# Variable Step Block Hybrid Method for Stiff Chemical Kinetics Problems
## 3 Stability Analysis of 3-Point VSBHM with Its Properties
### Zero-Stability  
**Definition 1. (Zero-stable)**. : _\"The Linear Multistep Method (LMM) is said to be zero-stable if no root of the first characteristic polynomial, \(p(t)\) has modulus greater than one, and if every root with modulus one is simple\"._  
**Definition 2. (A-stable)**. : _\"A method is said to be \(A\)-stable if all numerical approximations tend to zero when it is applied to the differential equation \(y^{\prime}=\lambda y\) with a fixed positive \(h\) and a complex constant \(\lambda\) with negative real part\"._  
Ref. [36] proposed the scalar test to determine the stability of the method for Table 1 as  
\[y^{\prime}=\lambda y,\;\lambda\;<\;0, \tag{9}\]  
where \(\lambda\) represents the complex constant with Re(\(\lambda\)) \(<0\). Equation (9) is substituted in Table 1, therefore it precedes for \(r=1\) as,  
\[y_{n+1}=\tfrac{3}{56}y_{n-1}-\tfrac{3}{5}y_{n}+3y_{n+2}-\tfrac{64}{35}y_{n+ \frac{5}{2}}+\tfrac{3}{5}y_{n+3}-\tfrac{3}{2}h\lambda y_{n+1}\]  
\[y_{n+2}=-\tfrac{1}{98}y_{n-1}+\tfrac{3}{35}y_{n}-\tfrac{3}{7}y_{n+1}+\tfrac{3 84}{245}y_{n+\frac{5}{2}}-\tfrac{3}{14}y_{n+3}-\tfrac{6}{7}h\lambda y_{n+2}\]  
\[y_{n+\frac{5}{2}}=-\tfrac{75}{9088}y_{n-1}+\tfrac{147}{2272}y_{n}-\tfrac{1225} {454}y_{n+1}+\tfrac{3675}{2272}y_{n+2}-\tfrac{3675}{9088}y_{n+3}+\tfrac{105}{14 2}h\lambda y_{n+\frac{5}{2}}\]  
\[y_{n+3}=\tfrac{3}{343}y_{n-1}-\tfrac{16}{245}y_{n}+\tfrac{12}{49}y_{n+1}- \tfrac{48}{49}y_{n+2}+\tfrac{3072}{1715}y_{n+\frac{5}{2}}+\tfrac{12}{49}h \lambda y_{n+3}\]  
Equation (10) is then inscribed into the matrix form to attain the matrix as follows  
\[\left[ \begin{array}{cccc}-\tfrac{3}{8}&\tfrac{64}{35}&-3&1+\tfrac{3}{2}h \lambda\\ \tfrac{3}{14}&-\tfrac{384}{245}&1+\tfrac{6}{7}h\lambda&\tfrac{3}{7}\\ \tfrac{3675}{9088}&1-\tfrac{105}{142}h\lambda&-\tfrac{3675}{2272}&\tfrac{1225} {4544}\\ 1-\tfrac{12}{49}h\lambda&-\tfrac{3072}{1715}&\tfrac{48}{49}&-\tfrac{12}{49} \end{array} \right]\left[ \begin{array}{c}y_{n+3}\\ y_{n+\frac{5}{2}}\\ y_{n+2}\\ y_{n+1}\end{array} \right]=\left[ \begin{array}{cccc}-\tfrac{3}{5}&0&\tfrac{3} {56}&0\\ \tfrac{3}{35}&0&-\tfrac{1}{98}&0\\ \tfrac{147}{2272}&0&-\tfrac{75}{9088}&0\\ -\tfrac{16}{245}&0&\tfrac{3}{343}&0\end{array} \right]\left[ \begin{array}{c} y_{n}\\ y_{n-\frac{1}{2}}\\ y_{n-1}\\ y_{n-2}\end{array} \right]\]  
whereas,  
which is equivalent to,  
\[C=\left[ \begin{array}{cccc}-\tfrac{3}{8}&\tfrac{64}{35}&-3&1+\tfrac{3}{2}h \lambda\\ \tfrac{3}{14}&-\tfrac{384}{245}&1+\tfrac{6}{7}h\lambda&\tfrac{3}{7}\\ \tfrac{3675}{9088}&1-\tfrac{105}{142}h\lambda&-\tfrac{3675}{2272}&\tfrac{1225} {4544}\\ 1-\tfrac{12}{49}h\lambda&-\tfrac{3072}{1715}&\tfrac{48}{49}&-\tfrac{12}{49} \end{array} \right]\text{, }D=\left[ \begin{array}{cccc}-\tfrac{3}{5}&0&\tfrac{3}{56}&0\\ \tfrac{3}{35}&0&-\tfrac{1}{98}&0\\ \tfrac{147}{2272}&0&-\tfrac{75}{9088}&0\\ -\tfrac{16}{245}&0&\tfrac{3}{343}&0\end{array} \right]\]\[CY_{N}=DY_{N-1}, \tag{11}\]  
where \(C\) and \(D\) in Equation (11) are appropriately selected \(m\times m\) matrix coefficients and \(N\) represent the number of blocks (note that the evaluation presented here is only for r = 1. For (\(r\) = 2, 10/19), the same procedure is considered). \(H=h\lambda\) is replaced in the matrices which are obtained by Equation (10). The stability polynomials \(R_{r}(t,H)\) correlated with Table 1 are attained by solving the characteristic equations \(|Ct-D|\) using different step size ratios \(\left(r=1,\;2,\frac{10}{19}\right)\).