# Variable Step Block Hybrid Method for Stiff Chemical Kinetics Problems
## 2 Formulation of 3-Point Variable Step Block Hybrid Method (3-Point VSBHM)  
This section discusses the formulation of 3-point VSBHM. The three values of \(y_{n+1}\), \(y_{n+2}\), and \(y_{n+3}\) with one off-step point \(y_{n+\frac{5}{2}}\) are calculated concurrently in a block using earlier blocks with each block containing three points (refer to Figure 1). The computed block contains \(3h\) as the step size and \(rh\) pointed out the previous block where \(r\) shows the step size ratio. To manage the step size, the step size ratio (\(r\)), is considered throughout the derivation process. The values of \(r\) used in [17] were \(r=1,2,\frac{5}{8}\) even though \(r=1,2,\frac{10}{19}\) used in [5]. Hence, values \(r=1\), \(r=2\) and \(r=\frac{10}{19}\) are considered in this research as the strategy to maintain, halve, or increase the step size by a factor of 1.9, respectively. The interpolating polynomial \(P_{k}(t)\) interpolates the values of a function \(f(t,y)\) in Equation (1) at \(k=6\) with interpolating points of \((t_{n-1},y_{n-1})\), \((\,t_{n},y_{n})\), \((\,t_{n+1},y_{n+1})\), \((\,t_{n+2},y_{n+2})\), \((\,t_{n+\frac{5}{2}},y_{n+\frac{5}{2}})\) and \((t_{n+3},y_{n+3})\). Lagrange polynomial \(P_{k}(t)\) is defined as:  
\[P_{k}(t)=L_{k,5/2}(t)y(t_{n+5/2})+\sum_{j=0}^{k-2}L_{k,j}(t)y\big{(}t_{n+3-j} \big{)} \tag{3}\]  
where  
\[L_{k,j}(t)=\left(\prod \begin{array}{cc}i=0&\frac{t-t_{n+3-j}}{t_{n+3-j}-t_{ n+3-i}}\\ i\
eq j&\text{and }L_{k,5/2}(t)=\prod_{i=0}^{k-2}\frac{t-t_{n+3-j}}{t_{n+5/2}-t_{ n+3-i}}.\end{array} \right)\text{ for each }j=0,1,\ldots,k-2,\]  
Let \(s=\frac{t-t_{n+1}}{h}\) and replace \(y(t)\) in Equation (1) by a polynomial Equation (3). To obtain the \(y^{\prime}\), the resultant polynomial in Equation (3) is differentiated for \(s\) at point \(t\) and substitute \(dt\) by \(hds\), so that  
\[ \begin{array}{c}y^{\prime}=f(t,y)\\ hP^{\prime}{}_{k}(t_{n+1}+sh)\ =\ \frac{y_{n+3}}{3(3+r)}[\frac{1}{2}(3+2s-15s^{2}-4s ^{3}+10s^{4}+r(3-4s-9s^{2}+8s^{3}))]\\ \\ -\frac{32y_{n+\frac{5}{2}}}{15(5+2r)}[5s^{4}-4s^{3}-9s^{2}+2s+2+r(4s^{3}-6s^{2 }-2s+2)]\\ \\ +\frac{y_{n+3}}{(2+r)}[3+5s-9s^{2}-6s^{3}+5s^{4}+r(3-s-\frac{15s^{2}}{2}+4s^{3 })]\\ \\ -\frac{y_{n+1}}{3(1+r)}[\frac{1}{2}(1+22s-9s^{2}-20s^{3}+10s^{4})+r(7+8s-21s^{ 2}+8s^{3})]\\ \\ +\frac{y_{n}}{15r}[-3+7s+6s^{2}-14s^{3}+5s^{4}+r(-3+13s-\frac{22s^{2}}{2}+4s^{3 })]\\ \\ -\frac{y_{n-1}}{r(r+1)(r+2)(r+\frac{5}{2})(r+3)}[-3+7s+6s^{2}-14s^{3}+5s^{4}] \end{array}  \tag{4}\]  
Figure 1: 3–Point Variable Step Block Hybrid Method (VSBHM). By substituting \(s=0,\ 1,\ \frac{3}{2}\) and \(2\) into Equation (4) we obtain,  
\[ \begin{array}{c}hf(t_{n+1})=\frac{(1+r)}{2(3+r)}y_{n+3}-\frac{64(1+r)}{15(5+2r )}y_{n+\frac{5}{2}}+\frac{3(1+r)}{2+r}y_{n+2}-\frac{(1+7r)}{6(1+r)}y_{n+1}\\ \hskip 142.26378 pt-\frac{3(1+r)}{15r}y_{n}+\frac{3y_{n-1}}{r(r+1)(r+2)\left(r+ \frac{5}{2}\right)(r+3)}\\ \\ hf(t_{n+2})=\frac{(-r-2)}{3(3+r)}y_{n+3}+\frac{64(2+r)}{15(5+2r)}y_{n+\frac{5} {2}}-\frac{(3r+4)}{2(2+r)}y_{n+2}-\frac{(2+r)}{3(1+r)}y_{n+1}\\ \hskip 142.26378 pt+\frac{(2+r)}{30r}y_{n}-\frac{1}{r(r+1)(r+2)\left(r+\frac{5} {2}\right)(r+3)}y_{n-1}\\ \\ hf\Big{(}t_{n+\frac{5}{2}}\Big{)}=\frac{(75+30r)}{48(3+r)}y_{n+3}+\frac{2(55+1 6r)}{15(5+2r)}y_{n+\frac{5}{2}}\\ \\ -\frac{(75+30r)}{16(2+r)}y_{n+2}\frac{(25+10r)}{48(1+r)}y_{n+1}-\frac{(15+6r)} {240r}y_{n}\\ \\ -\frac{15}{16(r+1)(r+2)\left(r+\frac{5}{2}\right)(r+3)}y_{n-1}\\ \\ hf(t)=\frac{(75+23r)}{3(3+r)}y_{n+3}-\frac{32(18+6r)}{15(5+2r)}y_{n+\frac{5}{2 }}+\frac{(9+3r)}{2+r}y_{n+2}-\frac{(9+3r)}{6(1+r)}y_{n+1}\\ \\ +\frac{(3+r)}{15r}y_{n}-\frac{3}{r(r+1)(r+2)\left(r+\frac{5}{2}\right)(r+3)}y_{ n-1}\\ \\ \end{array}  \tag{7}\]  
On substituting \(r=1\), \(r=2\) and \(r=\frac{10}{19}\) into Equations (5)-(8) gives the coefficients for 3-point VSBHM are presented as in Table 1. These \(r\) values are properly considered for zero stability and computational efficiency.