weno schemes & implementation in python -...

PYTHON IMPLEMENTATION OF WENO INTERPOLATION & RECONSTRUCTION

Presented by: Adrian Townsend

In collaboration with:Professor Randy LeVequeDavid KetchesonUniversity of Washington

OutlineWENO:

IntroductionThe Big PictureExamples, Advantages & Costs

Implementation:Motivation for Python ApproachA New Class: Piecewise Rational ClassPresent & Future Work

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Weighted Essentially NonOscillatorySchemesAn adaptive interpolation or reconstruction procedure.

Achieve arbitrarily high order of formal accuracy in smooth regionsMaintain stable, nonoscillatory and sharp discontinuity transitions

Popular with solving hyperbolic conservation laws

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WENO – The Big PictureGiven (or computed): ui or ui (average of cell between xi and xi+1)

Desired: Function that can be evaluated for any xInterpolate or Reconstruct (from cell averages)

Find rational interpolant, p(x)Arbitrary order of accuracy, N =5

x

ui

S

S1S2

S3

xi+1xi

Advantages& Costs

Arbitrarily high order of accuracy

Adaptive (nonlinear weights)

Ideal for convection dominated problems with sharp discontinuities and complicated smoothsolution structures.

Computationally Expensive!

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WENO versus scipy.interpolate.spline Interpolation

WENO Reconstruction from averages

Implementing WENOGiven order of accuracy, N = 2 n -1 , x i’s , ui’s

Find n polynomial interpolants, pk (x ), for u (xi ) of O(n) on each Sk

(pk degree n-1)

Find nonlinear weights

αk = αk (x i , N) is a polynomial

βk = βk (pk , x i , x i+1 ) is a smoothness factor that involves integrating derivatives of pk over the interval [ x i , x i+1 ]

A solution: return piecewise rational object

* Chi-Wang Shu, 20098

Python Implementation: PiecewiseRational.py

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Python Implementation: PiecewiseRational.py

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At Present & Looking Forward

Currently:Implementing boundary condition approaches Comparing two approaches for reconstructionDocumentation

Later:Submit module to SciPy

Thank You

Contact:Adrian Townsend adrian.townsend@gmail.comProf. Randy LeVeque rjl@amath.washington.edu

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References

Shu, Chi-Wang. "High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems." (2009) SIAM Review Vol. 51, No. 1, pp 82-126.

Carlini, E., Ferretti, R., Russo, G. "A Weighted Essentially Nonoscillatory, Large Time-Step Scheme for Hamilton-Jacobi Equations." (2005) SIAM J. Sci. Comput. Vol. 27, Iss. 3, pp 1071-1091.

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Stencil Growth

x

k = 2

Order 2k+1 = 5

k+1 = 3 Stencils

k = 1

Order 2k+1 = 3

k+1 = 2 Stencils

k = 3

Order 2k+1 = 7

k+1 = 4 Stencilsx

x

k = 4

Order 2k+1 = 9

k+1 = 5 Stencils

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