Consistency analysis of a 1D Finite Volume scheme for barotropic Euler models

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1 Consistency analysis of a 1D Finite Volume scheme for barotropic Euler models F Berthelin 1,3, T Goudon 1,3, and S Mineaud,3 1 Inria, Sophia Antipolis Méditerranée Research Centre, Proect COFFEE Inria, Sophia Antipolis Méditerranée Research Centre, Proect CASTOR 3 Univ Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, France Abstract This wor is concerned with the consistency study of a 1D (staggered inetic) finite volume scheme for barotropic Euler models We prove a Lax-Wendroff-lie statement: the limit of a converging (and uniformly bounded) sequence of stepwise constant functions defined from the scheme is a wea entropic-solution of the system of conservation laws 1 Introduction The model This wor is concerned with the consistency study of a (staggered inetic) Finite Volume (FV) scheme for barotropic Euler models t ρ + x (ρv ) =, (1) t (ρv ) + x (ρv + p(ρ)) = () The unnowns are the density ρ and the velocity V The pressure ( ρ p(ρ) ) is assumed to be C (, ) ) with p(ρ) >, p (ρ) >, p (ρ), ρ > Thus, the sound speed c : ρ p (ρ) is well defined and is an increasing function We consider the problem (1)-() on the bounded domain (, L), T with the boundary conditions V (, t) = = V (L, t), t > and the initial conditions ρ(x, ) = ρ (x), V (x, ) = V (x), x (, L) with ρ, V L (, L) Let Φ : ρ > Φ(ρ) such that ρφ (ρ) Φ(ρ) = p(ρ), ρ > The quantity S = 1 ρ V +Φ(ρ) is an entropy of the system: entropy solutions to (1)-() are required to satisfy: for any ϕ Cc ( ) (, L), T ) such that ϕ, FlorentBerthelin@unicefr thierrygoudon@inriafr mineaud@unicefr S t ϕ + ( S + p(ρ) ) V x ϕ (x, t) dx dt S (x)ϕ(x, ) dx (3) 1

2 The meshes We consider a set of J + 1 points = x 1 < x < < x J < x J+1 = L The x are the edges of the so-called primal mesh T We set x +1/ =x +1 x The centers of the primal cells, x +1/ = (x + x +1 )/ for {1,, J}, realize the dual mesh T We set x = (x 1/ + x +1/ )/ for {,, J 1} and x = size(t ) = max x +1/ The adaptive time step is t and we set t=max t The scheme We analyze the scheme introduced in 1 It wors on staggered grids: the densities, ρ +1/, {1,, J}, are evaluated at centers whereas the velocities, V, {1,, J + 1}, are evaluated at edges We set, for {1,, J} and i {,, J} ρ +1/ = 1 x+1 ρ (x) dx, Vi = 1 xi+1/ V (x) dx (4) x +1/ x x i x i 1/ The density is first updated with a FV approximation on the primal mesh ( x +1/ ρ +1/ +1/) ρ + t ( F+1 F ) =, {1,, J} (5) Then, the velocity is updated with a FV approximation on the dual mesh: ( x ρ V ρ V ) ( ) + t G+1/ G 1/ + π/ +1/ π/ 1/ =, (6) for {,, J}, while V 1 = V J+1 = The density on the edges ρ is defined by x ρ = x +1/ ρ +1/ + x 1/ρ 1/, {,, J} The definition of the fluxes relies on the inetic framewor We refer the reader to 1 for details Let us introduce the two following functions F + and F F ± (ρ, V ) = ρ ξ 1I c(ρ) ξ V c(ρ) d ξ ξ We adopt the following formulas for mass fluxes: F 1 = F J+1 =, F = F + (ρ 1/, V ) + F (ρ +1/, V ), {,, J}, (7) and, for momentum fluxes: G3/ = V F (ρ 5/, V ), G J+1/ = V J F + (ρ J 1/, V J ), G+1/ =V + V ( F + (ρ 1/, V ) + F + (ρ +1/, V +1) ) +1 ( F (ρ +1/, V ) + F (ρ +3/, V +1) ), {,, J 1} The discrete pressure gradient combines a space centered scheme and time semi implicit discretization, namely it uses π / +1/ = ρ +1/ Φ (ρ +1/ ) Φ(ρ +1/ ) (8)

3 Properties of the scheme The analysis is driven by the shapes of the functions F ±, see 1, Lemma 3 Here, we shall use the following properties (i) Smoothness: (ρ, V ) (, ) R F ± (ρ, V ) are of class C 1, (ii) Consistency: F + (ρ, V ) + F (ρ, V ) = ρv, V R, ρ Under CFL conditions, see 1, the scheme preserves the positivity of the discrete density and discrete inetic and internal energies evolution equations hold Lemma 11 Let N N Assume min i ( ρ i+1/ ) > For all {,, N 1}, there exists V >, which depends only on the state (ρ, V ), such that if then, min i ( ρ i+1/ ) >, {,, N} and = = x +1/ t e +1/ e +1/ (9) t min ( x+1/ )V 1, (1) D C, with D = 1 4 x ρ ( V V ), (11) + G +1 G + π / +1/ V+1 V D t, (1) x t EK, E K, + Γ +1/ Γ 1/ + π / +1/ π/ 1/ V + D, (13) t where EK, = 1 ( ) ρ V and e +1/ = Φ(ρ +1/ ) are the inetic and internal energies The fluxes are defined by G 1 = G J+1 = and G = Φ(ρ 1/ )V x ( ) 1/ Φ t ρ 1/ Φ(ρ 1/ ), {,, J}, Γ +1/ = 1 V V+1 F + F (V V+1) F, 1/ =ρ 1/ t ( F (ρ +1/ x, V 1/ ρ ) F (ρ 1/, V + F, +1, {1,, J}, ( V ) ρ 1/ V ), and F, 1 = F,, J+1 =, F = F + (ρ 1/, V ) F (ρ +1/, V ), {,, J} The function Φ is a C extension of the function Φ (see 1, Section 43) Results As in, we prove a Lax-Wendroff-lie statement: the limit of a converging (and uniformly bounded) sequence of stepwise constant functions defined from the scheme is a wea entropic-solution of the system of conservation laws Consistency analysis Notation Assuming that = t = T, we define the reconstructions (i =, 1) ρ (i) = = =1 ρ +i +1/ χ/ +1/, π = = =1 3 π / +1/ χ/ +1/, V = = = V χ /,

4 where χ / = χ x 1/, x +1/ t, t, χ / +1/ = χ x, x +1 t, t We also introduce the following discrete norms ρ,t = max max N 1 J ρ +1/, V,T = max N N ρ 1;BV,T = t J N ρ +1/ ρ 1/, V 1;BV,T = t J = = = =1 ρ BV;1,T = x +1/ ρ +1/ ρ +1/ =1 = max V, J V +1 V, For ϕ Cc ( ) (, L), T ), we set ϕ +1/ = ϕ(x +1/, t ) and ϕ = ϕ(x, t ) The interpolate ϕ T of ϕ on the primal mesh and its discrete derivatives are defined by ϕ T (, ) = ð t ϕ T = ϕ +1/ χ1/ +1/ (, ), ϕ =1 ϕ = =1 +1/ ϕ +1/ t (, t) = T χ / +1/, ð xϕ T = ϕ i+1/ χ/ +1/ = =1 = = ϕ +1/ ϕ x (, t), t >, 1/ χ / Similarly, the interpolate ϕ T of ϕ on T and its discrete derivatives are given by ϕ T (, ) = ð t ϕ T = = = = ϕ χ 1/ (, ), ϕ T (, t) = ϕ t ϕ χ /, ð xϕ T = ϕ χ / (, t), t >, = = ϕ +1 ϕ x = =1 +1/ χ / +1/ Assumptions Let ( T m )m 1 be a sequence of meshes s t size(t m) and a familly of time steps ( t ) m, m 1 verifying t m and (1) Assume that there exists N m N s t N m 1 = t m = T The scheme defines (ρ () m, V m ) m 1 Suppose that ρ () m,t + V m,t C, ρ () m 1;BV,T + V m 1;BV,T C BV (14) holds and, in the case ( ρ p (ρ)) ρ L 1 loc (, ), ( ρ () ) 1,T m there exists ( ρ, V ) L ((, T ) (, L)) such that C We assume that ( ρ () m, V m ) ( ρ, V ) in L r ((, T ) (, L)), 1 r < (15) Main results The uniform bounds imply that there exists constants such that sup A (ρ, V ) C A, with A = F ±, ρ F ± and V F ±, ρ, V C sup B(ρ) C B, with B = Φ, Φ, and Φ ρ C +4(C +C F ± ) 4

5 Note also that we have Φ ( ρ +1/) C Φ,ρ ρ +1/,, Furthermore, we show that ρ () m BV;1,T C by using (5) which allows to dominate x +1/ ρ +1/ ρ +1/ by t C ρf ± ( ρ +1/ ρ 1/ + ρ +3/ ρ +1/ ) + C V F ± V +1 V Consequently, ρ (1) m ρ and π m p( ρ) in L r ((, T ) (, L)); with (4) and since ρ, V L (, L), we get ρ () m (, ) ρ and V m (, ) V in L r ((, L)), 1 r < Finally, in the sequel, when a function ϕ Cc ( ) (, L), T ) is given, we assume that t m and x m are sufficiently small so that ϕ(x, ), x, x 3/ x J+1/, L and ϕ(, t), t t, t N Moreover, since ϕ is smooth, ϕ Tm, ϕ T m ϕ, ð t ϕ Tm, ð t ϕ T m t ϕ, and ð x ϕ Tm, ð xϕ T m x ϕ, in L r ((, T ) (, L)), 1 r Theorem 1 Assume (14) and (15) Then, ( ρ, V ) satisfies (1)-() in the distribution sense in ( Cc ( )), (, L), T ) that is ρ t ϕ + ρ V x ϕ (x, t) dx dt ρ (x)ϕ(x, ) dx =, (16) T ρ V t ϕ + ( ρ V + p( ρ) ) x ϕ Moreover, ( ρ, V ) satisfies the entropy inequality (3) (x, t) dx dt ρ (x)v (x)ϕ(x, ) dx= (17) Proof Let ϕ C c ( (, L), T ) ) For the sae of simplicity, the index m is dropped Mass balance We multiply (5) by ϕ 1 J to obtain +1/ x +1/ (ρ +1/ ρ +1/ )ϕ +1/ = =1 } {{ } :=T 1 and sum the results for N 1 and + t ( F +1 F ) ϕ +1/ = } =1 {{ } :=T For T 1, since ϕ N +1/ =, a discrete integration by part wrt time yields Noting that T 1 = = =1 t x+1 ρ () x t x +1/ ρ ( +1/ ϕ +1/ ϕ +1/ for {,, N 1}, {1,, J}, we get T 1 = = ) J x +1/ ρ +1/ ϕ +1/ =1 ð t ϕ T dx dt = x +1/ ρ ( +1/ ϕ +1/ ) ϕ +1/ ρ () ð t ϕ T dx dt ρ () (x, )ϕ T (x, ) dx For T, by integrating by part wrt space, we readily obtain T = t = J F = 5 ( ϕ +1/ ϕ 1/)

6 Bearing in mind that x = x 1/ + x +1/, we then combine the two following expressions of mass fluxes (see (9)-(ii)) to write F F = = ρ ±1/ V x 1/ x R,± ρ 1/ V with R,± + x +1/ x = F ± (ρ +1/, V ) F ± (ρ 1/, V ) ρ +1/ V x 1/ + R, x This expression of the mass fluxes leads to T = T,1 T, with T,1 = T, = = = t t = 1 x 1/ ρ 1/ + x +1/ρ +1/ = 1 x 1/ R, x +1/ R,+ We now observe that, for {,, N 1}, {,, J}, and t x t ρ () x 1/ t x+1/ ρ () t x V ð x ϕ T dx dt = t x 1/ V ð x ϕ T dx dt = t x +1/ Summing these equalities yields T xj+1/ T,1 = ρ () V ð x ϕ T dx dt = x 3/ since ð x ϕ T (x, ) for x, x 3/ x J+1/, L V V ρ 1/ V ρ +1/ V x +1/ x ϕ +1/ ϕ 1/ x, ϕ +1/ ϕ 1/ x ϕ +1/ ϕ x 1/ ϕ +1/ ϕ x 1/ ρ () V ð x ϕ T dx dt, R,+ With (15), we pass to the limit in T 1 and T,1 We prove that ( ρ, V ) satisfies the mass conservation equation (16) by showing that T, since T, C ρf ± xϕ L V,T ρ 1;BV,T x x Momentum balance We multiply (6) by ϕ and sum for N 1 and J to obtain T 3 +T 4 +T 5 = with ( T 3 = x ρ V ρ V ) ϕ, T 4 = = t = = = ( G +1/ G 1/ ) ϕ, T 5 = For T 3, integrating by part wrt time yields T 3 = = = = x ρ V ( ϕ ϕ t ) = =, ( / π +1/ ) π/ 1/ ϕ x ρ V ϕ 6

7 Next, we observe that t x+1/ ρ () t x 1/ V ð t ϕ T dx dt = V ϕ ϕ t = x ρ V t x+1/ ρ () t x 1/ ( ϕ ϕ ) Summing these equalities for {,, N 1}, {,, J} yields T 3 = = T xj+1/ ρ () x 3/ V ð t ϕ T dx dt ρ () V ð t ϕ T dx dt xj+1/ ρ () x 3/ dx dt (x, )V (x, )ϕ T (x, ) dx, ρ () (x, )V (x, )ϕ T (x, ) dx For T 4 and T 5, we first integrate by part wrt space and obtain T 4 = t = =1 G+1/ ( ϕ +1 ϕ ), T5 = We then use the following expression of the momentum flux G+1/ = 1 (V ) ρ ( ) +1/ + V +1 + Q +1/, Q +1/ = 1 V R,+ + 1 V +1 R, +1 to write T 4 = T 4,1 T 4, with T 4,1 = = t J π / ( +1/ ϕ +1 ) ϕ = =1 ( 1 V +1 V ) F + (ρ +1/, V +1 ) F (ρ +1/, V ) t =1 1 (V ρ ) ( +1/ + V ) (ϕ +1 T 4, = t J Q ( +1/ ϕ +1 ) ϕ = =1 +1 ) ϕ Next, we observe that t ( ) x+1 ρ () ( ) ð V ( ) t V x ϕ T dx dt = t ρ + V +1 ( +1/ ϕ +1 x ) ϕ and, consequently, summing for {,, N 1}, {1,, J}, Similarly, for T 5, we get T 4,1 = ρ () ( ) ð V x ϕ T dx dt T 5 = π ð xϕ T dx dt With (15), we pass to the limit in T 3, T 4,1 and T 5 We obtain that ( ρ, V) satisfies the momentum balance equation (17) by showing that T 4, Indeed, we have ( ) T 4, V,T x ϕ L C ρf ± ρ 1;BV,T + C F ± V 1;BV,T x x 7

8 Entropy inequality We now assume that ϕ Kinetic energy We multiply (13) by t ϕ and sum for N 1 and J We obtain to get T 6 +T 7 + T 8 with T 6 = x EK, E K, ϕ J, T 7 = t Γ +1/ Γ 1/ ϕ, = = T 8 = = t = Integrating by part wrt time yields T 6 = x EK, ϕ ϕ = = = = = π / +1/ π/ 1/ V ϕ + x EK,ϕ =1 1 ( ) ð L ρ() 1 V t ϕ T dx dt ρ() For T 7, we write Γ +1/ = 1 ) 4 +1/( ρ V 3 ( ) + V S +1/ = V V +1 = = D ϕ (x, ) ( V (x, ) ) ϕt (x, ) dx 4 S +1/ where R, +1 R,+ + (V+1 V ) F, + F, +1 ρ +1/ (V + V+1) Integration by part wrt space leads to T 7 = T 7,1 T 7, with T 7,1 = 1 ( ) 3ð ρ() V x ϕ T dx dt, T 7, = 1 4 Finally T 7, x since it is dominated by C ρf ± = t J S+1/ =1 ϕ +1 ϕ x x ϕ L V,T V,T ρ 1;BV,T + ( ) C F ± + V,T ρ,t V 1;BV,T Internal energy Multiply (1) by t ϕ to get T 9 + T 1 + T 11 with T 9 = = =1 T 11 = +1/ x +1/ e +1/ e +1/ ϕ = t =1 π / ( +1/ V and sum for N 1 and 1 J +1/, T 1 = +1 V Owing to integration by part wrt time, we get T 9 = = = =1 x +1/ Φ(ρ +1/ )( ϕ +1/ ϕ +1/ Φ ( ρ () ) ðt ϕ T dx dt = t ) ϕ +1/ =1 = =1 G +1 G ϕ +1/, D ϕ +1/ ) J x +1/ Φ(ρ +1/ )ϕ +1/, =1 Φ ( ρ () (x, ) ) ϕ T (x, ) dx 8

9 For T 1, we rewrite the flux as follows G = 1 x x 1/ Φ(ρ 1/ ) + x +1/Φ(ρ +1/ V ) + U 1, + U, + U 3,, U1, = ( e 1/ V V ), U, = x +1/ e x +1/ e 1/ V, U3, = x 1/ Φ( ρ t 1/) Φ(ρ 1/ ) It leads to T 1 = T 1, T 1,1 T 1, T 1,3 with T 1, = Φ ( ρ () ) V ð x ϕ T dx dt, T 1,i = = t = The term T 1,1 can be bounded as follows T 1,1 C Φ,ρ x ϕ t Ui, ( ϕ +1/ 1/) ϕ, i = 1,, 3 = = x ρ 1/ V V (18) Since a min(a, b)+ b a, we get ρ 1/ ρ + ρ +1/ ρ 1/ This leads to T 1,1 C Φ,ρ x ϕ ( L t J ) x ρ V V + V,T ρ 1;BV,T x = } = {{ } :=T Writing ρ =ρ (ρ ρ ) and using the Cauchy-Schwarz inequality yields ) 1/ ( T (T L ρ,t t = = D ) 1/+ V,T ρ BV;1,T t t 1/ It finally leads to T 1,1 t 1 + x The term T 1, can be bounded as follows T 1, C Φ V,T x ϕ L ρ 1;BV,T x x We now turn to T 1,3 We remar that 1/ ρ 1/ ρ t ( x 1/ C ρf ± ρ +1/ ρ 1/ + ρ 1/ Hence, using the same bound as for T 1,1 yields ( ( ) T 1,3 C Φ x ϕ L C ρf ± + V,T ) ρ 1;BV,T x + T V V ) t 1/ + x Pressure terms It remains to get the limit of T 8 + T 11 = T 1, T 1,1 T 1, T 1,3 9

10 with T 1, = π V ð xϕ T dx dt, T 1,1 = = = D ( ϕ +1/ ) ϕ, T 1, = 1 t J π / ( +1/ V V + V+1 V )( +1 ϕ +1 ) ϕ, = =1 T 1,3 = 1 = t =1 We bound T 1,1 and T 1,3 as follows T 1,1 ( xϕ L = = Note that π / +1/ D π / ( +1/ V ) x x, +1 V )( ϕ +1/ ϕ ( C Φ + C Φ,ρ ) ρ +1/ It readily leads to +1 ) ϕ T 1,3 C π 4 ( ) ( ) xxϕ L V 1;BV,T x x T 1, ( C Φ + C Φ,ρ ) x ϕ L T t 1 With (15), we pass to the limit in T 6, T 7,1, T 9, T 1, and T 1, We arrive at (3) since the other terms tend to References 1 F Berthelin, T Goudon, and S Mineaud Kinetic schemes on staggered grids for barotropic Euler models: entropy-stability analysis 14 in revision R Herbin, J-C Latché, and T T Nguyen Explicit staggered schemes for the compressible Euler equations ESAIM:Proc, 4:83 1, 13 1