Angled derivative approximation of the hyperbolic heat conduction equations

Numerical methods based upon angled derivative approximation are presented for a linear first-order system of hyperbolic partial differential equations (PDEs): the hyperbolic heat conduction equations. These equations model the flow of heat in circumstances where the speed of thermal propagation is finite as opposed to the infinite wave speed inherent in the diffusion equation. A basic angled derivative scheme is first developed which is second-order accurate. From this, an enhanced angled derivative scheme is then developed which is fourth-order accurate for Courant numbers of one-half and one. Both methods are explicit three-level schemes, which are conditionally stable. Careful treatment of initial and boundary conditions is provided which preserves overall order of accuracy and stability and suppresses any deleterious effects of spurious modes. After establishing a necessary and sufficient stability condition of Courant number less than or equal to one for both schemes, their dissipative and dispersive properties are investigated. A numerical example concerning the propagation of a thermal pulse train concludes this investigation.