Internal stability of Runge-Kutta methods
Things we have learned or proved:
- Internal stability depends on implementation
- Optimal SSP(s,2) methods are internally stable
- Optimal SSP(\(n^2\),3) methods are internally stable
The internal stability polynomials determine the full expression for the error in a Runge-Kutta method. Some key papers are
Verwer JG, Hundsdorfer WH, Sommeijer BP. Convergence properties of the Runge-Kutta-Chebyshev method. Numerische Mathematik. 1990;57:157–178.
Sanz-Serna JM, Verwer JG, Hundsdorfer W. Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations. Numerische Mathematik. 1986;50(4):405–418. Available at: http://www.springerlink.com/index/m3184355750v5541.pdf
Sanz-Serna JM, Verwer JG. Stability and convergence at the PDE/stiff ode interface. Applied Numerical Mathematics. 1989;5(1-2):117–132. Available at: http://dx.doi.org/10.1016/0168-9274(89)90028-7
More references: Mendeley group
Some interesting questions:
- The full error of a Runge-Kutta method depends on internal stability. Can we construct simple methods that demonstrate this? Perhaps by using a 2-stage 2nd-order method with very poorly chosen coefficients, applied to a PDE.
- What is the significance of the \(P_j\) polynomials?
- How to construct an internally stable method for a given polynomial? Orthogonal polynomials and recurrence relations?
- Are Assyr Abdulle’s methods internally stable?
- Can we stabilize van der Houwen’s diagonal methods?
IPython notebook on internal stability in NodePy (note: this contains some significant numerical errors and uses only the Butcher form):
How to compute the internal stability polynomials given either the Butcher’s form or the modified Shu-Osher form:
- LaTex source files This is a git repository stored in bitbucket.
Other things to do:
- Find \(Q_{sk}\) and \(P_k\) for other methods (extrapolation, IDC, etc.)
- Read the 3 papers above in detail and determine what is known/unknown/interesting
- Compute properties of optimized methods for SD
- Bound the internal stability constants for extrapolation methods – see this ipython notebook
