Nonoscillatory Central Schemes on Unstructured Triangulations...

Nonoscillatory Central Schemeson Unstructured Triangulations

for Hyperbolic Systems of Conservation Laws

Ivan Christov

Joint work with Bojan Popov.

Department of Mathematics,Texas A&M University, College Station, Texas 77843-3368

[email protected]

Department of Mathematics Colloquium,University of Louisiana at Lafayette

November 2, 2006

Ivan Christov et al. (TAMU) Nonoscillatory Central Schemes ... ULL Colloquium 1 / 25

http://www.math.tamu.edu/~christov/

http://www.math.tamu.edu/~popov/

mailto:[email protected]

The Big Picture

Motivation:

Why unstructured triangulations when (Cartesian) tensor product gridsare so simple and efficient to implement?How should we design “genuinely multidimensional” schemes andlimiters on unstructured grids?Need for a simple & fast predictor algorithm for the time-dependentextension of L1-minimization FEM (J.-L. Guermond & B. Popov).

Background:

On October 30 there were 18,630 articles on MathSciNet with primaryor secondary MSC matching 35L* (Partial differential equations ofhyperbolic type).High-resolution central schemes: Nessyahu & Tadmor (1992) → Jiang& Tadmor (1998) → Kurganov & Tadmor (2000) → Kurganov, Noelle,Petrova (2001) ...Extension to unstructured grids: Arminjon et al. (1997).Semi-discere central-upwind version: Kurganov & Petrova (2005).


A Picture is Worth a Thousand Words

Source: http://www.fluidmech.net/gallery/ss cars/sscars.htm


http://www.fluidmech.net/gallery/ss_cars/sscars.htm

Outline of the Talk

1 Intro to hyperbolic systems of conservation laws.2 Crash course on Godunov-type central schemes.

Derivation of a fully-discrete second-order-accurate scheme.

1D Example: the Nessyahu–Tadmor scheme.2D Example: our new scheme on unstructured triangulations.

The minimum-angle-plane reconstruction.How to construct the dual (or staggered mesh) of an unstructuredtriangulation.

3 Numerical results.

Scalar equation with convex flux: inviscid Burgers.System of equation with convex flux: Euler equations of gas dynamics.Scalar equation with nonconvex flux.


Statement of the Problem and Notation (I)

Consider the initial-boundary-value problem for a (d + 1)D hyperbolicsystem of conservation laws:

~qt +∇~x · ~F (~q) = ~s(~q,~x , t), (~x , t) ∈ Ω× (0,T ],

~q(~x , t = 0) = ~q0(~x), ~x ∈ Ω,

~q(~x , t) = ~qBC(~x , t), (~x , t) ∈ ∂Ω× (0,T ].

Ω ⊂ Rd is the interior of a polygonal domain and ∂Ω its boundary.

~q = (q1, . . . , qd)> is the vector of conserved quantities,~F = (~f1, . . . ,~fd)> is the tensor containing the corresponding fluxesand ~s(~q,~x , t) is a source term.


Statement of the Problem and Notation (II)

Bad news: classical solutions break down in finite time, weak solutionsare not unique. For d > 1 there is essentially no existence, uniquenessand regularity theory.

Good news: integral formulation requires less regularity, idea ofvanishing viscosity and the Kruzkov entropy solution concept can beused to select a unique weak solution.

We restrict to the case d = 2 and ~s(~q,~x , t) = 0:~qt + ~f (~q)x + ~g(~q)y = 0, (x , y , t) ∈ Ω× (0,T ],

~q(x , y , t = 0) = ~q0(x , y), (x , y) ∈ Ω,

~q(x , y , t) = ~qBC(x , y , t), (x , y , t) ∈ ∂Ω× (0,T ].


Integral Form and the Finite Volume Method

Idea: integrate the differential form of the conservation law overV × [tn, tn+1] and apply Gauss’ Divergence Theorem to obtain∫ tn+1

tn

(d

dt

∫V

~q(~x , t) dA +

∮∂V

~f (~q(~x , t))νx + ~g(~q(~x , t))νy ds

)dt = 0.

Using the Fundamental Theorem of Calculus, we get that

1

|V|

∫V

~q(~x , tn+1) dA =1

|V|

∫V

~q(~x , tn) dA

− 1

|V|

∫ tn+1

tn

∮∂V

~f (~q(~x , t))νx + ~g(~q(~x , t))νy ds dt.

Godunov’s idea: Approximate the cell-averages of ~q by apiecewise-constant function, which we can evolve “exactly.”

How do we find a “good” approximation to the flux integrals?


Godunov-type Schemes: 1D Example (I)

Consider the 1D conservation law ~qt + ~f (~q)x = 0.

Introduce a mesh size ∆x , and consider the Steklov sliding average

~q(x , t) :=1

∆x

∫Ix

~q(ξ, t) dξ, Ix =

ξ

∣∣∣∣ |ξ − x | ≤ ∆x

2

.

Then, integrating the conservation law over Ix gives the semi-discreteform

d

dt~q(x , t) +

1

∆x

[~f(~q(x +

∆x

2, t))− ~f(~q(x − ∆x

2, t))]

= 0.


Godunov-type Schemes: 1D Example (II)

Then, integrating over the temporal interval [tn, tn+1] gives thefully-discrete, conservative scheme

~q(x , tn+1) = ~q(x , tn)

− 1

∆x

[∫ tn+1

tn

~f(~q(x +

∆x

2, t))dt −

∫ tn+1

tn

~f(~q(x − ∆x

2, t))dt

].

Let xi = (i − 1/2)∆x be the center of each computational cell.

Then, there are two methods of sampling the F.V. formulation:

(Upwind) We sample at the cell-centers xi at all time steps (≡Godunov’s idea).(Central) We alternate between cell-interfaces xi+1/2 ≡ i∆x and thecell-centers xi .


Central Schemes in 1D (I)

Let w(x , t) ≈ ~q(x , t) and recall Godunov’s idea.

Anzats: At each discrete time tn the approximate solution is apiecewise polynomial

wn(x) := w(x , tn) =N∑

i=1

pni (x)χi (x), χi (x) := 1Ii ,

where pi (x) is an algebraic polynomial supported on the intervalIi ≡ Ixi , which is centered around the cell-center xi .

Sampling the sliding average equation at the cell-interfaces xi+1/2 weobtain

wn+1(xi+1/2) = wn(xi+1/2)

− 1

∆x

[∫ tn+1

tn

~f(w(xi+1, t)

)dt −

∫ tn+1

tn

~f(w(xi−1, t)

)dt

].


Central Schemes in 1D (II)

Given pni , we can compute exactly

wn(xi+1/2) =1

∆x

[∫ xi+1/2

xi

pni (x)dx +

∫ xi+1

xi+1/2

pni+1(x)dx

].

And under the CFL condition maxi ρ(

∂~f∂~q (wn(xi ))

)∆t < ∆x

2 thenumerical fluxes can be computed with an appropriate quadraturerule, say the midpoint rule:

1

∆x

∫ tn+1

tn

~f(w(xi±1, t)

)dt ≈ ∆t

∆x~f(w(xi±1, t

n+1/2)),

where tn+1/2 := 12(tn + tn+1).


The Second-Order Nessyahu–Tadmor Scheme (I)

1 Starting with the piece-wise constant approximation to the cellaverage wn(·), reconstruct a piece-wise linear function wn(·).

2 Compute the staggered averages wn+1i+1/2, which gives us wn+1(·).246 KURGANOV AND TADMOR

FIG. 2.1. Central differencing approach—staggered integration over the local Riemann fan.

[2, 3, 18]. We also would like to mention the corresponding central schemes for incom-

pressible flows in [24–26, 33] and applications to various systems in, e.g., [1, 7, 43]. Thus,

the use of higher-order reconstructions enables us to decrease the numerical dissipation

present in central schemes, and achieve a higher resolution of shocks, rarefactions, and

other spontaneous evolution of large gradient phenomena.

Remarks. 1. Characteristic vs componentwise approach. A key advantage of central

schemes is their simplicity–one avoids here the intricate and time-consuming character-

istic decompositions based on (approximate) Riemann solvers, which are necessary in

high-resolution upwind formulations. For systems of conservation laws, the numerical

derivatives (ux )nj can be implemented by componentwise extension of the scalar recipe for

nonoscillatory limiters. Similarly, the predicted values in (2.6) are based on approximate

derivatives of the flux, ( fx )nj . These values can be computed in terms of the exact Jacobian,

! f!u

(unj )(ux )nj . Alternatively, we can even avoid the use of the computationally expensive

(and sometimes inaccessible) exact Jacobian ! f!u. Instead, the approximate flux derivatives,

( fx )nj , are computed in a componentwise manner based on the neighboring discrete values

of f (u(x j!1, tn)), f (u(x j , t

n)) and f (u(x j+1, tn)). It was pointed out in [36, 18] that this

Jacobian-free version of the central scheme does not deteriorate its high resolution.

2. Cell averages vs point values. Note that here one realizes the approximate solution

by its cell averages, un+1j+1/2. In general, when dealing with first- and second-order schemes,

the cell averages, un+1j+1/2, can be identified with the corresponding point values, un+1j+1/2,

modulo a negligible second-order term. We therefore from now on omit the bar notation.

(Consult [4, 36], for example, for this distinction with higher-order central schemes).

3. Second- vs first-order. In the particular case of (ux )nj " 0, the second-order NT

scheme is reduced to the staggered form of the first-order LxF scheme. The nonstaggered

version of a second-order central scheme can be found in [17].


The Second-Order Nessyahu–Tadmor Scheme (II)

One of the simplest (yet most robust!) second-order, nonoscillatoryreconstructions is

pni (x) = wn

i +w ′

i

∆x(x − xi ), w ′

i = mm(wni+1 − wn

i , wni − wn

i−1),

where mm(a, b) is the minmod (i.e., minimum-modulus) function

mm(a, b) =1

2(sgn(a) + sgn(b))min(|a|, |b|).


The Second-Order Nessyahu–Tadmor Scheme (III)

1 Predict the temporal midvalues for the quadrature rule by a Taylorseries expansion:

wn+1/2i±1 := w(xi±1, t

n+1/2) = wni±1 −

∆t

2

∂~f

∂~q(wi±1)

w ′i±1

∆x.

2 Compute the new (staggered) averages

wn+1i+1/2 =

1

2(wn

i +wni+1)+

1

8(w ′

i−w ′i+1)−

∆t

∆x

[~f(w

n+1/2i+1

)− ~f(w

n+1/2i−1

)].


A Convergence Result for Central Schemes

For a scalar, one-dimensional, homogenous conservation law, the followingis true:

Theorem (B. Popov and O. Trifonov)

Let f be strictly convex and assume that the CFL condition

∆t

∆xmax |f ′| ≤ κ

holds with κ sufficiently small. Then, the Nessyahu–Tadmor schemeconverges to the unique (Kruzkov) entropy solution of the conservationlaw.

→ Numerische Mathematik, Vol. 104, Number 4, 2006.


Preliminaries for the 2D Godunov-type Scheme

T = τi is a conforming triangulation of Ω.

S = σk is the staggered grid — the “dual” of T .

Let wn be a piecewise-constant function that approximates the cellaverages of ~q on T at time tn.

wn(x , y) =N∑

i=1

wni χi (x , y),

Also, let wn be the analogue of wn on S.

The piecewise-linear reconstruction of wn will have the form

wn(x , y) =N∑

i=1

pni (x , y)χi (x , y),

pni (x , y) = wn

i +∇pni · (x − x∗i , y − y∗i )>.


Overview of the 2D Central Scheme

1. Perform a slope-limited piecewise-linear reconstruction on T .

wn −→ wn

2. Evolve the cell averages on the staggered grid S in time.

wn −→ wn, wn −→ wn+1

3. Project the solution from S back onto T .

wn+1 −→ wn+1, wn+1 −→ wn+1

The good, the bad and the ugly:

No need to solve a Riemann problem at each cell interface!

Need to define S in a “reasonable” manner.

Need to be able to perform a nonoscillatory reconstruction on S.


The Minimum-Angle-Plane Reconstruction

The algorithm:

1 Given an element τi ∈ Tand its neighbors τij ,1 ≤ j ≤ m, find all

(m+13

)possible planes.

2 Find the plane that makesthe smallest angle withthe horizontal, and use itto find a limited gradient.

Note that

!i2

(x!i2, y

!i2)

!i1

(x!i1, y

!i1)

!i3

(x!i3, y

!i3)

!i

(x!i , y

!i )

vi3

vi1 vi2

This “genuinely 2D” limiter behaves like minmod with a UNO flavor(i.e., ≈ Durlofsky–Engquist–Osher but > 1st order near extrema).

No particular geometry and/or connectivity is assumed in thedesign of the limiter, and there are no ad hoc parameters.

Limiter works exactly the same way on S as on T .


Evolution Through the Staggered Averages

As before ...

1 Predict the temporal midvalues for the quadrature rule with via Taylorseries expansion:

w(x , y , tn+1/2) ≈ wn(x , y)−∆t

2

(∂~f

∂~q

(wn(x , y)

)∂wn

∂x+

∂~g

∂~q

(wn(x , y)

)∂wn

∂y

).

2 Compute the new (staggered) averages:

wn+1k ≈ wn

k −∆t

|σk |

∮∂σk

~f(

~w(x , y , tn+1/2))νx +~g

(~w(x , y , tn+1/2)

)νy ds.


The Staggered Grid

The dual elements:

1 Triangles ∆i .

2 Polygons Λij .

3 Parallelograms Πij .

The usual CFL condition

∆t < 13 ·mini |τi |/Smax,

Smax = fastest wave’s speed,is good enough.

!i1!i2

!i3"i1 "i2

"i3

#i

In particular:

If |∆i | = |Πij | = 0, the staggered grid becomes the Voronoi diagram.

If the maximum local speed of propagation is used, we get theanalogue of Kurganov & Tadmor’s modified central differencingfor triangulations.


Numerical Results for a Convex Flux

Riemann problems for the 2D inviscid Burgers equation

ut +(

12u2)x

+(

12u2)y

= 0.

(L) 6,272 elements and 19,041 dual elements.(R) 12,800 elements and 38,721 dual elements.


Riemann Problem for the Euler Equations

~q = (ρ, ρu, ρv ,E )>, ~f (~q) =(ρu, ρu2 + p, ρuv , u(E + p)

)>,

~g(~q) =(ρv , ρuv , ρv2 + p, v(E + p)

)>,

For an ideal gas: p = (γ − 1)[E − 1

2ρ(u2 + v2)], γ = 1.4.

Very fine mesh (mini diam(τi ) =√

2/256) with CFL = 0.275.


Numerical Results for a Nonconvex Flux (I)

Riemann problem for the 2D scalar equation

ut + (sin u)x + (cos u)y = 0.

Adapted mesh with (only) 3,264 elements and 9,837 dualelements, CFL = 1

6 .


Numerical Results for a Nonconvex Flux (II)

Kurganov, Petrova & B. Popov reported that less compressive /higher order limiters (e.g., WENO5, MM2, SB) do not resolve theresulting composite wave correctly. The MAPR passes this test!

(L) 90,000 element Cartesian tensor-product mesh.


Selected Bibliography

D. Kroner, Numerical Schemes for Conservation Laws.Wiley–Teubner, 1997.

G.-S. Jiang & E. Tadmor, “Nonoscillatory Central Schemes forMultidimensional ... ”SIAM J. Sci. Comput. 19 (1998) 1892.

A. Kurganov & G. Petrova, “Central-Upwind Schemes onTriangular Grids ... ”Numer. Methods Partial Differential Eq. 21 (2005) 536.

A. Kurganov, G. Petrova & B. Popov, “Adaptive Semi-DiscreteCentral-Upwind Schemes ... ”SIAM J. Sci. Comput. (submitted).

J.-L. Guermond, “A Finite Element Technique for SolvingFirst-Order PDEs in Lp.”SIAM J. Numer. Anal. 42 (2004) 714.


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