Modelling non-linear heat conduction via a fast matrix-free implicit unstructured-hybrid algorithm

A matrix-free implicit algorithm is developed for the fast and efficient solution of the non-linear diffusion equation on unstructured-hybrid meshes. The aforementioned captures the physics prevalent in plasticity (structural mechanics), moisture transport in porous materials and non-linear heat conduction. This paper will be restricted to the latter. The boundary condition types supported are prescribed temperature, convection and radiation type Neumann conditions. A compact edge-based vertex-centered finite volume technique is employed for spatial discretisation purposes, while implicit temporal discretisation is effected. The resulting system of equations is Newton-linearized, where approximate analytical expressions for the Jacobian terms are developed. The discrete system is solved via a preconditioned Generalised Minimal Residual (GMRES) algorithm, where Lower–Upper Symmetric Gauss–Seidel (LU-SGS) is used as preconditioner. The computational performance of the developed algorithm is demonstrated by application to the modelling of steady 2D non-linear heat conduction problems. The calculated Jacobian terms are stored in the interest of computational speed, with an associated additional overall memory cost of less than 10%. Even for small problems, the developed solver outperforms Jacobi with local time-stepping in terms of computational time by a factor ranging between 30 and 1000, while having a total memory requirement of 1.9 times the aforementioned. Further, as compared to the LU-SGS and GMRES methods, the proposed solver methodology offers a speed-up in solution time of between one and two orders of magnitude.