is-shu-osher

A given Runge-Kutta method can be rewritten in infinitely many different Shu-Osher forms; this rewriting amounts to linear algebra and does not affect most of the theoretical properties of the method. However, the internal stability polynomials of a method depend on the particular Shu-Osher form used to implement it.

For example, consider the 2-stage, 2nd-order optimal SSP method. It is often written in the canonical Shu-Osher form: \[\begin{align*} y_0 & = u^n \\ y_1 & = y_0 + h f(y_0) \\ u^{n+1} = y_2 & =\frac{1}{2} u^n + \frac{1}{2}(y_1 + h f(y_1)). \end{align*}\] Let us work out the internal stability polynomial \(Q_{21}\) by taking \(f(y) = \lambda y\), \(z=h\lambda\), and inserting an error term \(r_j\) into each stage \(y_j\). Then we find (fill in details) \[Q_{21} = (1+z)/2\] However, in the Butcher form, the formula for \(y_2\) is written as \[u^{n+1} = u^n + \frac{1}{2} h (f(y_0) + f(y_1))\] which leads by a similar analysis to \[Q_{21} = z/2.\]

Note that the stability polynomial (unlike the internal polynomials) is independent of the choice of Shu-Osher form.