Hyperbolic Method

Hyperbolize, Discretize, and Solve.


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What is it?

Hyperbolic method (or first-order hyperbolic system method) is a general approach to numerically solving partial differential equations (PDEs). It is to reformulate all PDEs as a first-order hyperbolic system, discretize it, and then solve it. Consequently, numerical techniques well developed for hyperbolic systems can be directly applied to solve non-hyperbolic equations, e.g., parabolic. The hyperbolic method is different from other first-order-system-based methods in that

the hyperbolic system is constructed such that it recovers the original PDE exactly in the (pseudo) steady state: e.g.,
where tau is the pseudo time, Tr is the relaxation time of O(1), p and q are the gradient variables that will become the solution gradients in the pseudo steady state for any positive Tr.

Advantages include

- Discretization made simpler and unified.
- Higher-order advective/inviscid schemes.
- High-order and high-quality gradients on irregular grids.
- Accelerated convergence by elimination of second derivatives.
- Boundary conditions made simpler by the gradient variables.
- Diagonal dominance for anisotropic elliptic equations [ JCP2015 ]

These advantages have been demonstrated thus far for diffusion, advection-diffusion, and the compressible/incompressible Navier-Stokes equations for fully-unstructured Computational Fluid Dynamics (CFD) computations.

Example codes (fully annotated) :

- 1D hyperbolic diffusion scheme:

    First-order scheme: edu1d_oned_first_order_diffusion_v0.f90
    Second-order scheme: edu1d_oned_second_order_diffusion_v0.f90
    Third-order scheme: edu1d_oned_third_order_diffusion_v0.f90

- 2D hyperbolic advection-diffusion: edu2d-advdiff.zip
  (2nd-order edge-based scheme for unstructured grids.)

This website is created to provide a place where one can find the latest information about the first-order hyperbolic system method.

Copyright 2014- . Hiroaki Nishikawa. All rights reserved.