# Variable Step Block Hybrid Method for Stiff Chemical Kinetics Problems
## 1 Introduction  
Stiff chemical kinetic models are required for the modeling of a practically real-world chemical system, such as atmospheric chemistry, energy conversion and storage, chemical engineering, the environment and materials, and biomedical and pharmaceutical engineering [1, 2, 3]. The following Ordinary Differential Equations (ODEs) can be used to model a homogeneous chemical reaction system:  
\[y^{\prime}=f(t,y),\ \ \ t_{0}\leq t\leq t_{final} \tag{1}\]  
\[y(t_{0})=y_{0} \tag{2}\]  
The column vector of species concentration is represented by \(y=[y_{1},y_{2},\dots,\ y_{N}]^{T}\) and \(N\) represents the total quantity of chemical species where \(t\) denotes the time, and the initial and final time are symbolized as \(t_{0}\) and \(t_{final}\), respectively. The initial species concentrations are indicated by the column vector \(y_{0}\). Numerical solutions for stiff ODE systems defined by Equation (1) can be obtained using explicit or implicit ODE integrator [4, 5, 6, 7, 8]. Many ODEs have been used for chemical kinetic models; however, they are stiff [9], and solving stiff ODEs using an explicit technique necessitates very short time steps, making integrationcomputationally intensive. Alternatively, implicit ODE integrators, such as the Backward Differentiation Formula (BDF), can be employed. In general, the implicit methods usually involve Newton Iteration for solving stiff systems of ODEs which is time-consuming because it consists of a Jacobian matrix that needs to be solved for every integration step. Therefore, it is still a research focus to deal with the Jacobian part for solving stiff system of ODEs [10; 11], which is a fundamental aspect of several reaction-diffusion systems, such as in energy conversion, medicinal applications, and chemical engineering. Although it is intricate to provide a clear description of the stiffness of the chemical kinetic model, one of the conditions can be if different species have significantly different time scales. Some fast-evolving organisms, for example, have very small time scales, whereas others evolve slowly and take longer time scales. The computing cost of explicit ODE integrators created for non-stiff problems against implicit ones, developed for stiff problems on the application is a realistic way to quantify stiffness. The problem can also be classified as very stiff if the computational cost of using an implicit ODE integrator is significantly less than the cost of an explicit ODE integrator [12]. The backward differentiation formula (BDF), often known as [PERSON]'s approach and initially proposed in [13], is the foundation for many well-known strategies for addressing stiff problems. Various researchers, such as [1; 14; 15; 16; 17], proposed methods to increase the accuracy and computation time for stiff problems. Ref. [17] presented the idea of using the Block Backward Differentiation Formula (BBDF) to solve first-order stiff problems and since then the BBDF approach is increasingly being used to solve stiff ODEs. There are many solvers based on the BBDF method that are used to solve stiff ODEs that have been developed in the literature [18; 19; 20]. BDF has been extensively used because of its high stability properties. Based on the classical BDF method, several block methods have been proposed. The BBDF is one of the most prominent block approaches based on BDFs ([20; 21; 22]). \(r\)-point Block Backward Differentiation Formula (\(r\)-BBDF) presented by [23], are used and expanded later. The block method based on BDF is particularly efficient for solving stiff ODEs to reduce the computational time and enhance the accuracy [24; 25]. Despite having many advantages, the block method has the major drawback that the order of interpolation points could not surpass the order of differential equations [26; 27]. As a result of this drawback, the addition of off-step points in the block was introduced and named the Hybrid method. Hybrid block approaches for the solution of IVPs in ODEs were proposed by [28; 29; 30].