the coefficients.
– LutzL
Aug 20 at 17:00
Thats not true. In case of 4th order "The system involves 11 equations in 13 unknowns, so two of them could be chosen arbitrary." cfm.brown.edu/people/dobrush/am33/Matlab/RK4.html
– Soumen Basak
Aug 20 at 16:40
Thats not true. In case of 4th order "The system involves 11 equations in 13 unknowns, so two of them could be chosen arbitrary." cfm.brown.edu/people/dobrush/am33/Matlab/RK4.html
– Soumen Basak
Aug 20 at 16:40
That's if you want to choose not just $k_1, ldots, k_4$, but the points at which to evaluate the $f$'s. Once you make the classical choices for those, the $k_1,...k_4$ are determined.
– Robert Israel
Aug 20 at 16:49
That's if you want to choose not just $k_1, ldots, k_4$, but the points at which to evaluate the $f$'s. Once you make the classical choices for those, the $k_1,...k_4$ are determined.
– Robert Israel
Aug 20 at 16:49
For the specific sequence of Euler predictor, midpoint corrector twice and evaluation at the end-point, it is sufficient to compute the Taylor expansions for the special case $y'=f(x,y)=y$ to get the uniqueness of the coefficients.
– LutzL
Aug 20 at 17:00
For the specific sequence of Euler predictor, midpoint corrector twice and evaluation at the end-point, it is sufficient to compute the Taylor expansions for the special case $y'=f(x,y)=y$ to get the uniqueness of the coefficients.
– LutzL
Aug 20 at 17:00
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2888946%2f4-textth-order-runge-kutta-method%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Relevant: What's the motivation of Runge-Kutta method?, What is the motivation behind the Runge-Kutta method?.
– Frenzy Li
Aug 20 at 16:21
The old posts on similar topic do not have the answer of my question.
– Soumen Basak
Aug 20 at 16:24
Well, I did not start my comment with "Duplicate:", just "Relevant." In particular, the second post probably points in the correct direction but did not lay everything out explicitly.
– Frenzy Li
Aug 20 at 16:26
I suppose this particular choice of the coefficients has some advantages over other choices. I just wanted know those advantages with proper explanation.
– Soumen Basak
Aug 20 at 16:30