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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.,
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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.
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Hyperbolic Method