The ARS(2,2,2) additive tableau and its three stages

For the autonomous split problem $y^{\mathrm{\prime }}=a(y)+b(y)$ of Additive autonomous ODE partitions, set $\gamma :=1-\frac{1}{\sqrt{2}},\delta :=-\frac{1}{\sqrt{2}}\mathrm{.}$ The explicit tableau and the lower-triangular implicit tableau, using the separate stage conventions of Explicit Runge–Kutta tableaus, stage maps, and failure status and Implicit Runge–Kutta stage equations and update, are $\tilde{A}=\left(\begin{array}{ccc}0& 0& 0\\ \gamma & 0& 0\\ \delta & 1-\delta & 0\end{array}\right),\tilde{b}=(\delta ,1-\delta ,0),\tilde{c}=(0,\gamma ,1),$ $A=\left(\begin{array}{ccc}0& 0& 0\\ 0& \gamma & 0\\ 0& 1-\gamma & \gamma \end{array}\right),b=(0,1-\gamma ,\gamma ),c=(0,\gamma ,1)\mathrm{.}$ For a step $h>0$ from $y$, the method is defined by $Y_1=y,Y_2=y+\gamma ha(Y_1)+\gamma hb(Y_2),$ $Y_3=y+h[\delta a(Y_1)+(1-\delta )a(Y_2)]+h[(1-\gamma )b(Y_2)+\gamma b(Y_3)],y^{+}=Y_3\mathrm{.}$ Thus there are two implicit solves per step, one for each nonzero diagonal implicit stage. The unscaled stage-equation residual uses the update-residual convention of Implicit residuals and their update and derivative units. The tableaus and stages alone assert neither branch existence nor stability.