## 1 Introduction
Issues of stability and stiffness of the numerical method were considered in [24]. Integration of partial differential equations with the numerical method of lines is described widely in the literature by [PERSON] [25; 26] and [PERSON] [27]. For many problems described by nonstiff differential system equations, just about any reasonable method is basically stable, for example, the Runge-Kutta fourth-order method, even called the \"workhorse\" method in MATLAB(r). It is a very robust method and the best function to apply as a \"first try\" for most problems where computational efficiency is of no concern. For convenience or for low precision, adaptive step size Runge-Kutta dominates [28; 29]. In many problems, certain parameters change quickly and others change more slowly. For example, in packed-bed chemical reactors, the concentration varies quickly in time, but the temperature does not change rapidly because of the large heat capacity of the solid. Due to the high values of temperature and uptake gradients in an adsorption column (bed), equations describing the heat and mass balance operation of adsorption heat pumps are often described as \"stiff\" [25]. For stiff problems, many time steps would need to be taken with a small time step, and the calculations would be very slow. The Gear Backward Differentiation Formulae (BDF) method has been developed to overcome this problem [28; 29]. Crucial to the success of a stiff integration scheme is an automatic step size adjustment algorithm [30; 31]. In this paper, a single-stage adsorption heat pump with a silica gel adsorbent and water adsorbate was considered. The adsorption heat pump consisted of two adsorber/desorber columns, an evaporator, and a condenser. The mathematical model, developed to calculate temperature and uptake changes in the adsorber/desorber, was established comprising a set of heat and mass balance PDEs together with the initial and boundary conditions and was solved by the numerical method of lines (NMOL). The spatial discretization was performed with equally spaced axial nodes and the PDEs were reduced to a set of ordinary differential equations (ODEs). Although models of adsorption heat pumps and an adsorber/desorber have been extensively studied in the literature (examples above), very few studies have analyzed the selection of the numerical method and its influence on the calculated results as well as the error of the method. In addition, most of the studies refer to one numerical method, with only a general description. Therefore, a comparison of three numerical methods and their limitations in calculations of the adsorption model is presented in this paper. In the present paper, the authors focused on the comparison of results obtained when the set of heat and mass balance ODEs was solved using: (1) the Runge-Kutta fixed step size fourth-order method (RKfixed); (2) the Runge-Kutta-Fehlberg 4.5 th-order method (RK45) with a variable step size; and (3) the Gear BDF numerical method. The set of balance equations of the adsorber was solved by the previously mentioned numerical ODE subroutines available in the MATLAB(r) platform. Obtained simulation results from numerical methods were compared with the results from the experiment in [21]. The rest of the paper is organized as follows. The heat and mass balance model of adsorption heat pump performance is established in Section 2. The numerical method for the PDEs of the model is described in Section 3. Comparison of the three numerical methods for the obtained set of ODEs is presented in Section 4. Section 5 presents the numerical calculation results. For the validation of the numerical model, a comparison with the experimental data of [PERSON] et al. [21] was made. The main conclusions are summarized in Section 6.