Dimensional scaling and numerical similarity in hyperbolic method for diffusion

Abstract 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 ( 2 π ) − 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.

[1]  Hiroaki Nishikawa,et al.  Hyperbolic method for magnetic reconnection process in steady state magnetohydrodynamics , 2016 .

[2]  Hiroaki Nishikawa,et al.  First, second, and third order finite-volume schemes for advection-diffusion , 2013, J. Comput. Phys..

[3]  Alireza Mazaheri,et al.  First-Order Hyperbolic System Method for Time-Dependent Advection-Diffusion Problems , 2014 .

[4]  Hiroaki Nishikawa.  First, Second, and Third Order Finite-Volume Schemes for Diffusion , 2013 .

[5]  Hiroaki Nishikawa,et al.  Alternative Formulations for First-, Second-, and Third-Order Hyperbolic Navier-Stokes Schemes , 2015 .

[6]  Hiroaki Nishikawa.  New-Generation Hyperbolic Navier-Stokes Schemes: O(1=h) Speed-Up and Accurate Viscous/Heat Fluxes , 2011 .

[7]  Hong Luo,et al.  Reconstructed Discontinuous Galerkin Methods for Hyperbolic Diffusion Equations on Unstructured Grids , 2019, Communications in Computational Physics.

[8]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[9]  Yi Liu,et al.  Third-Order Inviscid and Second-Order Hyperbolic Navier-Stokes Schemes for Three-Dimensional Inviscid and Viscous Flows , 2016 .

[10]  Eleuterio F. Toro,et al.  Advection-Diffusion-Reaction Equations: Hyperbolization and High-Order ADER Discretizations , 2014, SIAM J. Sci. Comput..

[11]  Hong Luo,et al.  Cell-centered high-order hyperbolic finite volume method for diffusion equation on unstructured grids , 2018, J. Comput. Phys..

[12]  Hiroaki Nishikawa,et al.  A first-order system approach for diffusion equation. II: Unification of advection and diffusion , 2010, J. Comput. Phys..

[13]  Yi Liu,et al.  Third-Order Inviscid and Second-Order Hyperbolic Navier-Stokes Solvers for Three-Dimensional Unsteady Inviscid and Viscous Flows , 2017 .

[14]  Alireza Mazaheri,et al.  A first-order hyperbolic system approach for dispersion , 2016, J. Comput. Phys..

[15]  Philip L. Roe,et al.  Third-order active-flux scheme for advection diffusion: Hyperbolic diffusion, boundary condition, and Newton solver , 2016 .

[16]  Kimiya Komurasaki,et al.  A hyperbolic-equation system approach for magnetized electron fluids in quasi-neutral plasmas , 2015, J. Comput. Phys..

[17]  Hiroaki Nishikawa,et al.  Effects of high-frequency damping on iterative convergence of implicit viscous solver , 2017, J. Comput. Phys..

[18]  Hiroaki Nishikawa,et al.  First, Second, and Third Order Finite-Volume Schemes for Navier-Stokes Equations , 2014 .

[19]  Kimiya Komurasaki,et al.  A flux-splitting method for hyperbolic-equation system of magnetized electron fluids in quasi-neutral plasmas , 2016, J. Comput. Phys..

[20]  Ilya Peshkov,et al.  On a pure hyperbolic alternative to the Navier-Stokes equations , 2014 .

[21]  H. Nishikawa,et al.  Hyperbolic Navier-Stokes Solver for Three-Dimensional Flows , 2016 .

[22]  Hiroaki Nishikawa.  Robust and accurate viscous discretization via upwind scheme – I: Basic principle , 2011 .

[23]  o Yoshitaka Nakashima.  Development of an Effective Implicit Solver for General-Purpose Unstructured CFD Software , 2014 .

[24]  Carl Ollivier-Gooch,et al.  Accuracy analysis of unstructured finite volume discretization schemes for diffusive fluxes , 2014 .

[25]  Yi Liu,et al.  Hyperbolic advection-diffusion schemes for high-Reynolds-number boundary-layer problems , 2018, J. Comput. Phys..