Hyperbolize it. Hiro Nishikawa Future Directions in CFD Research, - - PowerPoint PPT Presentation

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Dec 18, 2023 253 likes •389 views

Hyperbolize it. Hiro Nishikawa Future Directions in CFD Research, August 6-8, 2012 Hampton Roads Convention Center Sushi is Diffusion A harmony of rice and topping: rice crumbles and dissolves with topping in your mouth. - Tasty diffusion

SLIDE 1

Hyperbolize it.

Hiro Nishikawa

Future Directions in CFD Research, August 6-8, 2012 Hampton Roads Convention Center

SLIDE 2

Sushi is Diffusion

Lack of Guiding Principle

Years of training required to make such good sushi. Know-how is in experts’ hands. The same for diffusion schemes, but the development is still in early stage.

A harmony of rice and topping: rice crumbles and dissolves with topping in your mouth.

Tasty diffusion

http://www.hanayuubou.com/

SLIDE 3

Guiding Principles

Upwind for advection - hyperbolic

A variety of schemes generated:

Flux-Vector/Difference Splitting Multidimensional upwind - RD, FS schemes Riemann Solvers, CUSP , AUFS, AUSM, LDFSS, HLL, Steger-Warming, SUPG, etc.

Isotropic for diffusion - parabolic ?

Not that successful, especially for unstructured and high-order. What can be a useful guiding principle for diffusion?

Behold, we already have it.

SLIDE 4

Algorithm Research Continues

Hyperbolic Parabolic Dispersion Source

Algorithm Well developed No so well Not so well Tricky

Really? It is already well developed for all terms if we ...

SLIDE 5

Hyperbolize Them

Dramatic simplification/improvements to numerical methods

Methods for hyperbolic system applicable to all terms

JCP2007, 2010, 2012, AIAA2009, 2010, 2011, 2013, CF2011

First-Order Hyperbolic System Method

SLIDE 6

Turn Every Food into a Burger!

Simple, Efficient, Accurate.

Sushi Burger!

It looks eccentric, but the taste is the same, or even better!

SLIDE 7

Hyperbolic Diffusion System

Equivalent to diffusion eq. in the steady state for any Tr. Tr is a free parameter.

This is hyperbolic, describing a symmetric wave:

SLIDE 8

Hyperbolic Diffusion Scheme

Upwind scheme for diffusion:

Same order of accuracy for solution and gradient.

Damped scheme for diffusion:

JCP2007, 2010 JCP2007

Rapid steady convergence with O(h) time step, not O(h^2). Strong damping with O(h^2) time step.

Consistent Dissipation

SLIDE 9

Traditional Diffusion Scheme

Traditional Diffusion Scheme: Simply ignore the second equation in hyperbolic scheme, and reconstruct the gradients.

See AIAA2010, CF2011

Consistent Damping (from dissipation)

Scalar diffusion scheme can be derived from hyperbolic scheme.

For every advection scheme, there is a corresponding diffusion scheme.

High-frequency damping term is introduced automatically. It is essential for accuracy and robustness.

SLIDE 10

Damping is Essential

Highly-skewed unstructured grid (unsteady diffusion problem)

Damping term is critical for unstructured computations

x u

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4

x u

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4

x u

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4

x 25y

1 0.125

Avg-LSQ Edge-normal See AIAA2010, CF2011, for many examples.

Results for cell-centered finite-volume method

SLIDE 11

Three Paths to Take

Hyperbolic Model for Diffusion Solution to Diffusion

Hyperbolic Scheme JCP2007, 2010, AIAA2011 Damped Scheme Traditional Scheme JCP2007

Steady, damping Dual-time for unsteady Accurate gradients O(h^2) Time Step

Steady and Unsteady, damping O(h^2) Time Step

Steady, propagation Dual-time for unsteady Accurate gradients O(h) Time Step

AIAA2010, 2011 CF2011

It all starts from the discretization of the hyperbolic model.

SLIDE 12

Navier-Stokes Results

500 1000 1500 2000 2500 1000 2000 3000 4000 5000 6000 7000

Number of Nodes

CPU Time (second)

CPU Time

Viscous Shock-Structure Problem

Conventional Hyperbolic

500 1000 1500 2000 2500 5 10 15 x 10

Number of Nodes

Iteration

1.6 1.4 1.2 1 0.8 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4

Log10(h) Log10( L1 error of xx )

Slope 2 Slope 1

Viscous Stress

1.6 1.4 1.2 1 0.8 2.5 2 1.5 1

Log10(h) Log10( L1 error of qx )

Slope 2 Slope 1

Heat Flux

Hyperbolic Conventional

2nd-order finite-volume schemes

AIAA2011-3043

The idea extended to nonlinear system.

SLIDE 13

Construct a hyperbolic system and discretize it:

Higher order accuracy for viscous/heat fluxes Orders of magnitude faster viscous computation by O(h) time step Compact stencil for high-order derivatives High-order advection schemes for diffusion Boundary conditions made simple (all Dirichlet; local characteristics) A greater variety of viscous discretizations Damping term incorporated automatically into viscous schemes

Large eccentricity leads to a hyperbolic trajectory, which enables us to escape towards the future.

Hyperbolize to the Future

Hitherto unexpected advantages being discovered: