Multidiszciplináris Tudományok, 14. kötet, 1. sz. (2024) pp. 60–70. https://doi.org/10.35925/j.multi.2024.1.6 60 COMPREHENSIVE INVESTIGATION OF THE EXPLICIT, POSITIVITY PRESERVING METHODS FOR THE HEAT EQUATION Part 2 Husniddin Khayrullaev PhD student, Institute of Physics and Electric Engineering, University of Miskolc 3515 Miskolc-Egyetemváros, e-mail: hxayrullayev@mail.ru Endre Kovács associate professor, Institute of Physics and Electric Engineering, University of Miskolc 3515 Miskolc-Egyetemváros, e-mail: endre.kovacs@uni-miskolc.hu Abstract In this paper-series, we investigate the performance of 12 explicit non-conventional algorithms in several 2D systems. All of them have the convex combination property, thus they are unconditionally stable and preserve the positivity of the solution when they are applied to the heat equation. In the first part of the series, we examined how the errors depend on the time step size and running times. Now we present additional numerical test results, where sweeps for parameters such as the stiffness and the wavelength of the initial function will be performed. Keywords: explicit numerical methods, unconditional stability, heat equation, parabolic PDEs 1. Introduction and the generalized form of the studied equation This paper is the second part of a paper-series where we examine the performance of our recently published methods (see e.g. [Kovács, 2020; Kovács et al., 2024]) and the unconditionally positive finite-difference (UPFD) method of Chen-Charpentier et al. (Chen-Charpentier and Kojouharov, 2013). The problem to be solved is the conduction of heat, which is modelled by the general form of the heat equation   u c k u cq t        , (1) where   u u r,t  , the unknown temperature function, depends on the time and the x and y coordinates,   k kr  ,  c cr  ,  r  are the heat conductivity, the specific heat, and the density, respec- tively. In principle, , , kc  are arbitrary functions of the space variables, except that ,, , kc  and the thermal diffusivity /( ) k c    are non-negative. The general analytical solution of the problem does not exist. In the first part (Khayrullaev and Kovács, 2024), we explained why solving these equations numerically is a nontrivial task. Then, we described the 12 numerical methods and we constructed five test cases with different parameters and examined how the errors depend on the time step size and the running times for each of the 12 methods. In this part, we conduct systematic testing by sweeping with specific parameters to investigate the performance of the algorithms. Based on the accumulated results