WENO Scheme with Eigensystem for Euler EquationsThe Annual Report in Advanced Research in Applied Mathematics and Scientific Computing 2018, OUC, China

Kangbo Tian, School of Mathematical Sciences, Ocean University of China

INTRODUCTIONConsider the Euler equation

∂Q

∂t+∂F

∂x= 0, (1)

where the conservative variables Q = (ρ, ρu,E)T , fluxF = (ρu, ρu2 + P, (E + P )u)T , ρ is the density, u is thevelocity, E is the total energy and P is the pressure. A fifth-order WENO scheme is used to numerically approximatethe solution of equation (1). The performance will be betterif the fluxes are projected on the characteristic fields beforeWENO reconstruction procedure is carried out.

PREPARE OF WENO SCHEME

xi xi+1 xi+2xi-1xi-2 xi+1/2

S2

S0

S1

S5The semi-discretized form of Eq. (1), by the method of lines,yields a system of ordinary differential equations

dQi(t)

dt= − ∂f

∂x

∣∣∣∣x=xi

, i = 0, . . . , N. (2)

By implicitly defining the numerical flux function h(x) as

f(x) =1

∆x

∫ x+ ∆x2

x−∆x2

h(ξ)dξ, (3)

Eq. (2) is exactly obtained by a conservative finite differ-ence formula at the cell boundaries,

dQi(t)

dt= − 1

∆x

(hi+ 1

2− hi− 1

2

). (4)

The reconstruction of hi± 12

and many of the details regard-ing WENO schemes, as well as a three-stage third orderTVD Runge-Kutta scheme for solving the ordinary equa-tion (2), can be found in the lecture note [2].

FLUX SPLITTINGOne would need to perform a flux splitting, for example,the global Lax-Friedrichs flux splitting, to split the flux Finto the positive F+ and negative F− fluxes, that is,

F± =1

2(F± αQ) , α = max

x∈ΩmaxQ

∣∣∣∣ ∂F∂Q∣∣∣∣ , (5)

where α is a vector of maximum absolute eigenvalues ofthe Jacobian.

ROE-AVERAGES EIGENSYSTEMThe system (1) can be written as

∂Q

∂t+∂F

∂x=∂Q

∂t+ A

∂Q

∂x= 0, (6)

where

ROE-AVERAGES EIGENSYSTEM

Q =

ρρUE

=

Q1

Q2

Q3

, (7)

F =

Q21

2Q1((3− γ)Q2

2 + 2(γ − 1)Q1Q3)1

2Q21((1− γ)Q3

2 + 2γQ1Q2Q3)

, (8)

and A = ∂F∂Q

is the Jacobian of the system

A =

0 1 0(γ − 3)U2 (3− γ)U γ − 1

((γ − 1)U2 − h)U h− (γ − 1)U2 γU

,

where c is the sound speed, h is the specific total enthalpyand U2 = 1

2U2 is specific kinetic energy.

Here are some of the relationships between the variables,

h =E + P

ρ, c =

√γP

ρ. (9)

The eigensystem of the Jacobian can be re-written as A =RΛL, where

Λ =

U − c 0 00 U 00 0 U + c

, (10)

R =

1 1 1U − c U U + ch− cU U2 h+ cU

, (11)

L =1

2c2

cU + bU2 −c− bU b2c2 − 2bU2 2bU −2b−cU + bU2 c− bU b

, (12)

where b = (γ − 1), L = R−1. For the content of the Roe-averaged states, we refer to [3] for the details.

CHARACTERISTIC PROJECTION

In the WENO reconstruction procedure, at each cell centerxi, i = 0, . . . N , we need to

• perform a characteristic projection of the fluxF±i+j , j = −r, . . . , r onto the characteristic space as

G±i+j = Li+ 12F±i+j , j = −r, . . . , r. (13)

• perform WENO reconstruction procedure with dataG±i+j , j = −r, . . . , r − 1 to obtain G±

i+ 12

.

• perform a characteristic projection of the flux G±i+ 1

2

back onto the physical space as

h±i+ 1

2= Ri+ 1

2G±

i+ 12. (14)

ACCURACY OF THE METHOD

Table 1: The initial condition is set as (ρ, u, P ) = (1 + 0.1 sin(πx), 1, 1) for the one-dimensional Euler equation, x ∈ [0, 2]

with periodical boundary condition, and ∆t is bounded by ∆x53 for the tests of accuracy at final time t = 2.

WENO-Z5 WENO-Z5-E WENO-Z5 WENO-Z5-E WENO-Z5 WENO-Z5-EN L∞ err. Order L∞ err. Order L2 err. Order L2 err. Order L1 err. Order L1 err. Order10 2.2e-3 — 2.2e-3 — 2.9e-3 — 2.9e-3 — 4.8e-3 — 4.8e-3 —20 7.1e-5 4.93 7.1e-5 4.93 7.6e-5 5.26 7.6e-5 5.26 1.1e-4 5.47 1.1e-4 5.4740 2.2e-6 4.98 2.2e-6 4.98 2.3e-6 5.07 2.3e-6 5.07 3.0e-6 5.18 3.0e-6 5.1880 7.0e-8 4.99 7.0e-8 4.99 7.0e-8 5.01 7.0e-8 5.01 9.1e-8 5.05 9.1e-8 5.05

160 2.2e-9 5.00 2.2e-9 4.95 2.2e-9 5.00 2.2e-9 5.00 2.8e-9 5.01 2.8e-9 5.01320 7.1e-11 4.95 7.1e-11 5.00 6.9e-11 5.00 6.9e-11 5.00 8.8e-11 5.00 8.8e-11 5.00

1D EULER EQUATIONS

Sod (N = 200) Lax (N = 200) Shu-Osher (N = 800)

x

Rh

o

-4 -2 0 2 40

0.2

0.4

0.6

0.8

1

ReferenceWENO-Z5WENO-Z5-E

-2.4 -2.2 -2 -1.8 -1.6 -1.40.25

0.3

0.35

0.4

x

Rh

o

-4 -2 0 2 40.2

0.4

0.6

0.8

1

1.2

1.4 ReferenceWENO-Z5WENO-Z5-E

2 2.5 31

1.1

1.2

1.3

x

Rh

o

-5 0 5 10 150.5

1

1.5

2

2.5

3

3.5

4

4.5

5 ReferenceWENO-Z5WENO-Z5-E

6.5 7 7.5 8

3.6

3.8

4

4.2

2D DMR PROBLEM

x

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1 22.421.420.419.418.417.416.415.414.413.412.411.410.49.48.47.46.45.44.43.42.41.4

x

y

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.80

0.1

0.2

0.3

0.4

0.5

FUTURE WORKDo more numerical experiments of the characteristic-wiseWENO scheme.

ACKNOWLEDGEMENTThanks to the support by Prof. Don Wai Sun, Prof. GaoZhen, Prof. Xie Shusen, Mr. Wang Baoshan, I got a chanceto study here this summer.

REFERENCES

[1] R. Borges, M. Carmona, B. Costa, W.S. Don, An improvedweighted essentially non-oscillatory scheme for hyperbolic conser-vation laws, J. Comput. Phys. (2008) 227(6) 3191-3211.

[2] W.S. Don, R. Borges, Introduction of WENO Finite Differ-ence Scheme for Hyperbolic Conservation Laws, https://www.researchgate.net/publication/309194721.

[3] P.L. Roe, Approximate Riemann Solvers, Parameter Vectors, andDifference Schemes, J. Comput. Phys. (1981) 43 357-372.