## 1 Introduction  
In the face of the growing need to search for ecologically safe sources of energy, research on increasing the use of available low-temperature heat sources by means of various physical processes has intensified. One of them is the adsorption process used, among others, in sorption heat pumps. During the past two decades, adsorption heat pumps have been exploited in theory and in practice to generate cooling [1] and heating [2; 3]. Adsorption heat pump applications in cooling refer to useful cooling [4]; chilled water circuits [5; 6]; electric vehicle air conditioners [7]; heat rejection systems, such as dry or wet cooling towers or ground coupled heat exchangers [8]; and solar systems as the driving heat source [9; 10]. Heating applications of adsorption heat pumps are found in heating systems, waste heat utilization [3; 5], low-temperature heat sources (e.g., ground heat exchangers and geothermal water installations [11]) and driving heat sources (e.g., gas furnace) [12]. Adsorptive storage of CO\({}_{2}\) can be an interesting example of another application [13]. adsorption heat pumps have, aside from environmental benefits, several advantages compared to conventional vapor compression systems, such as: simplicity, no moving parts, low maintenance requirements, and the use of stable, nontoxic reactants as adsorbents and adsorbates. They also have disadvantages: discontinuous operation, high design requirements for maintaining the vacuum, large size, and a relatively low coefficient of performance [4; 12]. The performance of an adsorption heat pump is controlled by many parameters, such as adsorbent and adsorbate properties, system design, and operating conditions. Recently, much research has been done on various types of adsorption heat pumps as an alternative to vapor compression systems [14]. Transient modeling of a two-bed silica gel-water adsorption chiller [15] and an adsorbent bed of a zeolite/water cooling system and its numerical modeling of combined heat and mass transfer [16] are examples of one- or two-bed single-stage heat pump modeling. Second law analysis of adsorption cycles with thermal regeneration [17] cannot be overestimated. In order to improve efficiency, more advanced four-bed [18] or even multibed [19] heat pumps have been studied. Different numerical methods to solve mathematical equations describing adsorption beds or adsorption heat pumps can be found in the literature. For example, a nonuniform temperature, nonuniform pressure dynamic model of heat and mass transfer in compact adsorbent beds was presented by [PERSON] et al. [20]. The model was described by a set of second-order partial differential equations (PDEs) and solved. Further steps and methods to obtain a numerical solution together with a discussion on parameters affecting the numerical resolution were carefully presented. Dimensionless equations were discretized using a forward difference scheme for time derivatives and boundary conditions for a spatial first-order derivatives quickest upstream difference scheme and a spatial second-order derivatives central difference scheme. To obtain the solution, the alternating direction implicit method was used. In [PERSON] et al. [21], information of the implicit finite difference method of Crank-Nicholson was used to solve the differential equations of the developed model. In a numerical study of a novel cascading adsorption cycle [22], equations representing the model were solved concurrently using the finite difference method. The next example [23] discussed heat transfer in the adsorbent of a waste heat adsorption cooling system, in which the model calculating domain was first discretized in angle, radial, and axial directions, and the three dimensional equations of the adsorbent were calculated by the alternating direction implicit method. The quadratic upstream differencing scheme was used to approximate the convection terms in the equations. Diffusion terms were replaced by the centered difference analogs.