- #1
monty37
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why is the 4th order Runge -Kutta method widely used than the 2nd or 3rd,for
solving ordinary differential equations?
solving ordinary differential equations?
Solution to ordinary differential equation
- Thread starter monty37
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In summary, the 4th order Runge-Kutta method is widely used for solving ordinary differential equations due to its greater accuracy compared to the 2nd and 3rd order methods. Although the 2nd and 3rd order methods are quicker, they are much less exact. For orders higher than 4, the computation time becomes too long. Additionally, the 4th order method is chosen for differential equations containing complex functions due to its greater accuracy. The reason why the 3rd order Runge-Kutta formula is not commonly seen in literature is because the method increases accuracy by two steps each time, and it is unclear if a 3rd order method is even possible.
- #2
arildno
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Cost-effectiveness.
Although 2. and 3.order Runge-Kutta are quicker than 4th order, they are much less exact.
For orders higher than 4, those take too long time to compute.
On another note:
Although I won't vouch for at which order this will become significant, the upper limit of an approximate scheme in terms of exactness will be when the finite arithmetic of the computer starts messing with the answers we want.
Although 2. and 3.order Runge-Kutta are quicker than 4th order, they are much less exact.
For orders higher than 4, those take too long time to compute.
On another note:
Although I won't vouch for at which order this will become significant, the upper limit of an approximate scheme in terms of exactness will be when the finite arithmetic of the computer starts messing with the answers we want.
- #3
monty37
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so even if the involved differential equations contains different complex functions,
due to greater accuracy ,the 4th order(RK) method is chosen.
due to greater accuracy ,the 4th order(RK) method is chosen.
- #4
matematikawan
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arildno said:
I agree with you. Just that I never see RK3 formula in the literatures
. Why is that so?- #5
arildno
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matematikawan said:I agree with you. Just that I never see RK3 formula in the literatures
Probably because R-K increases by two steps of accuracy each time.
It's been a long time since I had a glancing look at the procedure, and have forgotten if 3.order r-k is even possible.
What I remember is that r-k approximates changes due to the values of the derivatives by clever weighting of function values, using a nesting principle.
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1. What is a solution to an ordinary differential equation?
A solution to an ordinary differential equation is a function that satisfies the given equation and its initial conditions. It represents the relationship between a dependent variable and an independent variable.
2. How is a solution to an ordinary differential equation different from a general solution?
A solution to an ordinary differential equation is a specific function that satisfies the given equation and initial conditions, while a general solution is a family of solutions that includes all possible solutions to the equation. A general solution may contain arbitrary constants that can be determined by applying the initial conditions.
3. What are initial conditions and why are they important in finding a solution to an ordinary differential equation?
Initial conditions are values given for the dependent variable and its derivatives at a specific point. They are important because they help determine the specific solution to an ordinary differential equation from a general solution.
4. What are the steps involved in finding a solution to an ordinary differential equation?
The steps involved in finding a solution to an ordinary differential equation are:
- Identify the type of equation (e.g. linear, separable, etc.)
- Solve the equation using appropriate methods (e.g. separation of variables, integrating factor, etc.)
- Apply any given initial conditions to determine the specific solution
5. Can all ordinary differential equations be solved analytically?
No, not all ordinary differential equations can be solved analytically, especially for more complex equations. In some cases, numerical methods or approximations are used to find a solution.
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