User:Lothar.brendel/belt-notes

SEDNA: Ionization → Rate Equations

The rate equations (44) and (45) in Kuiper etal 2020 are non-linear, namely quadratic in \(x\) and \(y\), respectively. Solving them with an implicit Euler scheme due to stability reasons (the rate equations seemingly are not considerd in the \(\Delta t\) control), yields quadratic equations, (46) and (53), for the values at \(t+\Delta t\). For each, the correct solution must be chosen, namely the one for which \(x^{n+1}-x^n=\mathcal O(\Delta t)\).

For \(B/A>0\), the function Sedna1.1/RateEquations.c:QuadraticEquation(A,B,C,&x1,&x2) assigns the solution with larger absolute value, which happens to be negative, to \(x_2\). For \(B/A<0\), the solution with larger absolute value is positive and is assigned to \(x_1\). The sign of the other solution, with smaller absolute value, depends on the sign of \(C\).

\(x(t)\)

• \(0<b+c=A\propto\Delta t\)
• \(B=1+a-b>0\) (for not too large \(\Delta t\)!)
• \(C=-(x^n+a)<0\)

\[ x^{n+1}=\frac{-B+\sqrt{\dots}}{2A}~, \] which is the solution with smaller absolute value. QuadraticEquation() yields \(x_2<0<x_1<\vert x_2\vert~\Rightarrow~x^{n+1}=x_1\).

\(y(t)\)

• \(0<b+c=A\propto\Delta t\)
• \(B=-1-a-b-2c<0\)
• \(C=y^n+a>0\)

\[ y^{n+1}=\frac{-B-\sqrt{\dots}}{2A}~, \] which again is the solution of smaller absolute value. QuadraticEquation() yields \(0<x_2<x_1~\Rightarrow~y^{n+1}=x_2\).