Math 2.0

Numerical analysis Ordinary differential equations Proposition

Dormand–Prince method

Builds on Runge–Kutta methodsButcher tableauLocal truncation errorAdaptive step size

Statement

Consider the explicit seven-stage Butcher tableau with two weight rows

01515310340940454445−561532989193726561−253602187644486561−212729190173168−3553346732524749176−51031865613538405001113125192−218767841184bT3538405001113125192−2187678411840b^T5179576000757116695393640−920973392001872100140\begin{array}{c|ccccccc} 0 \\ \frac15 & \frac15 \\ \frac{3}{10} & \frac{3}{40} & \frac{9}{40} \\ \frac45 & \frac{44}{45} & -\frac{56}{15} & \frac{32}{9} \\ \frac89 & \frac{19372}{6561} & -\frac{25360}{2187} & \frac{64448}{6561} & -\frac{212}{729} \\ 1 & \frac{9017}{3168} & -\frac{355}{33} & \frac{46732}{5247} & \frac{49}{176} & -\frac{5103}{18656} \\ 1 & \frac{35}{384} & 0 & \frac{500}{1113} & \frac{125}{192} & -\frac{2187}{6784} & \frac{11}{84} \\ \hline b^{T} & \frac{35}{384} & 0 & \frac{500}{1113} & \frac{125}{192} & -\frac{2187}{6784} & \frac{11}{84} & 0 \\ \hat b^{T} & \frac{5179}{57600} & 0 & \frac{7571}{16695} & \frac{393}{640} & -\frac{92097}{339200} & \frac{187}{2100} & \frac{1}{40} \end{array}

Then:

  1. The Runge–Kutta method with weights bb has order 55, and the one with weights b^\hat b has order 44.
  2. The last row of the matrix equals bTb^{T} and c7=1c_7 = 1. So the seventh stage of a step equals f(tn+1,yn+1)f(t_{n+1}, y_{n+1}), which is the first stage of the next step.
  3. The difference yn+1−y^n+1y_{n+1} - \hat y_{n+1} of the two results equals the local truncation error of the fourth-order step up to a term O(h6)O(h^6), for ff smooth enough.

Remarks

Clause 1 is checked against Butcher's order conditions on rooted trees; for example ∑bi=∑b^i=1\sum b_i = \sum \hat b_i = 1 and ∑bici4=15\sum b_i c_i^4 = \frac15, while ∑b^ici4≠15\sum \hat b_i c_i^4 \ne \frac15. Clause 2 is the "first same as last" property: an accepted step needs only six new evaluations of ff. The method propagates the fifth-order value yn+1y_{n+1}, a practice called local extrapolation, and uses the error estimate of clause 3 for adaptive step size control. John Dormand and Peter Prince published the pair in 1980, choosing the free coefficients to make the error constants of the fifth-order formula small.

Going deeper

For a student

This pair, often called DOPRI5 or RK45, is the default non-stiff solver in many software packages, for example the ode45 function of MATLAB. Compared with the Runge–Kutta–Fehlberg method, it advances with the more accurate formula and has smaller error coefficients. It also admits a dense output, a polynomial interpolant that gives accurate values between the grid points. For stiff problems it needs very small steps, and implicit methods are preferred.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Adaptive step size, concept, no evidence levelAdaptive step sizeconceptButcher tableau, concept, no evidence levelButcher tableauconceptLocal truncation error, concept, no evidence levelLocal truncation errorconceptRunge–Kutta methods, concept, no evidence levelRunge–Kutta methodsconceptDormand–Prince method, statement, no evidence levelDormand–Prince methodstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Adaptive step size concept
  2. Butcher tableau concept
  3. Local truncation error concept
  4. Runge–Kutta methods concept
  5. Dormand–Prince method statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dormand%E2%80%93Prince_method
About this page
TagCDMXAH4A7G
URIm2://CDMXAH4A7G
Kindstatement, proposition
WritingW0 stub
MSC65L06, 65L05, 65L50
StatusThis copy of the site has no build data. Run m2 status CDMXAH4A7G.
AboutRunge–Kutta methods
UsesRunge–Kutta methods, Butcher tableau, Local truncation error, Adaptive step size
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