Implicit numerical schemes for generalized heat conduction equations

There are various situations where the classical Fourier's law for heat conduction is not applicable, such as heat conduction in heterogeneous materials (Both et al., 2016; Ván et al., 2017) or for modeling low-temperature phenomena (Kovács and Ván, 2015, 2016, 2018). In such cases, heat flux is not directly proportional to temperature gradient, hence, the role - and both the analytical and numerical treatment - of boundary conditions becomes nontrivial. Here, we address this question for finite difference numerics via a shifted field approach. Based on this ground, implicit schemes are presented and compared to each other for the Guyer-Krumhansl generalized heat conduction equation, which successfully describes numerous beyond-Fourier experimental findings. The results are validated by an analytical solution, and are contrasted to finite element method outcomes obtained by COMSOL.