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Ii

Ii-2

Ii-1 Ii+1 Ii+2 Ii+3

xi+1/2

S0

S2

S1

such that:

u i+ l =1

x i+ l I i + l p j (x)dx, l = k + j, , ju i+ l =

1 x

i+ l I i + l

Q (x)dx,l = k, , k

Find linear weight 0, , k :

Q(x i+1 / 2) =k

j =0

j p j (x i+1 / 2)

For k = 2, the fth order reconstruction, we have:

p0(x i+ 12 ) =13

u i 2 76

u i 1 +116

u i

p1(x i+12 ) =

16u i 1 +

56u i +

13u i+1

p2(x i+ 12 ) =13

u i +56

u i+1 16

u i+2

Q(x i+ 12 ) =130

u i 2 1360

u i 1 +4760

u i +920

u i+1 120

u i+2

and we obtain the linear weights:

0 =110

, 1 =610

, 2 =310

.

We compute the smoothness indicator, denoted by j , for each stencil S j , which mea-sures how smooth the function p j (x) is in the target cell I i . The smaller this smoothness

indicator j , the smoother the function p j (x) is in the target cell.

j =k

l=1 I i x2l 1i

l

x l p j (x)

2

dx (4)

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In the actual numerical implementation the smoothness indicators j are written out

explicitly as quadratic forms of the cell averages of u in the stencil, for example when

k = 2, we obtain:

0 =13

12(u i 2 2u i 1 + u i)2 +

1

4(3u i 2 4u i 1 + u i)2

1 =1312

(u i 1 2u i + u i+1 )2 +14

(3u i 1 u i+1 )2

2 =1312

(u i 2u i+1 + u i+2 )2 +14

(u i 4u i+1 + u i+2 )2

Compute the nonlinear weights based on the smoothness indicators:

j = j

j j, j =

j

j ( + j )2(5)

The nal WENO approximation is then given by:

u i+1 / 2 k

j =0 j p j (x i+1 / 2) (6)

The reconstruction to u +i 1/ 2 is mirror symmetric with respect to x i of the above pro-

cedure.

For systems of conservation laws, such as the Euler equations of gas dynamics, thereconstructions from {u i} to {u

i+1 / 2} are performed in the local characteristic directions

to avoid oscillation.

Time discretizations:

Using explicit, nonlinearly stable high order Runge-Kutta time discretizations .[Shu and

Osher,JCP,1988]

The semidiscrete scheme (3) is written as:

u t = L (u )

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is discretized in time by a nonlinearly stable Runge-Kutta time discretization, e.g. the third

order version (7).

u (1) = u n + tL (un )

u (2)=

3

4u n

+

1

4u (1)

+

1

4tL

(u (1)

) (7)u n +1 =

13

u n +23

u (2) +23

tL (u (2) ).