Derivatives of Systems in Divergence form in the Presence of Shocks

Derivatives of Systems in Divergence form in the Presence of Shocks

Claude Bardos and Olivier Pironneau^∗ September 9, 2002

Abstract

In this paper we present a synthetic method to differentiate with respect to a parameter partial differential equations in divergence form with shocks.

We show that the usual derivatives contain the differentiated interface condi- tions if interpreted by the theory of distributions. We apply the method to 3 problems: the Burger equation, Darcy’s law and the shallow water equations.

Calcul de la d´eriv´ee d’un syst`eme sous forme divergencielle avec chocs

R´esum´eOn pr´esente une m´ethode synth´etique pour calculer les ´equations aux d´eriv´ees partielles que v´erifient la d´eriv´ee par rapport `a un param`etre de la solu- tion d’un syst`eme sous forme∇ ·v= 0. On montre, pour les ´equations de Burger, Darcy et Saint-Venant que la d´eriv´ee au sens usuel, mais interpret´ee au sens des distributions, contient les conditions de saut, c’est `a dire les d´eriv´ees des conditions de transmission aux chocs. On retrouve ainsi les r ´esultats de Godlewski-Raviart et al.

Rapport de recherche R002??

∗Laboratoire Jacques-Louis Lions, Universit´e Paris VI - IUF ([email protected])

1 Introduction

Sensitivity and stability analyses lead to the differentiation of partial differential equations. For instance, if the solution of Burger’s equation, u, depends on a pa- rametera, then, one would like to write that

∂tu+∂x

u²

2 = 0 yields∂tu⁰+∂x(uu⁰) = 0 (1) where u⁰ denotes the derivative of u with respect to a. We will show that it is correct, but in the sense of distribution theory. Indeed in the usual sense there are serious problems when there are shocks which depend upona. This is becauseu has a jump at the shock, so its derivativeu⁰ is likely to have a Dirac mass at the shock and souu⁰is void of sense whenuis discontinuous.

This problem has been identified in a pionnering paper by Alazabal, Godlewski and Raviart[8] for (1) and also Hafez (see [4]) who pointed out that usual optimality conditions for control problems with discontinuous states would be incomplete since no variation of shock positions are allowed by standard calculus of variations;

Giles[5] proved later that it is not so if the criteria of the optimization problem does not involve the shock position explicitly. From the numerical point of view, it was also found that computer program implementing (1) compute Dirac masses indeed [14][4].

For Burger’s equation and to some extend for the Euler equations of fluids the prob- lem has been analyzed completely by Godlewski et al[10][9][14] and for Darcy’s law by Bernardi et al [3]. So what we propose here is only a new interpretation of the results; however this is done with a general tool to compute the divergence of distributions which have singularities tangent to the lines of singularities (Corol- lary 1).

The plan of the paper is as follows:

First we recall that a “simple” differentiation of the partial differential equation and of its interface conditions lead to the result. However such results would be difficult to justify mathematically; also they do not appear in a synthetic and compact form.

In the next session we show that by using the definition of derivatives in the sense of distributions, we can differentiate a functionvwhich has a line/surface discon- tinuity of normal nand we can also compute the divergence of the result when v·n= 0.

Finally in the third part we apply the result to 3 problems: Burger’s equation, Darcy’s law and the shallow water equations. We recover the results of [8] and [3] and show among other thing that the differentiated equations contain the jump conditions necessary for the well posedness of the differentiated problem.

2 Standard Analysis

2.1 Standard Sensitivity Analysis for Burger’s equation

Burger’s equation foruis

∂_tu+∂_x(u²

2 ) = 0 ∀{x, t} ∈Q:=R×(0,+∞),

u(x,0) =u⁰(x, a) ∀x∈R (2)

It may have discontinuous solutions (shocks) depending on the initial datau⁰. With compatible (entropy) discontinuous initial data at, say x = 0, the solution has a discontinuity atx=x(t, a), which propagates at velocity

˙

x:=∂tx= ¯u:= 1

2(u⁺(x(t, a), t, a) +u⁻(x(t, a), t, a)). (3) Here we assume that the initial data is function of a parameteraand we wish to find an equation foru⁰_a, the derivative ofuwith respect toa. We also assume that the shock is a unique curvex(t, a)defined att= 0by the discontinuity ofu⁰ at0.

Proposition 1 Burger’s equation differentiated with respect to a parameter,a, in the initial data is

∂tu⁰_a+∂x(uu⁰_a) = 0 inΩ\S

˙

x⁰= ¯u⁰_a+x⁰∂_xu¯ onS (4) wherex⁰ is the derivative with respect toaof the shock positionx(t),u¯⁰ = (u⁰⁺+ u^{0 −})/2 and∂_xu¯is the half sum of the left x of the rightx derivative ofuon the shock curve.

Proof:

Equation (4-a) is straightforward. To obtain (4-b) we must differentiate (3) with respect toa. All primes meaning derivatives with respect toa, (3) comes from the fact that∂af(x(t, a), t, a) =f_a⁰ +x⁰∂xf.

The fact that the original solution u(x, a) satisfies the entropy condition has for consequence thatu⁰_aand ( in most cases)x⁰are solutions of a well posed problem.

This is the object of the proposition (see Tai-Ping-Liu for instance [12]):

Proposition 2 Letu(x, t, a) be the entropic solution of the Burger equation with initial datau₀(x, a). Assume that it has a “simple” structure: there is a closed set Swhich is the union of a finite number of shocks curves and away from this setu

is continuously differentiable inxandt. Then the solution of (4-a) is uniquely and continuously determined in term of its initial value

u⁰_a(x, a,0) =v(x) := ∂u⁰

∂a (x, a) (5)

If in additionSis the union of a finite set of distincts (no intersection) shock curves xi(t, a) with origin at t = 0: xi(0, a); then the derivatives with respect to a,

∂x_i(t, a)/∂a depends continuously on v(x) and ∂x_i(0, a)/∂a according to the formula:

x⁰_i(t) =e^R

t

0∂xu(x¯ i(s),s)dsx⁰_i(0) + Z t

0

e^R

t

s ∂xu(x¯ i(τ),τ)dτu¯⁰_a(xi(s), s)ds (6) Proof

For any(x, t)∈/ Sthe solution of the ode dx(s)

ds =u(x(s), s), x(t) =x (7) is well defined as long as it does not intersectS. Sinceusatisfies also the entropy condition there are no characteristic coming from the shocks curves thereforex(s) does intersect S for 0 ≤ s ≤ t. Along this curve, equation (4-a) implies the relation:

du⁰_a

ds (x(s), s) +∂xu(x(s), s)(u⁰_a(x(s), s) = 0 (8) or after integration from0tot:

u⁰_a(x, t) =u⁰_a(x(0),0)e⁻ R^t

0∂xu(x(s),s)ds

(9) which established the first part of the proposition.

Consider a shock curve xi(t, a). Having determined u¯⁰_a(xi(t), t, a) one obtains x_i(t, a), from the relation

d dt

∂xi

∂a(t, a)

!

= ¯u⁰_a(xi(t), t, a) + ∂xi

∂a(t, a)∂xu(x¯ i(t), t, a) (10) Then one uses the hypothesis that the shock curves are distincts and that they starts fromxi(0, a)att= 0i.e. from points where the initial datau(x,0)is discontinu- ous.

With an integration of (10) from0totone obtains (6).

2.2 Standard Sensitivity Analysis for Darcy’s Law For hydrostatic flows through porous media,

v=κ∇φ and ∇ ·v = 0 inΩ⊂R^d (11) Asκ is discontinuous from one geological layer to the next let S be such an in- terface. Whenκis in L^∞(Ω) φthen is inH¹(Ω)and continuous inΩwhenκis piecewiseC¹, even if it is discontinuous across a surfaceS.

Proposition 3 Letφbe the unique solution of (11) and assume that κhas a dis- continuity,S(a), function of a parameterathen the derivative ofφ,φ⁰with respect toa, is discontinuous acrossS(a) ={x(s) : s∈I_S ⊂R^d−1}and satisifies

∇ ·(κ∇φ⁰) =−∇ ·(κ⁰∇φ) in Ω\S(a) [φ⁰] =−x⁰·[∇φ] onS

[κ∂φ⁰

∂n] =−x⁰· ∇[κ∂φ

∂n]−[κ⁰ κ](κ∂φ

∂n)−[κ∇φ]·n⁰ onS (12) where all primes indicate a derivative with respect toaand nis the normal toS oriented from minus to plus while the jump bracket is[φ⁰] =φ⁰⁺−φ⁰⁻.

Proof

From (11), (12-a) is obvious. Equation (12-b) comes from the differentiation ina of the condition[φ]S = 0. Equation (12-c) is obtained by differentiating the jump condition[κ∂_nφ]=0:

[κ∂φ

∂n]⁰+x⁰· ∇[κ∂φ

∂n] = 0 (13)

The first term is

[κ⁰∂φ

∂n] + [κ∂φ⁰

∂n] + [κ∇φ·n⁰] (14) and the whole can be rewritten as (12-b).

Remark 1 This derivation is somewhat formal and difficult to justify because a lot of regularity is required formφ. Nevertheless intuitively it should be clear that there are enough conditions to have a uniqueφ⁰.

3 Derivatives in the Sense of Distributions

We proceed now to rederive and reinterpret these two results above in the light of Distribution Theory.

3.1 Preliminaries

LetΩbe an open bounded set ofR^d and Aan interval of R. Consider a vector valued function fromΩtoR^dfunction of a parametera∈A:

(x, a)∈Ω×A→v(x, a)∈R^d (15) LetS(a)is a smooth surface which cutsΩinto two partsΩ⁺,Ω⁻. Assume that the restrictions ofv,v^± toΩ^±are continuously differentiable. We denote byC_S¹ the space of such functions.

Assume also thatvis differentiable with respect toaeverywhere except onS(a) and denote byv⁰_athis pointwise derivative.

We introduce the following notations:

• x=x(s, a),s∈R^d−1, is a parametric representation ofS(a)

• x⁰its partial derivative with respect toa.

• Q= Ω×A,Σ ={(x, a) : x∈S(a), a∈A}

• Q^±={(x, a) : x∈Ω^±(a), a∈A}

• n, n_Σ: the normals toS(a)andΣpointing insideΩ⁺andQ⁺.

• [v] :=v⁺−v−is the jump ofvacrossS(a).

Proposition 4 Letvbe a function inC_S¹. The derivative ofvwith respect toa, in the sense of distributionv⁰, is

v⁰ =v⁰_a−[v]x⁰·n δS (16) where v_a⁰ is the pointwise derivative of v with respect to a and δS is the Dirac function onSdefined by^R_Ωwδ_S =^R_Sw∀w∈ D(Ω).

Proof For clarity we do the proof ford = 3only. By definition of derivatives in the sense of distributions,

∀w∈ D(Q) : Z

Q

v⁰w=− Z

Q

vw⁰ (17)

Let us make use of the fact thatv^±:=v|Ω^±are smooth:

Z

Q

vw⁰ = Z

Q⁻

v⁻w⁰+ Z

Q⁺

v⁺w⁰

=− Z

Q⁻

v^{− 0}w− Z

Q⁺

v⁺⁰w+ Z

Σ

(v⁺−v⁻)n_Σd+1w (18) where n_Σd+1 is the last component of the normal to Σ. If Σ has for equation σ(x, a) = 0, then a normal is(∂xσ, ∂aσ)^T. Now ifx = x(s, a)is an parametric representation ofS(a)thenσ(x(s, a), a) = 0for allaso(∂_xσ)x⁰+∂_aσ= 0. This shows that the normal toσis also(n,−x⁰·n)because∂xσisn, a normal toS.

Corollary 1

(∇φ)⁰ =∇φ⁰_a−δS[∂φ

∂n]x⁰·n (19)

Proposition 5 Letvbe tangent toS(a), then

∇ ·(vδS) =δS∇^S·v (20) where∇^S·is the surface divergence onS.

Proof

Z

Ω

vδ_S· ∇w= Z

S

v· ∇w

=

d−1

X

i=1

Z

S

v·sⁱ∂_sⁱw=−

d−1

X

i=1

Z

S

∂_sⁱv·sⁱw (21) Consequently from (16) we have the following property.

Corollary 2 If[v·n] = 0and the pointwise derivative ina,v_a⁰ has a trace left and right ofS,

∇ ·v⁰=∇ ·v_a⁰ + [v⁰_a·n]δS−δS∇^S·([v]x⁰·n) (22) Remark 2 Notice that v⁰_a being discontinuous acrossS, its divergence contains also a Dirac mass and∇ ·v_a⁰ + [v_a⁰ ·n]δ_Sis∇ ·v⁰_ain the distribution sense.

Symbolic Notation

Letf, gbe functions ofC_S¹, differentiable with respect toainΩ^±. Let f¯=f^±inΩ^±and (f⁺+f⁻)

2 onS(a)and similarly forg. (23) Then

(f g)⁰=f⁰¯g+ ¯f g⁰ (24)

Justification

Derivatives of f g, f and g may have Dirac masses on S(a), but by the identity [f g] = ¯f[g] + [f]¯g, we have equality of the Dirac masses (see Proposition 4).

WithinΩ^±equation (24) is obviously true.

Remark 3 Such definition is of course symbolic because it makes no sense to de- fine a function on a set of measure zero, but it allows to go further in a symbolic calculus of derivatives for products as we shall see below for Darcy’s law and for the shallow water equations.

Furthermore note that we do not provide a rule to compute(gh)¯ ⁰ for instance, ex- cept in the case of Corollary 2, which makes it difficult to apply the above formula recursively to compute triple products such as(f gh)⁰. As is well known products of distributions are dangerous objects.

3.2 Sensitivity Analysis for Burger’s equation (II)

Letv= (v₁, v₂)^T, v₁ =u, v₂=u²/2. Callingx₁ =t, x₂=x, we see that (2) is

∇ ·v= 0.

Furthermorevis smooth except across the shockS= (t, x(t)), andn= (−x,˙ 1)^T/√ 1 + ˙x²; Proposition 4 says that

u⁰ =u⁰_a−[u] x⁰

√1 + ˙x²δS (u²

2 )⁰ =uu⁰_a−[u² 2 ] x⁰

√1 + ˙x²δS (25) Similarly Corollary 2 says that

0 =∇ ·v⁰ =∇ ·v⁰_a+ [v⁰]·nδS−dt([v·s]x⁰)δS

=∂tu⁰_a+∂x(uu⁰_a)−[u⁰_a] ˙x+ [uu⁰_a]−dt(x⁰[u] + [^u₂²] ˙x

1 + ˙x² )δS (26) where∂t and ∂x are classical derivatives ands = (1,x)˙ ^T/√

1 + ˙x². Recall that

˙

x= ¯uand that[uu⁰_a] = ¯u[u⁰_a] +baru⁰_a[u]. We have introduced the notation dtf := ∂f

∂t(x(t), t, a) =∂tf+ ˙x∂xf for the time derivative onx(t)(27) to emphasize the difference with∂_tf. As[u] + [^u₂²] ˙x= [u](1 + ˙x²)we have:

Proposition 6 Burger’s equation differentiated with respect to a parameter ain the initial data is

∂tu⁰_a+∂x(uu⁰_a) = 0 inΩ\S

−[u]¯u⁰_a+dt([u]x⁰) = 0 onS (28) withu¯⁰_a= (u⁺⁰_a+u⁻⁰_a)/2.

Proposition 7 Equation (28-b) is the derivative of the Rankine-Hugoniot condi- tions with respect to the parameter in the initial data.

Proof The Rankine-Hugoniot condition is

˙

x= ¯u|x=xS(t,a).Differentiated inait is: x˙⁰ = ¯u⁰_a+x⁰∂xu.¯ (29) Recall the identities

∂x(¯u[u]) = ¯u∂x[u] + [u]∂xu¯=∂x[u²

2 ] =−∂t[u] =−dt[u] + ¯u∂x[u](30) Multiplied by[u], (29) gives

[u] ˙x⁰ = [u]¯u⁰_a+ [u]x⁰∂xu¯= [u]¯u⁰_a−x⁰dt[u]

⇒ dt([u]x⁰) = [u]¯u⁰_a (31)

which is (4-b).

Proposition 8 Equation (4-a) read in the sense of Distribution theory contains (4-b).

Proof

As explained above,

∂tu⁰+∂x(uu⁰) = 0 is ∇ · u⁰

uu⁰

= 0 and also∇ · u

u² 2

⁰

= 0(32) Now by Proposition 2

0 =∇ · u

u² 2

⁰

=∇ · u⁰

uu⁰

+ ([u⁰] ˙x−[uu⁰]−dt([u]x⁰))δS (33) which is also

0 =∂tu⁰+∂x(uu⁰)−([u]¯u⁰+dt([u]x⁰))δS (34) So we must have (4-a) inΩ\Sand by density this shows that (4-b) holds onS.

Remark 4 Uniform BV estimates for nonlinear hyperbolic systems in one space variable with convenient hypotheses show that for any continuous test function with compact supportϕ,

δalim→0

Z

Q

u(x, t, a+δa)−u(x, t, a)

δa ϕ=

Z

Q

u⁰ϕ (35)

Numerical Consequences Finite volume methods are based on the weak form of the Burger equation and a finite difference approximation of the time deriva- tive. From the above we learn that a space-time approximation such as used by Johnson[2] would be preferable because it would be potentially capable of han- dling the singularities ofu⁰. A formulation based on

Z

Q

u⁰∂tw+uu⁰∂xw= 0 (36) would contain the differentiated Rankine-Hugoniot condition. Stillu⁰has a Dirac singularity onS(a) and special numerical care must be applied to handle it, such as explicitly writing the Dirac masses hidden in (36)

Z

Q

u⁰∂_tw+uu⁰∂_xw− Z T

0

[u]x⁰dw

dt (x(t), t) = 0 (37) This is an immediate consequence of Proposition 4. More generally:

Corollary 3 LetuvbeC_S¹ functions of(x, t)∈Q:= Ω×(0, T); letu⁰, v⁰be their derivative with respect toain the sense of distribution. Assume that, in the sense of distribution,

∂_tu+∂_xv= 0, ∂_tu⁰+∂_xv⁰= 0. (38) Then

Z

Q

(u⁰_a∂_tw+v_a⁰∂_xw)− Z T

0

[u]x⁰dw

dt(x(t), t) = 0 (39) Proof: By definition^R_Q(u⁰∂tw+v⁰∂xw= 0, therefore, by Proposition 4

Z

Q

(u⁰_a∂tw+v⁰_a∂xw− Z

Σ

[u](∂tw+ [v]∂xw)x⁰·n= 0. (40) Now by (41-a)[v] = [u] ˙xandx⁰·ndσ=x⁰dt.

3.3 Sensitivity Analysis for Darcy’s Law (II)

Proposition 9 Ifφsatisfies∇ ·(κ∇φ) = 0andκis discontinuous onS(a) then the derivative ofφ⁰with respect toais discontinuous acrossS(a)and satisifies

∇ ·(¯κ∇φ⁰+κ⁰∇φ) = 0 (41) withκ⁰ =κ⁰_a−δS[κ]x⁰·nwherex=x(s, a)is the equation ofS. The meaning of

∇ ·([κ]x⁰·n∇φ⁰δS)is given by Corollary 2.

Proof

By Proposition 4 we see that, withv=κ∇φ,

v⁰ = (κ∇φ)⁰_a−[κ∇φ·n]x⁰δS= (κ∇φ)⁰_a (42) because the normal composent ofκ∇φdoes not jump acrossS.

With the symbolic notations ( 24),

v⁰ = (κ∇φ)⁰=κ⁰∇φ+ ¯κ∇φ⁰ (43) which, by Proposition 4 and Corollary 1 is

v⁰ =κ⁰_a∇φ+κ∇φ⁰_a−δ_Sx⁰·n([κ] ¯∇φ·n+ ¯κ[∇φ]·n)

=κ⁰_a∇φ+κ∇φ⁰_a−δSx⁰·n[κ∇φ] (44) but the last term is zero. By Corollary 2:

0 =∇ ·v⁰ =∇ ·v⁰_a−∂_s([κ∂_sφ]x⁰) (45) which can be summed up as

∇ ·(¯κ∇φ⁰) =−∇ ·(κ⁰∇φ) (46) Ifκis piecewise constant, thenκ⁰_a = 0and[φ] = 0 ⇒ [∂sφ] = 0⇒ [κ∂sφ] = 0 therefore∇ ·v⁰ = 0andv⁰ =κ∇φ⁰. However it is not a standard problem because the solution is in L²(Ω) only and it jumps across S of a known quantity. The problem can be written in mixed form with the usual understanding of derivatives:

v⁰ ∈H(div,Ω), φ⁰∈L²(Ω) : ∇·v⁰ = 0, ∇φ⁰−1

κv⁰ = [1

κ]v·nx⁰·nδ_S. (47) So we have the following

Corollary 4 Equation (41) contains the derivative with respect toaof the trans- mission conditions onS(a): _da^d([φ]_x=x(a)) = 0and _da^d ([κ^∂φ_∂n]_x=x(a)) = 0

To illustrate the usefulness of the symbolic notation introduced with (24) let us show that (47) can be derived directly.

From the definition ofv,v=κ∇φ, whenκis piecewise constant on both sides of S(a), we find that

v⁰ = ¯κ∇φ⁰+κ⁰∇φ= ¯κ∇φ⁰−[κ]∇φx⁰δS (48) Now

0 = [κ∇φ] = [κ]∇φ+ ¯κ[∇φ] (49)

and

[∇φ] = [κ∇φ κ ] = (1

κ)[κ∇φ] + [1

κ]κ∇φ= [1

κ]κ∇φ (50) so,

[κ]∇φ= [1

κ]¯κκ∇φ (51)

and (47-b) is recovered.

Remark 5 An equation like (48) is hard to read; with the foreknowledge thatv⁰ does not have a Dirac mass it means thatφ⁰has one and its weight is given by the last term. Then, outsideS(a)it means thatv⁰ =κ∇φ⁰, and so if these have traces right and left ofS(a)then it contains also the equation[κ∇φ⁰·n] = [v⁰].

3.4 The Shallow Water Equations InR^d, d=1,2 or 3, consider

∂tρ+∇ ·(ρu) = 0

∂t(ρu) +∇ ·(ρu⊗u) +∇ρ= 0. (52) With appropriate initial and boundary conditions, the water heightρand its velocity uare uniquely defined, at least whend= 1(see Lions[1]).

Across shocksSthe Rankine-Hugoniot conditions are

[ρ] ˙x+ [ρu]·n= 0 [ρu] ˙x+ [ρu⊗u·n] + [ρ]n= 0 (53) Consider the mono dimensional case only,d= 1.

Differentiating (52-53) is easier to do in terms of the flux velocityv = ρu. For shocks normal to the flowu, letνbe the norm ofv:

∂_tρ⁰+∂_xv⁰ = 0

∂tv⁰+∂x(ρ⁻¹(v⊗v⁰+v⁰⊗v)−ρ⁰ρ⁻²v⊗v) +∂xρ⁰ = 0 [ρ⁰] ˙x+ [ρ] ˙x⁰+ [ν⁰] + ˙xx⁰·∂x[ρ] +x⁰·∂x[ν] = 0

[ν⁰] ˙x+ [ν] ˙x⁰+ ([ρ⁻¹ν²+ρ])⁰+ ˙xx⁰∂_xν

+x⁰∂x(ρ⁻¹ν²+ρ) = 0 (54)

Alternatively, in the variable (t,x), equation (52-a) is a divergence ofV = (ρ, v)^T and so the same result can be obtained in the sense of Distribution theory by Corol- lary 2. However in order to differentiate the products we need to write the equations as:

∂tρ+∂xv= 0 ∂tv+∂x(vu) +∂xρ= 0 v=ρu (55)

Then its derivative is

∂tρ⁰+∂xv⁰ = 0

∂tv⁰+∂x(v⁰u¯+ ¯vu⁰+ρ⁰) = 0

v⁰=ρ⁰u¯+ ¯ρu⁰ (56)

Interpretation As for Burger’s equation (56) contains the standard derivative of (52) and 3 jump conditions involving[u],[v],[ρ], x⁰.

System (52) is of the form

∂tW +∂xF(W) = 0 (57)

withW = (ρ, ρu)^T, therefore the linearized equation forV and the differentiated equations have the same jacobian matrixAij =∂WiF(W)j, i, j= 1,2. The anal- ysis of shocks is based on the computation of eigenvalues and eigenvectors ofA.

For One-Shocks for the shallow water system one Riemann invariant is computed from characteristics on the right side of the shock with a transmission condition for the left side one while the other invariant is computed with left and right charac- teristics independently on both sides of the shock, without transmission condition across the shock while . This means that the jump of this second invariant is fixed by (56) understood in the classical sense. So we have the 3 jump conditions yielded by (56) in the distribution sense for[u],[v],[ρ], x⁰but with one more equation dues to the knowledge of the jump of the second Riemann invariant.

Remark 6 If we had in the momentum equationρ^γinstead ofρ, the same analysis applies. The momentum equation is the divergence of(v, vu+ρ^γ)^T so the terms which appears in the jump condition is[v] ˙x+ [vu+ρ^γ] and it is also the jump found when

∂tv⁰+∂x(u⁰v¯+ ¯uv⁰+a⁰) = 0

but we must findasuch thata[rho] =¯ ρ^γ. Whenγ = 3/2(air) it can be done as follows:

b²=ρ, a=bρ ⇒ 2¯bb⁰ =ρ⁰, a⁰ = ¯bρ⁰+b⁰ρ¯

⇒ a⁰ =√ρρ⁰+ ¯ρ ρ⁰

2√ρ (58)

4 Euler’s Equations

Perfect inviscid fluids are governed also by (52) butp is not a function ofρ and there is an additional equation for the conservation of energy:

∂_tθ+∂_x(v(u²

2 +θ)) = 0 withθ= γ γ−1

p

ρ (59)

To decompose all products into binary multiplications we multiply byρthe second equation and setw=u²/2. Differentiation leads to

∂_tθ⁰+∂_x(v⁰w¯+ ¯vw⁰+θ⁰) = 0 γ

γ−1p⁰ = ¯θρ⁰+ ¯ρθ⁰ w⁰ = ¯uu(60)⁰ Therefore the result is

∂tρ⁰+∂x(¯ρu⁰+ ¯uρ⁰) = 0

∂t(¯ρu⁰+ ¯uρ⁰) +∂x(¯u²ρ⁰+ (¯ρu¯+ρu)u⁰+p⁰) = 0 γ

γ−1∂_t(p⁰

¯ ρ −(p

ρ)ρ⁰) +∂_x( γ γ−1((p⁰

¯ ρ −(p

ρ)ρ⁰)ρu+ (p

ρ)(¯ρu⁰+ ¯uρ⁰)) +(¯uρu+u²ρ¯

2 )u⁰+u²u¯

2 ρ⁰) = 0

5 Conclusion

We have seen that equations in divergence form can be differentiated even in the presence of shocks. The differentiated equation, taken in the sense of distribu- tion, contains the transmission condition across the shock which fixes its motion.

Therefore integration by parts of the differentiated equation is valid and numer- ical methods based on space time finite volume or finite element method should work. Other equations can be analyzed also by this method, such as the transonic equation [11] [6]. This will be done in a forthcoming publication.

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