First-order Schemes

2.2 Numerical Methods for Conservation Laws

2.3.1 First-order Schemes

i. Properties (i.) and (ii.) assures no oscillations in the approximation in a sense that the operator

L

in Eq. (2.3.2) is monotonicity preserving (see proposition

2.1.1).

ii. TVD condition (2.3.5) is crucial for preventing oscillations from happening, but is strict and makes the scheme badly performs near critical points, i.e., at most first-order (see properties (iii.) and (iv.) and [112] for high-order TVD schemes). It will be relaxed in ENO and WENO schemes. See the below section.

In the following sections, we mention commonly discussed TVD schemes, starting from the first-order ones. We discuss in detail the Godonov scheme, which is the fundamental approxima- tion for all the others in the upwind approach. The term “upwind” refers to the incorporation of the characteristic information of the solution in the numerical discretizations. Here, we note that another approach based on central discretizations stands alone as an active research field till the present. Central schemes stem from the first-order Lax-Friedrichs and was pioneered in the paper of Nessyahu and Tadmor ([106]) for hyperbolic conservation laws on a staggered grid.

We refer to, e.g., [10], [94], or [117] for high-order central schemes. We then proceed to the

treatments on the coefficients C and D in definition

2.3.1

so that the accuracy is improved to

second order, except at critical points as indicated in proposition

2.3.1. Here, the essential idea

is to design such limiters that no oscillations occur in the numerical approximations. Typical

limiters will also be listed. We limit the discussion to only the case of scalar conservation laws.

Since v

L

, v

R

are constants, problem (2.3.10) can be solved exactly as follows, assuming that f

^′′

(˜ v)

≥

0, and that there are no interactions of waves, i.e., we limit their distance of propagation at most half of the interval,

max

v˜ |

f

^′

(˜ v)

|

∆t

≤

1

2 ∆x. (2.3.11)

Eq. (2.3.11) is often referred as the CFL condition.

•

Case 1: v

_L

> v

_R

(shock-wave)

˜

v(x, t) =

(

v

_L

, x/t < s,

v

_R

, x/t > s, (2.3.12)

where

s = f (v

R

)

−

f (v

L

)

v

_R−

v

_L

, (2.3.13)

is the propagation speed of the shock by the Rankine-Hugoniot condition.

•

Case 2: v

L≤

v

R

(rarefaction-wave)

˜

v(x, t) =







v

_L

, x/t < f

^′

(v

_L

),

f

⁻¹

(x/t), f

^′

(v

_L

)

≤

x/t

≤

f

^′

(v

_R

), v

_R

, x/t > f

^′

(v

_R

).

(2.3.14)

Thanks to condition (2.3.11), there are no interactions of waves, thus ˜ v(x

_j+¹

2

, t) remains constant for all t

_n ≤

t < t

_n+1

. The Godonov’s flux is then correspondingly written compactly as below (see [109]),

f ˆ

^G

(¯ v

_jⁿ

, ¯ v

ⁿ_j+1

) = 1

∆t

Z tn+1

tn

f (˜ v(x

_j+¹

2

, t))dt = f (˜ v(x

_j+¹

2

, t))

=

(

min

_v¯ⁿ

j≤˜v≤¯v_j+1ⁿ

f (˜ v), if ¯ v

_jⁿ≤

¯ v

ⁿ_j+1

, max

_¯vⁿ_j+1≤v˜≤¯vⁿ_j

f (˜ v), if ¯ v

_jⁿ

> v ¯

ⁿ_j+1

.

(2.3.15)

We check that Godonov’s scheme is conservative, TVD, and satisfies the discrete entropy condition (2.2.35).

Proposition 2.3.2. (Godonov’s scheme) The Godonov scheme with the flux defined in Eq.

(2.3.15) is conservative, TVD, and converges to the unique physical weak solution.

Proof. i. (Conservativeness) It suffices to show that the Godonov flux (2.3.15) is consistent

and Lipschitz continuous. The former property follows directly from the definition of the

flux. For the latter, since f (u) is

C²

, by the mean value theorem there exists a constant L such that, for any u

₁

, u

₂

we have that

|

f (u

₁

)

−

f (u

₂

)

| ≤

L

|

u

₁−

u

₂|

. (2.3.16)

Suppose that ¯ v

_jⁿ≤

v ¯

_j+1ⁿ

, then

f ˆ

^G

(¯ v

_jⁿ

, ¯ v

ⁿ_j+1

) = min

¯

vⁿ_j≤˜v≤¯vⁿ_j+1

f (˜ v) =: f (ˆ v), (2.3.17) for some ˆ v

∈

[¯ v

ⁿ_j

, v ¯

_j+1ⁿ

].

Thus,

|

f ˆ

^G

(¯ v

ⁿ_j

, v ¯

ⁿ_j+1

)

−

f (u)

|

=

|

f (ˆ v)

−

f (u))

| ≤

L

1|

v ˆ

−

u

|

, (2.3.18) for sufficiently small

|

v ˆ

−

u

|

. Similarly, for ¯ v

ⁿ_j

> ¯ v

_j+1ⁿ

, there exists L

₂

such that Eq. (2.3.18) holds. Choosing L = max(L

₁

, L

₂

) proves relation (2.3.16).

ii. (TVD’ness) Comparing the conservative form (2.2.7) and the TVD one (2.3.4), we deduce that

∆x

∆t C =

−

f ˆ

_j+^G 1

2 −

f (¯ v

ⁿ_j

)

¯

v

ⁿ_j+1−

¯ v

ⁿ_j ≥

0, ∆x

∆t D =

−

f ˆ

_j−^G 1

2 −

f (¯ v

_jⁿ

)

¯

v

_jⁿ−

v ¯

_j−1ⁿ ≥

0, (2.3.19) by the definition (2.3.15) of the Godonov flux (2.3.15). We next check that

C + D = ∆t

∆x

f (¯ v

ⁿ_j

)

−

f ˆ

_j+^G 1 2

¯

v

_j+1ⁿ −

¯ v

ⁿ_j

+

f (¯ v

_jⁿ

)

−

f ˆ

_j−^G1 2

¯

v

ⁿ_j −

¯ v

ⁿ_j−1

≤

1, (2.3.20)

thanks to the CFL condition (2.3.11). Hence, Godonov’s scheme is TVD.

iii. (Entropy condition) Since Godonov’s scheme solves the Riemann problem (2.3.10) exactly, the following entropy condition holds for ˜ v,

∂

∂t U (˜ v(x, t)) + ∂

∂t F (˜ v(x, t))

≤

0, (2.3.21)

where (U, F ) is an entropy pair.

Integrating (2.3.21) over Ω := I

j×

[t

n

, t

n+1

), we obtain that 1

∆x

Z x_{j+ 1}

2

x_j−1 2

U (˜ v(x, t

_n+1

))dx

−

1

∆x

Z x_{j+ 1}

2

x_j−1 2

U (˜ v(x, t

_n

))dx + 1

∆x

Z tn+1

tn

F (˜ v(x

_j+¹

2

, τ ))dτ

−

1

∆x

Z tn+1

tn

F (˜ v(x

_j−¹

2

, τ ))dτ

≤

0.

(2.3.22)

Thanks to condition (2.3.11), ˜ v is constant along its characteristic, i.e., for all (x, t)

∈

Ω,







˜

v(x, t

n

) = ¯ v

ⁿ_j

,

˜ v(x

_j±¹

2

, t) = ˜ v(x

_j±¹

2

, t

_n

),

(2.3.23)

we deduce that 1

∆x

Z _x

j+ 12

x_j−1 2

U (˜ v(x, t

_n+1

))dx

≤

U (¯ v

_jⁿ

)

−

σ(F (˜ v(x

_j+¹

2

, t

_n

))

−

F (˜ v(x

_j−¹

2

, t

_n

))). (2.3.24) Since U (˜ v(x, t)) is convex, by Jensen’s inequality, we have that

U (¯ v

_jⁿ⁺¹

) = U

1

∆x

Z x_{j+ 1}

2

x_j−1 2

˜

v(x, t

_n+1

)dx

≤

1

∆x

Z x_{j+ 1}

2

x_j−1 2

U (˜ v(x, t

_n+1

))dx; (2.3.25)

hence,

U (¯ v

ⁿ⁺¹_j

)

≤

U (¯ v

_jⁿ

)

−

σ(F (˜ v(x

_j+¹

2

, t

_n

))

−

F (˜ v(x

_j−¹

2

, t

_n

)), (2.3.26) which is the discrete entropy condition (2.2.35). The uniqueness then follows immediately.

2.3.1.2 Other Schemes

The Godonov scheme, due to the exact solving of a Riemann problem in order to approximate

¯

v

∆x

(x

_j+¹

2

, t

n

), is costly and complicated, especially when dealing with a system case. For the the latter, an iterative solver must be facilitated. For example, see [138] for the solver of Riemann problems for the Euler system. The below schemes, except for the Lax-Friedrichs, are derivations of the Godonov one where ¯ v

_∆x

(x

_j+¹

2

, t

_n

) is approximated from the Riemann

problem (2.3.10).

•

Lax-Friedrichs’ scheme:

f ˆ

^LF

(¯ v

_jⁿ

, v ¯

_j+1ⁿ

) = 1

2 (¯ v

_jⁿ

+ ¯ v

_j+1ⁿ

)

−

α(¯ v

_j+1ⁿ −

v ¯

_jⁿ

), (2.3.27) where α = max

_j|

f

^′

(¯ v

_jⁿ

)

|

.

•

Engquist-Osher’s scheme: (see [28]) f ˆ

^EO

(¯ v

_jⁿ

, ¯ v

ⁿ_j+1

) =

Z ¯vⁿ_j+1 0

min(f

^′

(s), 0)ds +

Z ¯vⁿ_j

0

max(f

^′

(s), 0)ds + f (0). (2.3.28)

•

Roe’s scheme: ([118])

f ˆ

^R

(¯ v

ⁿ_j

, v ¯

ⁿ_j+1

) =

(

f(¯ v

ⁿ_j

), if s > 0,

f(¯ v

ⁿ_j+1

), if s < 0, (2.3.29) where

s = f (¯ v

_j+1ⁿ

)

−

f (¯ v

_jⁿ

)

¯

v

_j+1ⁿ −

v ¯

_jⁿ

. (2.3.30)

Remark 2.3.2.

i. The Godonov flux (2.3.15) is also applicable for a non-convex flux function f (u). (See [109].)

ii. The Engquist-Osher flux and Roe flux are approximations of the Godonov one in a sense of using only rarefaction-waves or shock-waves, respectively. (See [147], [128], [111], [56], etc.)

iii. Roe’s scheme is especially efficient for a system case, e.g., the Euler equations ([118]), thanks to its simplicity. A tradeoff is that the scheme does admit nonphysical weak solutions, i.e., expansion shock-waves (see [55]), due to the use of shock-waves only in defining the flux. A remedy for this drawback is called an entropy fix, for example (see [92]),

f ˆ

^RF

(¯ v

ⁿ_j

, v ¯

ⁿ_j+1

) =















f (¯ v

_jⁿ

), if s

≥

0, f (¯ v

_jⁿ⁺¹

), if s

≤

0, f ˆ

^LF

(¯ v

_jⁿ

, v ¯

_j+1ⁿ

),

(2.3.31)

where min(¯ v

ⁿ_j

, v ¯

ⁿ_j+1

)

≤

v ˜

≤

max(¯ v

_jⁿ

, v ¯

_j+1ⁿ

).

iv. Except for the Roe flux, the other ones are called monotone fluxes, i.e., for all j’s,

∂ f ˆ

∂¯ v

_jⁿ ≥

0, ∂ f ˆ

∂ v ¯

ⁿ_j+1 ≤

0, (2.3.32)

and are consistent and Lipschitz continuous; thus satisfy the discrete entropy condition (2.2.35). See [21] for detailed discussions on monotone fluxes and schemes. See also [109]

for E schemes whose fluxes satisfy the entropy condition.

v. It is worthy to mention the Godonov theorem ([37]) which states that all monotone

schemes are at most first-order. This implies all linear schemes applying to conserva-

tion laws are also at most first-order (see [46]). Hence, in order to improve the accuracy,

one must design schemes whose coefficients depend on the solution themselves. This leads

to the concept of “limiters” which play roles as an adapting mechanism of high- and low-

order fluxes, or a controller of the slope of stencil-wise approximating polynomials so that

condition (2.3.5) holds, i.e., no oscillations occur in the approximation. We discuss these

approaches in the below section.