# Variable Step Block Hybrid Method for Stiff Chemical Kinetics Problems
## 5 Test Problems  
In this section, some stiff chemical kinetics problems are presented. These test problems are solved using the 3-point VSBHM and compared with the numerical results with ode15s. Graphical representations of results are presented in Section 6. **Example 1**. : _Belousov-Zhabotinskii reaction._  
The following system of homogeneous chemical reactions can be used to illustrate the Belousov-Zhabotinskii reaction [2],  
\[ \begin{array}{c}A+Y\to X,\,k_{1}=4.72s^{-1}\\ X+Y\to P,\,k_{2}=3\times 10^{9}s^{-1}\\ B+X\to 2X+Z,\,k_{3}=1.5\times 10^{4}s^{-1}\\ 2X\to Q,\,k_{4}=4\times 10^{7}s^{-1}\\ Z\to Y,\,k_{5}=1s^{-1}\end{array} \]  
The letters \(A,\,\ldots,\,Z\) represent the species involved in the reactions, while the constants \(k_{i}\) stands for the reaction rates. We just need to analyze the fluctuations in concentrationsover time \(t\), since the Belousov-Zhabotinskii reaction is homogeneous (all species are evenly distributed in the reaction space). The reaction rate constant characterizes each reaction step. The rate constants differ by several orders of magnitude, indicating that the associated ODE system is likely to be stiff. The initial conditions are determined by the species concentrations at \(t=0\):  
\[A=B=0.066,\ Y=X=P=Q=0,\ Z=0.002.\]  
The reaction scheme is modeled by the following system of ODEs:  
\[\frac{dA}{dt}=y_{1}^{\prime}=-k_{1}y_{1}y_{2}\]  
\[\frac{dY}{dt}=y_{2}^{\prime}=-k_{1}y_{1}y_{2}-k_{2}y_{3}y_{2}+k_{5}y_{6},\]  
\[\frac{dX}{dt}=y_{3}^{\prime}=-k_{2}y_{3}y_{2}+k_{3}y_{3}y_{5}-2k_{4}y_{3}^{2}+ k_{1}y_{1}y_{2},\]  
\[\frac{dP}{dt}=y_{4}^{\prime}=k_{2}y_{3}y_{2},\]  
\[\frac{dB}{dt}=y_{5}^{\prime}=-k_{3}y_{5}y_{3},\]  
\[\frac{dZ}{dt}=y_{6}^{\prime}=-k_{3}y_{5}y_{3}-k_{5}y_{6},\]  
\[\frac{dQ}{dt}=y_{7}^{\prime}=-k_{4}y_{3}^{2},\]  
where the initial concentrations of the species are \([A_{0}]=y_{1}(0)=0.066M,\ [Y_{0}]=y_{2}(0)=0M,\ [X_{0}]=y_{3}(0)=0M,\ [P_{0}]=y_{4}(0)=0M, [B_{0}]=y_{5}(0)=0.066M,[Z_{0}]=y_{6}(0)=0.002M\) and \([Q_{0}]=y_{7}(0)=0M\ (1M=1 mol.L^{-1})\) at the time interval in seconds \(t\in\ [0,40]\). **Example 2**. : _Stiff Chemical Problem._  
Consider a non-linear system of differential equations of one of the chemical kinetic problems [37]:  
\[y_{1}^{\prime}=\lambda y_{1}+y_{2}^{\prime}y_{2}^{\prime}=-y_{2}\]  
where \(\lambda=10,000\). The exact solution is \(y_{1}(x)=\frac{e^{-2x}}{(\lambda+2)}\) and \(y_{2}(x)=e^{-x}\). **Example 3**. : _Hires._  
This HIRES (High Irradiance Responses) problem is a first-order differential equation system of mild stiffness. It is a chemical process that simulates how light influences plant morphogenesis. [PERSON] [38] hypothesized this chemical process involving eight reactants to explain plant tissue development and differentiation in the absence of photosynthesis at high levels of light irradiance. It was previously used as a test case for a block-oriented simulation system by [PERSON] [39]. Figure 3: Concentration of \(A,\ B,X,Y,\ P,Q\), and \(Z\) computed using the 3\(-\) Point VSBHM for Problem 1. From left to right and top to bottom, Figure (**a**–**g**) shows the numerical solution of the Belousov–Zhabotinskii reaction for t \(\in[0,40]\) in seconds for the concentrations \((y_{1}-y_{7})\) with the error tolerance \(10^{-8}\). \[\frac{dA}{dt}=y_{1}^{\prime}=-k_{1}y_{1}y_{2}\]0 Comparison graphs with the solution from ode15s for Problem 1 with \(tol\,=\,10^{-8}\) for 3-point VSBHM. \[\frac{dA}{dt}=y_{1}^{\prime}=-k_{1}y_{1}y_{2}\]1 Comparison graphs with the solution from ode15s for Problem 2. \[\frac{dA}{dt}=y_{1}^{\prime}=-k_{1}y_{1}y_{2}\]2 Concentration of eight species using 3–point VSBHM for Problem 3. From **left** to **right** and **top** to **bottom**, Figure (**a–h**) shows the numerical solution of the HIRES problem for \(t\in~{}[0,50]\) in minutes for the concentrations \((y_{1}-y_{8})\) with the error tolerance \(10^{-6}\). Figure 4 displays the comparisons of the concentrations by using the 3-point VSBHM and ode15s. The 3-point VSBHM can approximate the solution of stiff Problem 1.