Tridiagonal matrix algorithm

Abstract: Physical phenomena with critical blow-up regimes simulated by the 3D nonlinear diffusion equation in a spherical shell are studied. For solving the model numerically, the original differential operator is split along the radial coordinate, as well as an original technique of using two coordinate maps for solving the 2D subproblem on the sphere is involved. This results in 1D finite difference subproblems with simple periodic boundary conditions in the latitudinal and longitudinal directions that lead to unconditionally stable implicit second-order finite difference schemes. A band structure of the resulting matrices allows applying fast direct (non-iterative) linear solvers using the Sherman-Morrison formula and Thomas algorithm. The developed method is tested in several numerical experiments. Our tests demonstrate that the model allows simulating different regimes of blow-up in a 3D complex domain. In particular, heat localisation is shown to lead to the breakup of the medium into individual fragments followed by the formation and development of self-organising patterns, which may have promising applications in thermonuclear fusion, nonlinear inelastic deformation and fracture of loaded solids and media and other areas.