Intermediate stability polynomial for Assyr Abdulle’s methods

Following the idea of van der Houwen and Sommeijer (1980), the three-term recurrence relation can be used to define the internal stages of an \(s\)-stage explicit method as follows:

\[\begin{equation} \begin{aligned} y_0 & = u^n \\ y_1 & = u^n + h \mu_1 f(y_0) \\ y_j & = h \mu_j f(y_{j-1}) - \nu_j y_{j-1} - \kappa_j y_{j-2}, \quad j = 2, \ldots ,s-2 \quad \text{for $2^{nd}$-order methods or} \quad j = 2, \ldots ,s-4 \quad \text{for $4^{th}$-order methods}, \end{aligned} \end{equation}\]

where \(u^n\) represents the approximation to the exact solution \(u\) at time \(t=t^n\) and \(h=t^{n+1}-t^{n}\).

The method above applied to the scalar test problem \(u\prime=\lambda u\) leads to the linear, one-step recursion

\[\begin{equation} u^{n+1} = P_s(z) u^n, \quad z = h \lambda, \end{equation}\]

where the stability function \(P_s\) is a polynomial of degree \(s\). \(P_s\) can also be defined recursively as follows: \[\begin{equation} \begin{aligned} P_0(z) & = 1, \\ P_1(z) & = 1 + \mu_1 z\\ P_j(z) & = (\mu_j z - \nu_j) P_{j-1} - \kappa_j P_{j-2} \quad j = 2, \ldots ,s-2 \quad \text{for $2^{nd}$-order methods or} \quad j = 2, \ldots ,s-4 \quad \text{for $4^{th}$-order methods} \end{aligned} \end{equation}\]

Internal stability polynomial for Assyr Abdulle’s methods

By applying the procedure to compute the internal stability polynomial \(Q_{j,k}\) one arrives at \[\begin{equation} \begin{aligned} Q_{j,k}(z) & = 1, \\ Q_{k+1,k}(z) & = \mu_{k+1} z - \nu_{k+1}\\ Q_{j,k}(z) & = (\mu_j z - \nu_j) Q_{j-1,k} - \kappa_j Q_{j-2,k} \quad j = k+2, \ldots ,s-2 \quad \text{for $2^{nd}$-order methods or} \quad j = k+2, \ldots ,s-4 \quad \text{for $4^{th}$-order methods} \end{aligned} \end{equation}\]

The coefficients $j, j and _j are not reported in Assyr Abdulle’s papers. We get them from his Fortran codes ROCK2 and ROCK4