Dimensional scaling and numerical similarity in hyperbolic method for diffusion

This paper discusses issues encountered by the hyperbolic method for diffusion (Nishikawa, 2007) [1] in dimensional heat conduction problems, and proposes a practical resolution. It is shown that the relaxation length must be scaled by a reference length of a domain of interest for solving dimensional equations, and the corresponding non-dimensionalized length should be given an optimal value for fast iterative convergence. To achieve both, a practical formula is proposed for computing a reference length for a given computational grid, such that (2π)−1 gives an optimal value for rectangular domains and also serves as an effective approximation for general domains. Numerical results confirm that the proposed scaling is critically important for rendering hyperbolic diffusion schemes independent of the grid unit and for achieving optimal performance of a hyperbolic diffusion solver.