All speed scheme for the low mach number limit of the Isentropic Euler equation

Transcription

1 arxiv:98.99v [math-ph] 3 Aug 9 All speed scheme for the low mach number limit of the Isentropic Euler equation Pierre Degond,, Min Tang, -Université de Toulouse; UPS, INSA, UT, UTM ; Institut de Mathématiques de Toulouse ; F-36 Toulouse, France. -CNRS; Institut de Mathématiques de Toulouse UMR 59 ; F-36 Toulouse, France. pierre.degond@math.univ-toulouse.fr, tangmin@gmail.com Abstract An all speed scheme for the Isentropic Euler equation is presented in this paper. When the Mach number tends to zero, the compressible Euler equation converges to its incompressible counterpart, in which the becomes a constant. Increasing approimation errors and severe stabilit constraints are the main difficult in the low Mach regime. The ke idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated eplicitl and the other one implicitl. Moreover, the flu of the equation is also treated implicitl and an elliptic tpe equation is derived to obtain the. In this wa, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods [, 3, 7], nonphsical oscillations can be suppressed. We develop this semi-implicit time discretization in the framework of a first order local La-Friedrich (LLF) scheme and numerical tests are displaed to demonstrate its performances. AMS subject classification: 65M6,65Z5,76N99,76L5 Kewords: Low Mach number; isentropic euler equation; compressible flow; incompressible limit; asmptotic preserving; La-Friedrich scheme.

2 Introduction Singular limit problems in fluid mechanics have drawn great attentions in the past ears, like low-mach number flows, magneto-hdrodnamics at small Mach and Alfven numbers and multiple-scale atmospheric flows. As mentioned in [7], the singular limit regime induces severe stiffness and stabilit problems for standard computational techniques. In this paper we focus on the simplest Isentropic Euler equation and propose a numerical scheme that is uniforml applicable and efficient for all ranges of Mach numbers. The problem under stud is the Isentropic Euler equation { t ρ ε + (ρ ε u ε )=, ) t (ρ ε u ε )+ (ρ ε u ε u ε + ε p ε =. where ρ ε,ρ ε u ε is the and momentum of the fluid respectivel and ε is the scaled Mach number. This is one of the most studied nonlinear hperbolic sstems. For standard applications, the equation of state takes the form () p(ρ)=λρ γ, () where Λ,γ are constants depending on the phsical problem. It is rigorousl proved b Klainerman and Majda [5, 6] that when ε, i.e. when the fluid velocit is small compared with the speed of sound [3], the solution of () converges to its incompressible counterpart. Formall, this can be obtained b inserting the epansion ρ ε = ρ + ε ρ () +, u ε = u + ε u () +, (3) into () and equate the same order of ε. The limit reads as follows [5, 8]: ρ = ρ, u =, t u + (u u )+ p =. (4a) (4b) (4c) Here p is a scalar pressure that can be viewed as the Lagrange multiplier which enforces the incompressibilit constraint. Phsicall, this limit means that in slow flows (compared with speed of sound), the factor /ε in the momentum equation in front of the pressure gradient generates fast pressure waves, which makes the pressure and therefore, the, uniform in the domain [3, 4].

3 For atmosphere-ocean computing or fluid flows in engineering devices, when ε is small in (), standard numerical methods become unacceptabl epensive. Indeed, () has wave speeds of the form λ = u ε ± ε p (ρ ε ), where p (ρ ε ) is the derivative with respect to ρ ε. If a standard hperbolic solver is used, the CFL requirement is = O(ε ). Moreover in order to maintain stabilit, the numerical dissipation required b the hperbolic solver is proportional to λ. If λ =O( ε ), in order to control the diffusion, we need to have =o(εr ), where r is some appropriate constant. Thus the stabilit and accurac highl depend on ε. Our aim is to design a method whose stabilit and accurac is independent of ε. The idea is to find an asmptotic preserving (AP) method, i.e. a method which gives a consistent discretization of the isentropic Euler equations () when, resolve ε, and a consistent discretization of the incompressible limit (4) when ε (, being fied). The efficienc of AP schemes at the low Mach number regime can be proved similarl as in [9]. The ke idea of our all speed scheme is a specific semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one part being treated eplicitl and the other one implicitl. Moreover, the flu of the equation is also treated implicitl. For the space discretization, when ε is O(), even if the initial condition is smooth, shocks will form due to the nonlinearit of the div ( ρ ε u ε u ε ) term and shock capturing methods should be emploed here. In the literature, lots of efforts have been made to find numerical schemes for the compressible equation that can also capture the zero Mach number limit [, 6,, 3, 4]. In [], Bijl and Wesseling split the pressure into thermodnamic and hdrodnamic pressure terms and solve them separatel. Similar to this approach, the multiple pressure variable (MPV) method was proposed b Munz et al. in [3, 4]. There is also some recent work b J. Hauck, J-G. Liu and S. Jin []. Their approach involves specific splitting of the pressure term. We avoid using this splitting, the proper design of which seems ver crucial in some cases. Some similar ideas can be found in the ICE method, which is designed to adapt incompressible flow computation techniques using staggered meshes to the simulation compressible flows. The method was first introduced b Harlow and Amsdan in 965 and 97 [,3] and is called Implicit Continuous-fluid Eulerian (ICE) technique. It is used to simulate single phase fluid dnamic problems with all flow speeds. The introduce two parameters in the continuit equation and the 3

4 momentum equation to combine information from both previous and forward time steps. However this method is not conservative, which leads to discrepancies in the shock speeds. Additionall it suffers from small wiggles when there are moving contact discontinuities. The first problem was solved b an iterative method, for eample SIMPLE [5], or PISO [4]. In some recent work, Heul and Wesseling also find a conservative pressure-correction method [7]. All these methods are based on the so called MAC staggered mesh in order to be consistent with the staggered grid difference method for the incompressible Euler equations [3]. Specificall, if we write the simplified ICE technique presented in [] in conservative form, we are led to the semi-discrete framework: { ρ ε ρ n ε + (ρ ε u ε ) n =, (ρ ε u ε ) (ρ ε u ε ) n + (ρ n ε uε n u n (5) ε)=, { ρ n+ ε ρ ε + ((ρ ε u ε ) n+ (ρ ε u ε ) ) =, (ρ ε u ε ) n+ (ρ ε u ε ) + p(ρ n+ ε ε )=. B substituting the gradient of the second equation of (6) into its first equation and using the results of the first equation (5), ρ ε can be updated b solving an elliptic equation which does not degenerate when ε. We use a similar idea in our method. However, we do not use the predictorcorrector procedure but we rather discretize the problem in a single step. We use standard shock capturing schemes which allows to guarantee the conservativit and the desired artificial viscosit. We onl use implicit evaluations of the mass flu and pressure gradient terms to ensure stabilit and provide an etremel simple wa to deal with the implicitness. Additionall, we propose a modification of the implicit treatment of the pressure equation. Indeed, using a similar idea as in [], we split the pressure into two parts and put α p(ρ ε ) into the hperbolic sstem. This makes the first sstem no longer be weakl hperbolic and much more stable. The numerical results show the advantage of our method in the following sense: The method is in conservative form and can capture the right shock speeds. The non-phsical oscillations [] can be suppressed b choosing the proper value of the parameter which determines the fraction of implicitness used in the evaluation of the pressure gradient term. The choice of this parameter depends on the time and space step and on the specific problem. (6) 4

5 In this paper we onl use the first order LLF scheme. Higher order space and time discretizations will be subject of future work. The main objective of this work is to show that the semi-discrete time discretization provides a framework for developing AP methods for singular limit problems. Similar ideas can be etended to the full Euler equation and more complicated fluid model and have also been used in other contets such as quasineutralit limits [4, 7] and magnetized fluids under stong magnetic fields [5]. The organization of this paper is as follows. Section eposes the semiimplicit scheme and its capabilit to capture the incompressible limit is proved. The detailed one dimensional and two dimensional full discretized schemes and their AP propert are presented in section 3 and 4 respectivel. In section 5, how to choose the ad-hoc parameter is discussed and finall, some numerical tests are given in section 6 to discuss the stabilit and accurac of our scheme. The efficienc at both the compressible and low mach number regime are displaed. Finall, we conclude in section 6 with some discussion. Time Semi-discrete scheme Let be the time step, t n = n,n=,, and let the n superscript denote the approimations at t n. The semi-discrete scheme for the nth time step is ρ n+ ε ρ n ε (ρ ε u ε ) n+ (ρ ε u ε ) n + (ρ ε u ε ) n+ =, (7) + div ( ρ n ε un ε un ε + α p(ρn ε )) + αε ε p(ρ n+ ε ) =, (8) where α is an ad-hoc parameter which satisfies α /ε. The choice of α depends on the space and time steps and on the fluid speed. When the shock is strong, α should be bigger, which means that the sstem should be more eplicit to follow the discontinuit more closel. We discuss the choice of α for specific equations of state in this paper and test its effect numericall. It depends on the required accurac, the small parameter ε and the shock amplitude in a sometimes quite comple wa. Rewriting the momentum equation (8) as (ρ ε u ε ) n+ =(ρ ε u ε ) n (ρ n ε un ε un ε + α p(ρn ε ) ) αε ε P(ρ n+ ε ) 5

6 and substituting it into the equation, one gets ρ n+ ε αε ε P(ρ n+ ε )=φ(ρε,u n n ε) (9) which is an elliptic equation that can be solved relativel easil. Here φ(ρ n ε,un ε )=ρn ε (ρn ε un ε )+ ( ρ n ε un ε un ε + α p(ρn ε )). () The Laplace operator in (9) can be approimated b ( P (ρ n ε) ρ n+ ) ε and (9) becomes ρ n+ ε αε (P (ρ n ) ε ) ρn+ ε = φ(ρ n ε,u n ε ), () ε Though shocks will form for the original sstem (7)(8), we alwas add some numerical diffusion terms so that φ(ρε,u n ε) n is smooth. Then so is ρ n+ ε. When we implement this method, ρ n+ ε can be obtained from (9) first and u n ε is then updated b the momentum equation (8) afterwards. Therefore, apart from the resolution of the elliptic equation (), the scheme onl involves eplicit steps. We now show that the scheme (7)(8) is asmptotic preserving. We introduce the formal epansion ρ n ε ()=ρn c + ερn () ()+ε ρ n () ()+, u n ε = un ()+εun () ()+. () In the sequel, the c in the inde means that the quantit is independent of space. When, are fied and ε goes to in (9), we formall have P(ρ n+ ) =, which implies that ρ n+ is independent of space, where ρ n+ is the limit of ρ n+ ε when ε. Thus we have ρc n+ ρc n + (ρ c u ) n+ = (3) b equating the O() terms in the equation (7). Integrating (3) over the computational domain, one gets Ω ρn+ c ρc n = ρc n+ Ω (u ) n+ = ρc n+ n u n+. (4) Ω As discussed in [], for wall boundar condition, periodic boundar condition and open boundar condition, (4) gives ρ n+ c = ρ n c, (5) 6

7 that is ρ is also independent of time. Thus (3) also implies u n+ =. (6) Then, b using the fact that the curl of the gradient of an scalar field is alwas zero, the curl of the O() terms of the momentum equation (8) becomes un+ u n + ( u n ) un =. (7) Thus u n+ u n + ( u n ) un + p n () =, (8) where p n () is some scalar field. Equations (5), (6), (8) are the semi-discretization in time of (4) and thus the scheme (7), (8) is consistent with the low Mach number limit ε of the original compressible Euler equation. This statement is eactl saing that the scheme is AP. We can see that, in order to obtain the stabilit and AP properties, it is crucial to treat the flu in the equation (7) implicitl. Letting U =(ρ ε,ρ ε u ε ) T, we can write (7), (8) abstractl as where ( F(U n+/ )= U n+ U n + F(U n+ )+QU n+ =, (9) (ρ ε u ε ) n+ ρ n ε u n ε u n ε + α p(ρ n ε) ) (, Q= αε ε P ). () Here P is an operator on ρ ε and U n+/ reminds that the flu is partl implicit and partl eplicit. This semi-discretization gives us a framework for developing AP schemes that can capture the incompressible limit. Now we are left with the problem of how discretizing the space variable. Because shocks can form, considerable literature has been devoted to the design of high resolution methods that can capture the correct shock speed. Upwind schemes and central schemes are among the most widel used Godunov tpe schemes [9 ]. In the present paper, the hperbolic operator U n+ U n + F(U n+ ) 7

8 is approimated b an upwind hperbolic solver and the stiff /ε factor in front of the pressure term is treated implicitl. The implicitness of the flu is treated b combining it with the momentum equation. For simplicit, in the present work we onl consider the first order modified La-Friedrich scheme with local evaluation of the wave-speed in the current and neighboring cell. 3 Full time and space discretization: One dimensional case For simplicit, we consider the domain Ω=[,]. Using a uniform spatial mesh with =/M, M being an positive integer, the grid points are defined as j := j, j=,,,m. The flu and Jacobian matri of (9) become ( ) ( ρ F(U)= ε u ε ρ ε u ε + α p(ρ, F (U)= ε) so, the wave speeds are Let U j be the approimation of U( j ) and let ) u ε + α, () p (ρ ε ) u ε λ = u ε ± α p (ρ ε ). () A j+ (t)=ma ( λ j, λ j+ ). (3) These are the local maimal wave-speeds in the current and neighboring cells. We discretize (9) in space as follows: Uj n+ Uj n where F j±/ (U n+ ) is the numerical flu + F j+ (U n+ ) F j (U n+ ) + Q j U n+ =, (4) and F j+ (U n+ ( )= F + (U n+ j+ )+F (U n+ ) j+ ) ( F + (U n+ j+ )= (ρ ε u ε ) n+ j + A n j+ ρ n ε j (ρ ε u ε u ε ) n j + α p(ρn ε j )+An j+ (ρ ε u ε ) n j ), (5) 8

9 ( F (U n+ j+ )= (ρ ε u ε ) n+ j+ An j+ ρ n ε j+ (ρ ε u ε u ε ) n j+ + α p(ρn ε j+ ) An j+ (ρ ε u ε ) n j+ ) and Let QU n+ j = ( αε ε ( P(ρ n+ ε j+ ) P(ρn+ ε j )) q=ρu, (6) and F (),F () denote the first and second element of F respectivel, we can rewrite the momentum discretization in (4) as follows: Here q n+ ε j = q n ε j D jf ()( ρε,u n n ) αε ( ε p(ρ n+ ε ε j+ ) p(ρn+ ε j )). (7) D ju= u j+/ u j /. B substituting (7) into the equation in (4), one gets ρ n+ ε j ( αε ) ( 4ε p(ρ n+ ε j+ ) p(ρn+ ε j )+ p(ρ n+ ε j )=Dφ(ρ ) ε,q n ε), n (8) where Dφ(ρ n ε,qn ε )=ρn ε D j F() (ρ n ε,un ( ε )+ D j+ D ) j F () (ρ n ε,un ε ) (9) is a discretization of φ(ρ n ε,un ε ) in (). We notice that (8) is a discretization of the elliptic equation (). We can update q n+ ε through (7) afterwards. To obtain ρ n+ ε in (8), a nonlinear sstem of equations needs to be solved. One possible wa to simplif it is to replace P(ρ n+ ε ) b P (ρ n ε) ρ n+ ε, so that the following linear sstem is obtained: ρ n+ ε j ( αε ) ( 4ε p (ρ n ε j+ )( ρ n+ ε j+ ) ρn+ ε j p (ρ n ε j )( ρ n+ ε j ρ n+ ) ) ε j = Dφ(ρ n ε,qn ε ). (3) This is a five point scheme which is too much diffusive, especiall near the shock. One possible improvement is that instead of (3), we use the following three ). 9

10 points discretization ρ n+ ε j ( αε ) ( ε p (ρ n ε j+ )( ρ n+ ε j+ ) ρn+ ε j p (ρ n ε j) ( ρ n+ ε j ρ n+ ) ) ε j = Dφ(ρ n ε,q n ε). (3) After obtaining ρ n+ ε, we can substitute it into (7) to get q n+ ε j. To summarize, three schemes are proposed here: (8), (7); (3), (7) and (3), (7). To investigate the AP propert, we take (3), (7) as an eample. The proofs for the other two schemes are similar. B substituting the following epansion ρ n ε j = ρn c + ε ρ() n j +, qn ε j = qn c + εqn () j +, (3) into (3), the O( ) terms give that ρ n+ ε periodic boundar condition, and thus: () j = ρn+ c ρ n+ ε j = ρ n+ ()c + ε ρ n+ () j +. Summing (3) over all the grid points, one gets is constant in space b using the ρ n+ c = ρ n c = ρ c, (33a) which implies that ρ is independent of time and space. Thus, the O() terms of (3) are p (ρc n )( ρ n+ () j+ ) ρn+ () j p (ρc n )( ρ n+ () j ρn+ () j ) =, b recalling that the O() terms of both ρ n ε and qn ε are constant in space. Then the periodic boundar condition gives ρ n+ () j = ρn+ ()c, (33b) which gives that ρ n+ () is also independent of space. Therefore from (), (7), q n+ j = q n j = qn c. (33c) In one dimension, (33) is the discretization of (5), (6), (8) when periodic boundar conditions appl and thus is consistent with the incompressible limit. In fact all the three methods proposed here are AP.

11 4 Full time and space discretization: Two dimensional case We consider the domain Ω=[,] [,]. For M,M two positive integers, we use a uniform spatial mesh =/M, =/M. The grid points are Let and ( i, j ) :=(i, j ), i=,,m ; j=,,m Now U =(ρ ε,q () ε,q () ε ) T and U i, j is the numerical approimation of U( i, j ). G (U)= ρ ε u ε ρ ε u + α p(ρ ε) ρ ε u u Eq. (9) can be written as Denote G (U n+ )= Q= αε ε, G (U)= P P ρ ε u ε ρ ε u u ρ ε u + α p(ρ ε). t U+ G (U)+ G (U)+QU =. (ρ ε u ε ) n+ ρε(u n n ε ) + α p(ρε) n ρ n ε un ε un Q= D i ju= u i j+ u i j, G (U n+ )= αε ε D ˆP αε ε D ˆP,, D i j u= u i+ j u i j.. (34) (ρ ε u ε ) n+ ρ n ε u n ε un ε ρ n ε (un ε ) + α p(ρ n ε ) Now the eigenvalues of the two one-dimensional hperbolic equations are λ () = u,u ± α p (ρ ε ), λ () = u,u ± α p (ρ ε ).,

12 The full discrete scheme for the two dimensional problem is where and Ui n+ j Ui n j + D i jg (U n+/ )+ ) Ai (, jd i j A i+, j D i j+ U n +D i j G (U n+/ )+ ( Ai, j D i j A i, j+ D ) i j+ U n + QU i n+ j =, (35) D i j u= u i j u i j D i j u= u i j u i j, (D i j+u)= u i+ j u i j,, D i j+ u= u i j+ u i j, A n i+, j = ma{ λ() i j, λ () i+, j, λ() i j, λ () i+, j }, = ma{ λ () i j, λ () i, j+, λ() i j, λ () i, j+ }. (36) A n i, j+ Let q ε be like in (7). Like in one dimension, we can substitute the epressions of q n+ i j,q n+ i j into the equation and get the following discretized elliptic equation, where ρεi n+ j ( αε ) ( ( P(ρ n+ 4ε εi+, j ) P(ρn+ εi, j )+P(ρn+ εi, j )) + (37) ( P(ρ n+ εi, j+ ) P(ρn+ εi, j )+P(ρn+ εi, j ))) = Dφ i j (ρε,q n n ε,qn ε ), Dφ i j (ρ n ε,q n ε,qn ε ) = ρ n ε (D i j qn ε + D i j qn ε + ( Ai D i j A i+, jd i j+ + A i, j D i j A i, j+ D ) ) i j+ ρ n ε + ( D i j D i j( ρ n ε (uε n ) + α p(ρ n ε )) + D ( i j D i j ρ n ε (uε n ) + α p(ρ n ε )) +(D i j D i j + D i j D i j )ρn ε un ε un ε + D i j (A i, jd i j A i+, jd i j+ )qn ε + D i j (A i, jd i j A i+, jd i j+ )qn ε + D i j (A i, j D i j A i, j+ D i j+ )qn ε + ) D i j (A i, j D i j A i, j+ D i j+ )qn ε. (38)

13 After obtaining ρi n+ j b (37), q n+ i j,q n+ i j can be updated b the momentum equation afterwards. Similar to the one-dimensional case, the modified diffusion operator using a reduced stencil is as follows: αε ε ( ( P (ρεi, n j+ )( ρεi, n+ j+ ρn+ εi, j ρεi n+ j + ( P (ρεi+, n j )( ρεi+, n+ j ρn+ εi, j ) P (ρεi, n j )( ρεi, n+ j ρεi, n+ ) ) j ) P (ρεi, n j) ( ρεi, n+ j ρεi, n+ ) )) j = φ(ρ n ε,qn ε,qn ε ). (39) Now we prove the AP propert of our full discrete scheme. Here onl wellprepared initial conditions are considered, which means that there will be no shock forming in the solution. Then α can be chosen to be to minimize the introduced numerical viscosit. Assuming that the epansions of ρ ε,u ε in () hold at time t n, when ε, the O( ) terms of (39) give ε ( P (ρi, n j+ )( ρi, n+ j+ ) ρn+ i, j P (ρi, n j )( ρi, n+ j ρi, n+ ) ) j + ( P (ρi+, n j )( ρi+, n+ j ρn+ i, j ) P (ρ n i, j )( ρ n+ i, j ρ n+ i, j) ) =. When using periodic boundar conditions, one gets ρi n+ j = ρc n+ from (). The time independence of ρ n+, similar to the one dimensional case, can be obtained b summing (39) over all the grid points. Accordingl we have ρεi n+ j = ρc n + ε ρ n+ ()i j +. (4) To prove the limiting behavior of u ε,u ε, we do not want to use the equation because the diffusion operator with reduced stencil does not allow us to find the corresponding equation. Therefore, we consider the O() term of (39), ρ n+ i j ( ( P (ρ n c )( ρ n+ ()i, j+ ρn+ ()i, j + ( P (ρ n c )( ρ n+ ()i+, j ρn+ ()i, j ) P (ρc n )( ρ n+ ()i, j ) ) ρn+ ()i, j ) P (ρc n )( ρ n+ ()i, j ) )) ρn+ ()i, j = φ(ρ n,qn,qn ). (4) 3

14 Moreover, noting the fact that and similarl, D i j P(ρn+ ε ) = D i j P(ρn c + ε ρ n+ () + o(ε )) = D i j P(ρn c )+ε D i j = ε D i j ( ρ n+ () P (ρ n c )) + o(ε ), ( ρ n+ () P (ρ n c )) + o(ε ), D i j P(ρn+ ε )=ε D ( i j ρ n+ () P (ρc n )) + o(ε ), the O() terms of the momentum equations of (35) become q n+ i j qn ( i j q + D ) n+ ( q q ) n+ ( i j D i j Ai ρ ρ q n+ i j qn i j + D i j +A i, j D i j A i, j+ D i j+) q = D i j ( q q ρ ) n+ ( q ) D n+ ( i j Ai ρ +A i, j D i j A i, j+ D i j+) q = D i j Comparing (4) with (D i j (4a)+D i j (4b)), one gets, jd i j A i+, j D i j+ ( P (ρc n+ )ρ n+ ) (), (4a), jd i j A i+, j D i j+ ( P (ρc n+ )ρ n+ ) (). (4b) D i j qn+ + D i j qn+ = O( ), (43) which is an approimation of (6). Moreover, it is obvious that (4) is a discretization of (8). Thus we obtain a full discretization of (4) in the limit ε. Therefore, the two-dimensional scheme is also AP. 5 The ad-hoc parameter In this section we illustrate how to choose and the parameter α b considering the simple state equation P(ρ ε ) = ρ ε. In this contet, the full discrete scheme (4) can be written as { ρ n+ ε ρ n ε q n+ ε + q n+ ε q n ε + q n ε =, q n ε + ( ρ n ε u n ε u n ε+ αρ n ) ε + αε ε ρ n+ ε =., (44) 4

15 where is the centered difference while stands for the difference of flues. The latter is defined as follows (in one space-dimension for simplicit): ( F(U n )) j = F j+ (U n ) F j (U n ), where the flu F j±/ is defined as in (5). B substituting (q n+ ε q n) from the momentum equation of (44) into its equation, one gets ρ n+ ε ρ n ε + q n ε αε ε ρ n+ ε ( ρ n ε u n ε u n ε + αρε) n =. (45) The O() terms behave like a diffusion term which suppresses the oscillations at the discontinuit. Assuming that we use a first order eplicit LLF scheme, the diffusions needed to damp out the oscillations in the mass and momentum equations are respectivel: ( un ε + ε ) ρn ε, ( un ε + ε ) qn ε. (46) Here in (45), besides the O() terms, q n ε also includes some numerical dissipation. B noting ρ n+ ε = ρ n ε qn+ ε + O( ), the diffusion for ρ ε now is ( ( uε + α ) + ) ε ρ n ε + (ρn ε un ε un ε ) (47) plus some higher order terms. Moreover, the diffusion for q ε is ( uε + α ) q n ε + αε ε q n ε (48) and some higher order terms. Comparing (46) and (47), (48), in order to suppress the oscillations at discontinuities we onl need to have that is ( u n ε + α ) + αε ε u n ( ε + ), ε ε + αε. (49) 5

16 Moreover the CFL condition for the eplicit part is σ ma{ u ε + α}, (5) where σ is the Courant number which is less than. We usuall choose σ to be.5. Then the parameter α should satisf ε α σ ma{ u n ε } (5) according to (49), (5). Then the following constraint on should hold if we want the scheme to be stable and non-oscillator ma{ u n ε }+ σ + ε. (5) The reason for the occurence of nonphsical oscillations when α = lies in the fact that the diffusion is not large enough. In this case, with a simple reduction of, it is likel that the diffusion can no longer suppress the oscillations. This is wh we need to introduce α to control the oscillations. But, from the analsis, no matter the value of α, as long as it is less than /ε, the diffusion can never be sufficient when 4ε. In summar there is no specific wa of choosing α < /ε that can guarantee that the nonphsical oscillations will disappear in an case. For well-prepared initial conditions in the low Mach number regime, because there is no shock formation in the solution, it is better to choose α as small as possible to get better accurac, but if strong shocks eist in the solution, α should be big enough to suppress the oscillations. This is wh the choice of α depends on the considered problem. 6 Numerical results Three numerical eamples will allow us to test the performances of the proposed schemes. In fact, three schemes are proposed in section 3 and 4, for eample in one dimension: the scheme (4) without linearizing P(ρ n+ ε ) is denoted b NL. We need to use Newton iterations to solve the nonlinear sstem. When P(ρ n+ ε ) is approimated b P (ρ n ε ) P(ρn+ ε ), the unknowns become a linear sstem. This scheme is represented b L. LD denotes the scheme with the narrower stencil (3). Here we use well-prepared initial conditions of the form (3) and α = for all the test cases. 6

17 In one dimension, let the computational domain be[a,b] and the mesh size be. The grid points are j = a+( j ). In the following tables, the L norm of the relative error between the reference solutions u and the numerical ones U e(u)= U u L u L = ( M j U j u( j ) ) M e ( i u( i ) ) are displaed. Eample P(ρ ε ) and the initial conditions are chosen as P(ρ ε )=ρ ε, ρ ε (,)=, p ε (,)= ε / [,.] [.8,]; ρ ε (,)=+ε, p ε (,)= (.,.3]; ρ ε (,)=, p ε (,)=+ε / (.3,.7] ρ ε (,)= ε, p ε (,)= (.7,.8] This eample consists of several Riemann problems. Shocks and contact discontinuities are stronger when ε is bigger. We first check the difference of the three schemes (8), (3) and (3). The CFL condition for the linearized reduced stencil scheme (3) is discussed in (ii) and a fied Courant number independent of ε is found numericall. Compared with the first order ICE method using local La-Friedrich discretization for (5), the improvement of removing nonphsical oscillation of our scheme is shown. We investigate the effect of α for different values of ε in (iii). In (iv), when α =, we numericall test the uniform convergence order. Finall, the AP propert and its advantages are demonstrated in (v) b comparing with the full eplicit La-Fridrich scheme for the initial Isentropic Euler equation (). When ε =., the initial and momentum are displaed in Figure and we can see the discontinuities clearl. (i) In this eample, we choose ε =.8,.3,.5 corresponding to the compressible, intermediate and incompressible regimes. The numerical results at T =.5 of NL, L and LD are represented in Figure. Here is chosen to make all these three schemes stable and diminishing onl will not improve much the numerical accurac. The reference solution is calculated b an eplicit La-Friedrich method [9, ] with = /5, = 7

18 momentum Figure : Eample. When ε =., the initial and momentum are displaed. /. We can see that all these three methods can capture the right shock speed. The results of the three schemes are quite close, which implies that the linearization idea does simplif the scheme but the LD scheme does not reall introduce less diffusion. When ε is small, though we can no longer capture all the details of the waves, the error is of the order which is the maimum information one can epect. Numericall, for different scales of ε, there is not much difference between these three methods. Thus in the following one dimensional eamples, we onl test the performance of the LD scheme. (ii) Because of the eplicit treatment of the flu terms in the momentum equation, the stabilit of the LD scheme can be onl guaranteed under the following CFL condition σ min i u i + αp (ρ ε ). (53) Here <σ < is the Courant number and is set up at initialization. Consistentl with the fact that these three methods are AP, the Courant number does not depend on ε. Indeed, below, we numericall verif that σ is independent of ε. For ε =.8,.3,.5, the numerical Courant numbers are displaed in Table and we can see numericall that the biggest allowed ma{u} are close to for all ε s. Therefore, σ =.9 is enough to guarantee stabilit and is numericall shown to be independent of ε. B contrast, the eplicit local La-Friedrich scheme for the original Euler equation has a stabilit condition which becomes more and more restrictive as 8

19 momentum a) momentum b) momentum c) Figure : Eample. When T =.5, = /, = /, the and momentum of the NL, L and LD schemes for isentropic Euler equation are represented respectivel b dashed, dash dotted, and dotted lines. The solid line is the reference solution calculated b an eplicit La-Friedrich method [9, ] with = /5, = /. a): ε =.8; b): ε =.3; c): ε =.5. Left: ; Right: momentum. For all ε s, these three lines are so close to each other that -.-. and... are not visible in the figure. 9

20 ε goes to zero. Thus the CFL condition of the standard hperbolic solver = O(ε ) is considerabl improved. u ε ma λ stable / / / / /4 / /8 / / / / / /4 / /8 / / / / / /4 / /8 /9.4.3 Table : Eample. The numerical Courant numbers for different ε. Here ma{λ} denotes the maimum of ma{λ j } defined in () until T =. for all time steps. (iii) The classical ICE method even in its conservative form introduces some nonphsical oscillations, no matter how small the time step is. These oscillations cannot be diminished b decreasing the time step. Their amplitude becomes smaller as the mesh is refined as long as the scheme is stable. In this part we show that our method can suppress these oscillations numericall b choosing α =. When T =., for ε =.8,.3,.5, the numerical results of both our method with α = and ICE calculated b =/, = / are displaed in Figure 3. The oscillations are more important for the ICE method and smooth awa when α =. We can see that numerical nonphsical oscillations occur in the results of the ICE method when ε =.8,.3, but disappear when ε becomes small. This can also be seen from (47), (48). When ε is small the diffusion introduced b the implicitness is bigger. These oscillations also go awa as time goes on due to dissipation. (iv) When α =, the relative errors of the LD scheme for different,

21 momentum.5 a) momentum b) c) momentum Figure 3: Eample. When T =., the and momentum for different ε are presented. The solid and dashed lines are the numerical results of our scheme and ICE with = /, = / respectivel. a) ε =.8; b) ε =.3; c)ε =.5.

22 at time T =. are shown in Table. Here, do not need to resolve ε and the reference solution is obtained b the eplicit LLF scheme calculated with a ver fine mesh =/8, = /8. We can see that good numerical approimations can be obtained without resolving the small ε. The convergence order is / when / is fied, uniforml with respect to ε. This convergence order when there are discontinuities is the same as the eplicit LLF []. We can see from Table that refinement in the time step does not improve the accurac much (provided the Courant number is appropriatel small, like σ =.7). Take ε =.8 as an eample. When =/3, in order to obtain stabilit, should be less than /9. It is demonstrated in Table that the error calculated with = /3 does not decrease much when is changed from /88 to /8. Thus as long as the scheme is stable, we cannot use a smaller to obtain a better accurac. This feature is the same as for standard hperbolic solvers. ε e(ρ ε ) ratio e(p ε ) ratio.8 / / /4 / /8 / /6 / /3 / /3 / / / /4 / /8 / /6 / /3 / /3 / Table : Eample. T =., the L norm of the relative error between the reference solution which is calculated with a ver fine mesh = /8, = /8 and the numerical results for different ε with different, are displaed. (v) We emphasize the AP propert in this final part. For ε =.5, the numerical results at T =. with unresolved mesh =/, = /5 and

23 momentum Figure 4: Eample. B using the LD scheme, the (left) and momentum (right) for ε =.5 at T =. are represented. The circles are the results for =/,= /5 and the solid line is calculated with =/,= /5. resolved mesh =/, = /5 are displaed in Figure 4, while the full eplicit La-Fridrich scheme is not stable with the same mesh size. We do capture the incompressible limit when, do not resolve ε. Eample : In this eample we simulate the evolution of two collision acoustic waves b the LD scheme and test the convergence. We choose α =,ε =., = /, = /. Here is chosen to stabilize the scheme and decreasing alone will not improve much the numerical accurac. Similar to Klein s paper [7], P(ρ ε ) and the initial conditions are chosen as P(ρ ε )=ρ.4 ε, for [,] ρ ε (,)=.955+ ε ( cos(π) ), uε (,)= sign().4 ( cos(π) ). The initial and momentum are displaed in Figure 5. For ε =., the numerical results of the LD scheme at different times T are shown in Figure 6. The initial data approimate two acoustic pulses, one rightrunning and one left-running. The collide and their superposition gives rise to a maimum in the. Then the pulses separate again. This procedure is demonstrated clearl in Figure 6. 3

24 momentum Figure 5: Eample. When ε =., the initial and momentum are displaed. Eample 3 In this eample, we show numerical results for the two dimensional case. Let P(ρ)=ρ and the computational domain be (,) [,] [,]. Because no shock will form in this eample, we choose α = and the initial condition as follows: ρ(,,)= +ε sin (π(+)), q (,,)= sin(π( ))+ε sin(π(+)), q (,,)=sin(π( ))+ε cos(π(+)). The initial conditions for ε =.8 and numerical results at T = calculated with =/, = /8 are shown in Figure 7. Numerical tests show that a similar CFL condition is required as for the one-dimensional case. When ε =.5 at time T =, the numerical results with an unresolved mesh =/, = /8 and a resolved mesh = /8, = /3 are displaed in Figure 8. We can see that the results using the coarse mesh are much smoother than the one using the refined mesh. In this eample the amplitude deca due to numerical diffusion cannot be ignored. When a coarse mesh is used, the first order method is known to have dissipation. This is mainl due to the numerical diffusion term, which smoothes out the solution. This phenomenon not onl happens when ε is small but also when ε is O(). We can also see from Figure 8 that when ε is small, D p ε + D p ε is close to. As a comparison, the numerical solutions of the incompressible limit (4) with and without numerical viscosit are shown in Figure 9. The latter is obtained b a difference method based on a staggered grid configuration [3]. This staggered difference method is attractive for incompressible flows, since no artificial terms 4

25 Figure 6: Eample. When ε =., the and momentum of the LD scheme at different times are represented: a) T =.; b) T =.; c) T =.4; momentum a) momentum b) momentum c) momentum d) momentum e)

26 are needed to obtain stabilit and suppress the oscillations. Because of the stable pressure-velocit coupling, solutions with almost no viscosit can be obtained. The viscosit introduced here is of the form A where A is given b (36). We can see that the amplitude of the wave decas as time evolves even though the viscosit is onl O( ). In the limit of ε, (4) generates a discretization of the incompressible limit with O( ) numerical diffusion terms. This is wh the results for = /, = /8 in Figure 8 are close to those with viscosit in Figure 9. When the meshes are refined, less diffusion is introduced and the solution becomes closer to the solution with no viscosit. The scheme indeed catches the incompressible Euler limit and good numerical approimations can be obtained without resolving ε, which confirms the AP propert that is proved in section 4. However we need to take care of the numerical diffusion when coarse meshes are used. One possible wa to improve this is to use less diffusive shock capturing schemes at first order or higher order schemes using the MUSCL strateg for instance [8, 9, ], or to use staggered grid discretizations. 7 Conclusion We propose an all speed scheme for the Isentropic Euler equation. The ke idea is the semi-implicit time discretization, in which the low Mach number stiff pressure term is divided into two parts, one being treated eplicitl and the other one implicitl. Moreover, the flu of the equation is also treated implicitl. The parameter which tunes the eplicit-implicit decomposition of the pressure term allows to suppress the nonphsical oscillations. The numerical results show that the oscillations around shocks of O(ε ) strength can be suppressed b choosing α =. The low Mach number limit of the time semi-discrete scheme becomes an elliptic equation for the pressure term, so that the becomes a constant when ε. In this wa, the incompressible propert is recovered in the limit ε. Implemented with proper space discretizations, we can propose an AP scheme which can capture the incompressible limit without the need for, to resolve ε. In this paper we demonstrate the potential of this idea b using the first order La-Friedrich scheme with local evaluation of the wave speeds. Though this first order method is quite dissipative, we can observe that the scheme is stable independentl of ε and that the CFL condition is = O( ) uniforml in ε. It can also capture the right incompressible limit without resolving the mach number. Higher order space discretizations like the MUSCL method [8,9,] can be built 6

27 a) p p b) p p c) Figure 7: Eample 3. When ε =.8, the initial and momentum (left) and the numerical result with = /, = /8 at time T = (right) are represented. a) ρ ε ; b) p ε ; c) p ε. 7

28 a) b) p p c) p p d) Figure 8: Eample 3. When ε =.5, the numerical result with = /, = /8 (left) and = /8, = /38 (right) at time T = are represented respectivel. a): ρ ε ; b): D p ε + D p ε ; c): p ε ; d): p ε.

29 .5.5 u u a) u u b) Figure 9: Eample 3. The numerical results of the incompressible Euler limit using =/, = /8 with (left) and without (right) viscosit at time T = are represented. a): the first element of the velocit u ; b): the second element of the velocit u. 9

30 into this framework. This is the subject of current work. This paper provides a framework for the design of a class of all speed schemes. Compared with the ICE method [, 3] and some recent work b Jin, Liu and Hauck [], the idea is simpler and more natural. This framework can also be easil etended to the full Euler equation and flows with variable densities and temperatures. These etensions and applications [] will be the subject of future work. Acknowledgments This work was supported b the french Commissariat à l Energie Atomique (CEA) (Centre de Sacla) in the frame of the contract ASTRE, # SAV References [] H. Bijl, P. Wesseling, A unified method for computing incompressible and compressible flows in boundar-fitted coordinates, J. Comput. Phs., 4: 53-73, (998) [] M. P. Bonner, Compressible subsonic flow on a staggered grid, Master thesis, The Universit of British Columbia. (7) [3] P. Constantin, On the Euler equations of incompressible fluids, Bulletin of the American Mathematical Societ, Vol. 44, No. 4, 63-6, (7) [4] P. Crispel, P. Degond, M-H. Vignal, An asmptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit, J. Comput. Phs., 3, 8-34, (7) [5] P. Degond, F. Deluzet, A. Sangam, M-H. Vignal, An asmptotic preserving scheme for the Euler equations in a strong magnetic field, J. Comput. Phs, Vol. 8, No., , (9) [6] P. Degond, S. Jin and J-G. Liu, Mach-number uniform asmptoticpreserving gauge schemes for compressible flows, Bulletin of the Institute of Mathematics, Academia Sinica, New Series,, No. 4, 85-89, (7) 3

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