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.

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.

Based on many previous experiments, the most efficient explicit and stable numerical method to solve heat conduction problems is the leapfrog-hopscotch scheme. In our last paper, we made a successful attempt to solve the nonlinear heat conduction–convection–radiation equation. Now, we implement the convection and radiation terms in several ways to find the optimal implementation. The algorithm versions are tested by comparing their results to 1D numerical and analytical solutions. Then, we perform numerical tests to compare their performance when simulating heat transfer of the two-dimensional surface and cross section of a realistic wall. The latter case contains an insulator layer and a thermal bridge. The stability and convergence properties of the optimal version are analytically proved as well.

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.