In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 10⁵ different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well.

Our goal was to find more effective numerical algorithms to solve the heat or diffusion equation. We created new five-stage algorithms by shifting the time of the odd cells in the well-known odd-even hopscotch algorithm by a half time step and applied different formulas in different stages. First, we tested 10⁵ = 100,000 different algorithm combinations in case of small systems with random parameters, and then examined the competitiveness of the best algorithms by testing them in case of large systems against popular solvers. These tests helped us find the top five combinations, and showed that these new methods are, indeed, effective since quite accurate and reliable results were obtained in a very short time. After this, we verified these five methods by reproducing a recently found non-conventional analytical solution of the heat equation, then we demonstrated that the methods worked for nonlinear problems by solving Fisher’s equation. We analytically proved that the methods had second-order accuracy, and also showed that one of the five methods was positivity preserving and the others also had good stability properties.

Calculating heat transfer in building components is an important and nontrivial task. Thus, in this work, we extensively examined 13 numerical methods to solve the linear heat conduction equation in building walls. Eight of the used methods are recently invented explicit algorithms which are unconditionally stable. First, we performed verification tests in a 2D case by comparing them to analytical solutions, using equidistant and non-equidistant grids. Then we tested them on real-life applications in the case of one-layer (brick) and two-layer (brick and insulator) walls to determine how the errors depend on the real properties of the materials, the mesh type, and the time step size. We applied space-dependent boundary conditions on the brick side and time-dependent boundary conditions on the insulation side. The results show that the best algorithm is usually the original odd-even hopscotch method for uniform cases and the leapfrog-hopscotch algorithm for non-uniform cases.

Using efficient methods to calculate heat transfer in building components is an important issue. In the current work, 14 numerical methods are examined to solve the heat transfer problem inside building walls. Not only heat conduction but convection and radiation are considered as well, in addition to heat generation. Five of the used methods are recently invented explicit algorithms, which are unconditionally stable for conduction problems. First, the algorithms are verified in a 1D case by comparing the results of the methods to an analytical solution. Then they are tested on real-life cases in the case of surface area (made of brick) and cross-sectional area (two-layer brick and insulator) walls with and without thermal bridging. Equidistant and non-equidistant grids are used as well. The goal was to determine how the errors depend on the properties of the materials, the mesh type, and the time step size. The results show that the best algorithms are typically the leapfrog-hopscotch and the modified Dufort–Frankel and odd–even hopscotch algorithms since they are quite accurate for larger time step sizes, even for 100 s as well.

New analytical solutions of the heat conduction equation obtained by utilizing a self-similar Ansatz are presented in cylindrical and spherical coordinates. Then, these solutions are reproduced with high accuracy using recent explicit and unconditionally stable finite difference methods. After this, real experimental data from the literature regarding a heated cylinder are reproduced using the explicit numerical methods as well as using Finite Element Methods (FEM) ANSYS workbench. Convection and nonlinear radiation are also considered on the boundary of the cylinder. The verification results showed that the numerical methods have a high accuracy to deal with cylindrical and spherical bodies; also, the comparison of the temperatures for all approaches showed that the explicit methods are more accurate than the commercial software.