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BackgroundWENO Schemes

Modified Weights

Weighted ENO Schemes

Xiaolei ChenAdvisor: Prof. Xiaolin Li

Department of Applied Mathematics and StatisticsStony Brook University, The State University of New York

February 7, 2014

Xiaolei Chen Advisor: Prof. Xiaolin Li Weighted ENO Schemes


Modified Weights

1 Background

2 WENO SchemesBasic IdeaChoice of the WeightsFlux Splitting

3 Modified WeightsMapped WENO SchemesWENO-Z Scheme



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Background

1D Scalar Hyperbolic Equation:

ut + (f(u))x = 0

Assume the grid is uniform and solve the hyperbolic equation directlyusing a conservative approximation to the spatial derivative.

dui(t)

dt= − 1

∆x(f̂i+ 1

2− f̂i− 1

2),

where ui(t) is the numerical approximation to the point value u(xi, t),and f̂i+ 1

2is called numerical flux.

Question: How to approximate the numerical flux f̂i+ 12

?



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Basic IdeaChoice of the WeightsFlux Splitting

Basic Idea

xi xi+1 xi+2xi-1xi-2 xi+1ê2xi-1ê2

vi+1ê2-vi-1ê2

+

S

S0

S 1

S 2

Figure: Stencil of fifth order WENO scheme



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Basic Idea

At smooth region, each sub-stencil gives third (k-th) order numericalfluxes v−

i+ 12

and v+i− 1

2

.

For example, on stencil S1,

v(1)−i+ 1

2

= −1

6v̄i−1 +

5

6v̄i +

1

3v̄i+1

v(1)+

i− 12

=1

3v̄i−1 +

5

6v̄i −

1

6v̄i+1

Similarly, we have

v(0)−i+ 1

2

and v(0)+

i− 12

on stencil S0

v(2)−i+ 1

2

and v(2)+

i− 12

on stencil S2

*The coefficients come from reconstruction process.



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Basic Idea

Apply a weight to each stencil and a fifth ((2k-1)-th) order WENOscheme is obtained. Assume the weights are w0, w1, w2. Then, werequire

wr ≥ 0,

k−1∑s=0

ws = 1,

for stability and consistency.The fifth order fluxes are given by

v−i+ 1

2

=

k−1∑r=0

wrv(r)−i+ 1

2

, v+i− 1

2

=

k−1∑r=0

wrv(r)+

i− 12

Question: How to choose the weights?



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Choice of the Weights

If the function v(x) is smooth in all of the candidate stencils, there areconstants dr such that

v−i+ 1

2

=

k−1∑r=0

drv(r)−i+ 1

2

= v(xi+ 12) +O(∆x2k−1),

v+i− 1

2

=

k−1∑r=0

d̃rv(r)+

i− 12

= v(xi− 12) +O(∆x2k−1).

For example, when k=3,

d0 =3

10, d1 =

3

5, d2 =

1

10,

d̃0 =1

10, d̃1 =

3

5, d̃2 =

3

10.



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Choice of the Weights

In this smooth case, we would like to have

wr = dr +O(∆xk−1).

Form of the Weights:

wr =αr∑k−1s=0 αs

, r = 0, · · · , k − 1

αr =dr

(ε+ βr)2

where βr are the so-called ”smooth indicators” of the stencil Sr.We require

if v(x) is smooth in the stencil Sr, then βr = O(∆x2)

if v(x) has a discontinuity inside the stencil Sr, then βr = O(1)



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Smooth Indicators

Let the reconstruction polynomial on the stencil Sr be denoted by pr(x).Then, define

βr =

k−1∑l=1

∫ xi+1

2

xi− 1

2

∆x2l−1(dlpr(x)

dxl)2dx

When k=3,

β0 =13

12(v̄i − 2v̄i+1 + v̄i+2)

2 +1

4(3v̄i − 4v̄i+1 + v̄i+2)

2,

β1 =13

12(v̄i−1 − 2v̄i + v̄i+1)

2 +1

4(v̄i−1 − v̄i+1)

2,

β2 =13

12(v̄i−2 − 2v̄i−1 + v̄i)

2 +1

4(v̄i−2 − 4v̄i−1 + 3v̄i)

2



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Flux Splitting

Lax-Friedrichs Splitting:

f±(u) =1

2(f(u)± αu),

where α = maxu|f ′(u)| over the relevant range of u.FD WENO Procedure with Flux Splitting:

Identify v̄i = f+(ui) and use WENO procedure to obtain v−i+ 1

2

, and

take f̂+i+ 1

2

= v−i+ 1

2

Identify v̄i = f−(ui) and use WENO procedure to obtain v+i+ 1

2

, and

take f̂−i+ 1

2

= v+i+ 1

2

Form the numerical flux as f̂i+ 12

= f̂+i+ 1

2

+ f̂−i+ 1

2



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Mapped WENO SchemesWENO-Z Scheme

Mapped WENO Schemes

gr(w) =w(dr + d2r − 3drw + w2)

d2r + w(1− 2dr)

α∗r = gr(wJSr )

wMr =

α∗r∑2s=0 α

∗s

Advantage: fifth order accuracy at critical pointsDisadvantage: the weight of the stencil that contains discontinuitybecomes larger after the map; more computational cost.



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WENO-Z Scheme

τ5 = |β0 − β2|

αzr = dr(1 + (

τ5βk + ε

)q)

wzr =

αzr∑2

s=0 αzs

Advantage: fifth order accuracy at critical points with q=2;computationally more efficient than mapped WENO.



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Bibliography I

[1] B. Cokcburn, C. Johnson, C.-W. Shu, and E. Tadmor, Advancednumerical approximation of nonlinear hyperbolic equations, Ed. A.Quarteroni, Lecture Notes in Mathematics, vol. 1697, Springer,1998.

[2] C.-W. Shu, High order weighted essentially non-oscillatoryschemes for convection-dominated problems, SIAM Review, v51(2009), pp.82-126.

[3] A. K. Henrick, T. D. Aslam, J. M. Powers, Mapped weightedessentially non-oscillatory scheme: Achieving optimal order nearcritical points, Journal of Computational Physics 207 (2005),pp.542-567.



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Bibliography II

[4] R. Borges, M. Carmona, B. Costa, W. S. Don, An improvedweighted essentially non-oscillatory scheme for hyperbolicconservation laws, Journal of Computational Physics, 227, (2008)pp.3191-3211.


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