In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.


  1. Runge--Kutta methods
  2. internal stability
  3. roundoff error
  4. strong stability preservation
  5. extrapolation
  6. ordinary differential equations

MSC codes

  1. 65L05
  2. 65L06
  3. 65L70
  4. 65G50

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2227 - 2249
ISSN (online): 1095-7170


Submitted: 9 September 2013
Accepted: 8 April 2014
Published online: 11 September 2014


  1. Runge--Kutta methods
  2. internal stability
  3. roundoff error
  4. strong stability preservation
  5. extrapolation
  6. ordinary differential equations

MSC codes

  1. 65L05
  2. 65L06
  3. 65L70
  4. 65G50



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