List of Runge–Kutta methods

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Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation

Explicit Runge–Kutta methods take the form

Stages for implicit methods of s stages take the more general form, with the solution to be found over all s

Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows:

For adaptive and implicit methods, the Butcher tableau is extended to give values of , and the estimated error is then

.

Explicit methods[edit]

The explicit methods are those where the matrix is lower triangular.

Forward Euler[edit]

The Euler method is first order. The lack of stability and accuracy limits its popularity mainly to use as a simple introductory example of a numeric solution method.

Explicit midpoint method[edit]

The (explicit) midpoint method is a second-order method with two stages (see also the implicit midpoint method below):

Heun's method[edit]

Heun's method is a second-order method with two stages. It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method. (Note: The "eu" is pronounced the same way as in "Euler", so "Heun" rhymes with "coin"):

Ralston's method[edit]

Ralston's method is a second-order method[1] with two stages and a minimum local error bound:

Generic second-order method[edit]

Kutta's third-order method[edit]

Generic third-order method[edit]

See Sanderse and Veldman (2019).[2]

for α ≠ 0, 23, 1:

Heun's third-order method[edit]

Van der Houwen's/Wray third-order method[edit]

Ralston's third-order method[edit]

Ralston's third-order method[1] is used in the embedded Bogacki–Shampine method.

Third-order Strong Stability Preserving Runge-Kutta (SSPRK3)[edit]

Classic fourth-order method[edit]

The "original" Runge–Kutta method.[3]

3/8-rule fourth-order method[edit]

This method doesn't have as much notoriety as the "classic" method, but is just as classic because it was proposed in the same paper (Kutta, 1901).[3]

Ralston's fourth-order method[edit]

This fourth order method[1] has minimum truncation error.

Embedded methods[edit]

The embedded methods are designed to produce an estimate of the local truncation error of a single Runge–Kutta step, and as result, allow to control the error with adaptive stepsize. This is done by having two methods in the tableau, one with order p and one with order p-1.

The lower-order step is given by

where the are the same as for the higher order method. Then the error is

which is . The Butcher Tableau for this kind of method is extended to give the values of

Heun–Euler[edit]

The simplest adaptive Runge–Kutta method involves combining Heun's method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is:

The error estimate is used to control the stepsize.

Fehlberg RK1(2)[edit]

The Fehlberg method[4] has two methods of orders 1 and 2. Its extended Butcher Tableau is:

0
1/2 1/2
1 1/256 255/256
1/512 255/256 1/512
1/256 255/256 0

The first row of b coefficients gives the second-order accurate solution, and the second row has order one.

Bogacki–Shampine[edit]

The Bogacki–Shampine method has two methods of orders 3 and 2. Its extended Butcher Tableau is:

0
1/2 1/2
3/4 0 3/4
1 2/9 1/3 4/9
2/9 1/3 4/9 0
7/24 1/4 1/3 1/8

The first row of b coefficients gives the third-order accurate solution, and the second row has order two.

Fehlberg[edit]

The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4; it is sometimes dubbed RKF45 . Its extended Butcher Tableau is:

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four. The coefficients here allow for an adaptive stepsize to be determined automatically.

Cash-Karp[edit]

Cash and Karp have modified Fehlberg's original idea. The extended tableau for the Cash–Karp method is

0
1/5 1/5
3/10 3/40 9/40
3/5 3/10 −9/10 6/5
1 −11/54 5/2 −70/27 35/27
7/8 1631/55296 175/512 575/13824 44275/110592 253/4096
37/378 0 250/621 125/594 0 512/1771
2825/27648 0 18575/48384 13525/55296 277/14336 1/4

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

Dormand–Prince[edit]

The extended tableau for the Dormand–Prince method is

0
1/5 1/5
3/10 3/40 9/40
4/5 44/45 −56/15 32/9
8/9 19372/6561 −25360/2187 64448/6561 −212/729
1 9017/3168 −355/33 46732/5247 49/176 −5103/18656
1 35/384 0 500/1113 125/192 −2187/6784 11/84
35/384 0 500/1113 125/192 −2187/6784 11/84 0
5179/57600 0 7571/16695 393/640 −92097/339200 187/2100 1/40

The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order accurate solution.

Implicit methods[edit]

Backward Euler[edit]

The backward Euler method is first order. Unconditionally stable and non-oscillatory for linear diffusion problems.

Implicit midpoint[edit]

The implicit midpoint method is of second order. It is the simplest method in the class of collocation methods known as the Gauss-Legendre methods. It is a symplectic integrator.

Crank-Nicolson method[edit]

The Crank–Nicolson method corresponds to the implicit trapezoidal rule and is a second-order accurate and A-stable method.

Gauss–Legendre methods[edit]

These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau:

The Gauss–Legendre method of order six has Butcher tableau:

Diagonally Implicit Runge–Kutta methods[edit]

Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems; [5] the advantage of this approach is that here the solution may be found sequentially as opposed to simultaneously.

The simplest method from this class is the order 2 implicit midpoint method.

Kraaijevanger and Spijker's two-stage Diagonally Implicit Runge–Kutta method:

Qin and Zhang's two-stage, 2nd order, symplectic Diagonally Implicit Runge–Kutta method:

Pareschi and Russo's two-stage 2nd order Diagonally Implicit Runge–Kutta method:

This Diagonally Implicit Runge–Kutta method is A-stable if and only if . Moreover, this method is L-stable if and only if equals one of the roots of the polynomial , i.e. if . Qin and Zhang's Diagonally Implicit Runge–Kutta method corresponds to Pareschi and Russo's Diagonally Implicit Runge–Kutta method with .

Two-stage 2nd order Diagonally Implicit Runge–Kutta method:

Again, this Diagonally Implicit Runge–Kutta method is A-stable if and only if . As the previous method, this method is again L-stable if and only if equals one of the roots of the polynomial , i.e. if .

Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method:

Three-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method:

with

Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau:

with one of the three roots of the cubic equation . The three roots of this cubic equation are approximately , , and . The root gives the best stability properties for initial value problems.

Four-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method

Lobatto methods[edit]

There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto. All are implicit methods, have order 2s − 2 and they all have c1 = 0 and cs = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta.

Lobatto IIIA methods[edit]

The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule:

The fourth-order method is given by

These methods are A-stable, but not L-stable and B-stable.

Lobatto IIIB methods[edit]

The Lobatto IIIB methods are not collocation methods, but they can be viewed as discontinuous collocation methods (Hairer, Lubich & Wanner 2006, §II.1.4). The second-order method is given by

The fourth-order method is given by

Lobatto IIIB methods are A-stable, but not L-stable and B-stable.

Lobatto IIIC methods[edit]

The Lobatto IIIC methods also are discontinuous collocation methods. The second-order method is given by

The fourth-order method is given by

They are L-stable. They are also algebraically stable and thus B-stable, that makes them suitable for stiff problems.

Lobatto IIIC* methods[edit]

The Lobatto IIIC* methods are also known as Lobatto III methods (Butcher, 2008), Butcher's Lobatto methods (Hairer et al., 1993), and Lobatto IIIC methods (Sun, 2000) in the literature.[6] The second-order method is given by

Butcher's three-stage, fourth-order method is given by

These methods are not A-stable, B-stable or L-stable. The Lobatto IIIC* method for is sometimes called the explicit trapezoidal rule.

Generalized Lobatto methods[edit]

One can consider a very general family of methods with three real parameters by considering Lobatto coefficients of the form

,

where

.

For example, Lobatto IIID family introduced in (Nørsett and Wanner, 1981), also called Lobatto IIINW, are given by

and

These methods correspond to , , , and . The methods are L-stable. They are algebraically stable and thus B-stable.

Radau methods[edit]

Radau methods are fully implicit methods (matrix A of such methods can have any structure). Radau methods attain order 2s − 1 for s stages. Radau methods are A-stable, but expensive to implement. Also they can suffer from order reduction. The first order Radau method is similar to backward Euler method.

Radau IA methods[edit]

The third-order method is given by

The fifth-order method is given by

Radau IIA methods[edit]

The ci of this method are zeros of

.

The third-order method is given by

The fifth-order method is given by

Notes[edit]

  1. ^ a b c Ralston, Anthony (1962). "Runge-Kutta Methods with Minimum Error Bounds". Math. Comput. 16 (80): 431–437. doi:10.1090/S0025-5718-1962-0150954-0.
  2. ^ Sanderse, Benjamin; Veldman, Arthur (2019). "Constraint-consistent Runge–Kutta methods for one-dimensional incompressible multiphase flow". J. Comput. Phys. 384: 170. arXiv:1809.06114. Bibcode:2019JCoPh.384..170S. doi:10.1016/j.jcp.2019.02.001. S2CID 73590909.
  3. ^ a b Kutta, Martin (1901). "Beitrag zur näherungsweisen Integration totaler Differentialgleichungen". Zeitschrift für Mathematik und Physik. 46: 435–453.
  4. ^ Fehlberg, E. (1969-07-01). "Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems". {{cite journal}}: Cite journal requires |journal= (help)
  5. ^ For discussion see: Christopher A. Kennedy; Mark H. Carpenter (2016). "Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review". Technical Memorandum, NASA STI Program.
  6. ^ See Laurent O. Jay (N.D.). "Lobatto methods". University of Iowa

References[edit]

  • Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0.
  • Hairer, Ernst; Wanner, Gerhard (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-60452-5.
  • Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2006), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-30663-4.