SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS. Min Tang. (Communicated by the associate editor name)

Transcription

1 Manuscript submitted to AIMS Journals Volume X, Number X, XX 2X Website: pp. X XX SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS Min Tang Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, No. 8 Dong Chuan Road, Minhang, Shanghai 224, P. R. China (Communicated by the associate editor name) Abstract. Standard hyperbolic solvers for the compressible Euler equations cause increasing approimation errors and have severe stability requirement in the low Mach number regime. It is desired to design numerical schemes that are suitable for all Mach numbers. A second order in both space and time all speed method is developed in this paper, which is an improvement of the semi-implicit framework proposed in [5]. The second order time discretization is based on second order Runge-Kutta method combined with Crank-Nicolson with some implicit terms. This semidiscrete framework is crucial to obtain second order convergence, as well as maintain the asymptotic preserving (AP) property. The AP property indicates that the right limit can be captured in the low Mach number regime. For the space discretization, the pressure term in the equation is divided into two parts. Two subsystems are formed correspondingly, each using different space discretizations. One is discretized by Kurganov-Tadmor central scheme (KT), while the other one is reformulated into an elliptic equation. The proper subsystem division varies with time and the scheme becomes eplicit when the time step is small enough. Compared with previous semi-implicit method, this framework is simpler and natural, with only two linear elliptic equations needed to be solved for each time step. It maintains the AP property of the first order method in [5], improves accuracy and reduces the diffusivity significantly.. Introduction. One of the main challenge of fluid simulations is when the compressible and weakly compressible regions occur simultaneously in the computational domain. To calculate weakly compressible flows with based compressible formulations, the efficiency and accuracy highly depend on the Mach number. In the low Mach number regime, the computational costs of standard compressible methods increase significantly due to the control of numerical diffusion and scheme stability requirements. 2 Mathematics Subject Classification. Primary: 65M6,65Z5; Secondary: 76L5. Key words and phrases. Mach number; isentropic Euler equations; compressible flow; incompressible Euler equations; incompressible limit; asymptotic preserving; Kurganov-Tadmor central scheme; Runge-Kutta method; Crank-Nicolson method. The author would like to thank Prof. Pierre Degond for useful suggestions and corrections; the anonymous referees for valuable comments.

2 2 MIN TANG In this paper we discuss a second order improvement of the first order all speed scheme proposed in [5]. The semi-discrete framework for the isentropic Euler equations proposed in [5] is accurate and efficient for all Mach numbers. However, due to the first order Rusanov space discretization, too much numerical diffusion is introduced, no matter whether the Mach number is small or not. In this paper, an all speed method that is second order in both space and time is developed. Here the word all speed indicates that the scheme is applicable for all Mach numbers, ranging from very small to order one values, and its stability and accuracy are independent of ϵ. Following [5], we use isentropic Euler equations as a laboratory model. This is because the design of an all speed scheme is mainly a mathematical and numerical issue. Though isentropic flows occur only when the changes of flow variables are small and gradual, numerically the equations themselves carry similar mathematical difficulties and properties as the full Euler system. They ehibit both finite and small Mach number regimes, have shocks involved when the Mach number is finite and possess the incompressible limit. Thus we use this simplified model to eplain and test the basic scheme designing ideas, for AP and for second order convergence, even at the price of a less physically realistic description. This framework is not restricted to the isentropic Euler equation. Its etension to the full Navier-Stokes equations for practical simulations is on going. Mathematically the basic ideas of scheme designing for achieving AP and second order properties are similar, but they involves more equations and compleity [2]. The isentropic Euler equations are t ρ + (ρu) =, ( ) t (ρu) + ρu u + p =. Depending on the physical problem, the equation of state usually takes the form () p(ρ) = Λρ γ, (2) where Λ, γ are constants. Let, t, ρ, ũ and p be the characteristic scales of the variables in the equations. Then after rescaling and using the fact that ũ = / t, () becomes t ρ ϵ + (ρ ϵ u ϵ ) =, ) t (ρ ϵ u ϵ ) + (ρ ϵ u ϵ u ϵ + ϵ 2 p ϵ =, where ρ ϵ, u ϵ, p ϵ are the resclaed variables and ϵ is the rescaled Mach number that is given by ϵ 2 = ρũ2 p. If the state equation is as in (2), the rescaled state equation is (3) p ϵ = Λ ργ p ργ ϵ = Λ ϵ ρ γ ϵ. (4) Physically, the sound speed is c 2 = γ p ρ, then ϵ = γ ũ c which indicates that the Mach number is the ratio of the flow velocity to the sound speed. From now on, we focus ourselves on the nondimensionalized form (3) and (4). However mathematically, the scheme itself is not restricted to this specific equation of state. The computational costs are related to the Mach number scale. For instance, in p (ρ ϵ ) dimension one, the eigenvalues of the Jacobian of (3) are u ϵ ± ϵ, which are the characteristic wave speeds of the flow[23]. To maintain stability and accuracy, standard hyperbolic solvers, like La-Friedrich method, Kurganov and Tadmor central

3 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS 3 scheme (KT) [9, 2], Polynomial upwind scheme [4] require, t to be smaller than ϵ. This requirement makes the costs even more epensive in higher dimensions. In [6, 7], Klainerman and Majda gave the limit of (3) when ϵ. The coupling between and pressure vanishes in the limit and the solution satisfies the incompressible Euler equations: ρ = ρ, u =, t u + (u u ) + p 2 =. Here p 2 is a scalar dynamic pressure depending on u and its subscript 2 is from the asymptotic epansion that will be developed in section 2. The difficulty of designing all speed methods is to fulfill the following two requirements simultaneously: on the one hand, capture the shocks which can develop for finite Mach numbers and on the other hand, have uniform stability and accuracy when the Mach number tends to zero. In order to meet these two somehow antagonist demands, people use various strategies in the literature. Most of them can be grouped into two classes: one class collects segregated pressure based methods that start from the incompressible limit, such as the Implicit Continuous-fluid Eulerian technique (ICE) [3, 2, ], trying to adapt incompressible techniques on the staggered meshes towards modeling compressible flow [4, 5, 26]. The other class collects based methods which start from the compressible equation, and use specific treatments on the pressure to capture the incompressible limit, like the semi-implicit method in [], the multiple pressure variable method in [24, 25], the pressure correction method [4, 28, 29] etc.. Also there are some other specific procedures to improve the accuracy at low Mach number regime [3, 3]. In this paper, we start from a shock capturing method for the compressible equation and fulfill the other requirements by proving that the scheme is asymptotic preserving (AP). This means that when, t resolve ϵ, it is a discretization of (3) and when ϵ goes to zero in the fully discrete scheme, it becomes a discretization of the incompressible limit (5). The AP property guarantees the accuracy and efficiency in the incompressible regime such that it can capture the incompressible limit with unresolved, t [8]. The concept of an AP scheme was first proposed by S. Jin in 999 [9]. It is proved useful for designing numerical schemes involving different scales in many different contets. The semi-implicit method proposed in [5] is simple, natural and proved to be AP. Let the time step be t, t n = n t, n =,,. The n superscripts denote the approimations at time t n. The time semi-discrete scheme is ϵ ρ n ϵ ) + (ρ ϵ u ϵ ) n+ =, t (ρn+ t ((ρ ϵu ϵ ) n+ (ρ ϵ u ϵ ) n ) + (ρ n ϵ u n ϵ u n ϵ + αp(ρ n ϵ )) + ϵ 2 ( αϵ2 )p(ρ n+ ϵ ) =. where α /ϵ 2 is a positive ad-hoc parameter. That the pressure term is divided into two parts, one being eplicit and the other implicit, is to prepare for different space discretizations. Compared with previous semi-implicit methods, firstly, nonphysical oscillations can be suppressed by choosing proper α; besides, linearizing the implicit pressure term as in (2) reduces much computational cost, for which (5) (6)

4 4 MIN TANG only one linear elliptic equation needs to be solved implicitly in each time step [5]. Comparing the efficiency with pressure correction method in [4], the cost is the same as solving the pressure correction equation in [4], while the eplicit parts are much simpler. Only the simplest Rusanov space discretization is tested in [5]. The main defect of this first order Rusanov method is that there is too much diffusion, no matter what the Mach number is. It is desirable to develop higher order schemes that can maintain the same AP property. This is the main purpose of this paper. A new framework that is second order in time is proposed, in which the specific implicit treatments are crucial to obtain second order convergence, as well as the AP property. We will also investigate the nonphysical oscillations and the choice of α. The second order time discretization is based on a second order Runge-Kutta method combined with a Crank-Nicolson method. This semi-discrete time discretization maintains the AP property so that it can capture the right limit. For the space discretization, similarly to [5], we divide the pressure term in the equations into two parts and obtain two subsystems. Different space discretizations are used for these two subsystems: one is discretized by Kurganov and Tadmor central scheme (KT) [9, 2], and the other one is reformulated into an elliptic equation. These developments improve the accuracy and reduce the diffusions significantly. The organization of this paper is as follows. Section 2 gives the semi-implicit framework of time discretizations that are second order and AP. The detailed second order space discretizations in both one and two dimensions and their AP properties are illustrated in section 3. In section 4, we discuss the nonphysical oscillations and investigate some shock tests with non-well prepared initial conditions. The efficiency of our scheme in both compressible and incompressible regimes is demonstrated by four numerical eamples in section 5. Finally, we conclude the paper in section Second order time-discrete scheme. In order to obtain second order convergence in time, inspired by [25], we use a specific combination of second order Runge-Kutta method and Crank-Nicolson method. In the subsequent part, we denote it by the acronym RK2CN. To maintain the AP property, the minimum requirement is to put part of the flu and part of the pressure term implicit. The details of the second order semi-discrete scheme are as follows: Firstly find ρ n+ 2 ϵ, q n+ 2 ϵ from 2 2 ϵ ρ n ϵ ) + q n+ 2 ϵ =, t (ρn+ 2 t (qn+ 2 ϵ q n ϵ ) + (ρ n ϵ u n ϵ u n ϵ + αp(ρ n ϵ )) + ϵ 2 ( αϵ2 ) p(ρ n+ 2 ϵ ) = ; Then update ρ ϵ, q ϵ by ϵ ρ n ϵ ) + 2 (q n ϵ + q n+ ) ϵ =, where t (ρn+ t (qn+ ϵ q n ϵ ) + (ρ ϵ u ϵ u ϵ + αp(ρ ϵ )) n ϵ 2 ( αϵ2 ) ( p(ρ n ϵ ) + p(ρ n+ ϵ ) ) =, q ϵ = ρ ϵ u ϵ. (7) (8)

5 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS 5 The first step (7) is eactly the same as (6) but with a time step t/2. Let us recall how (6) in [5] is simplified and linearized: Firstly, by introducing q ϵ = ρ ϵ u ϵ, (6) can be written as t (ρn+ ϵ ρ n ϵ ) + q n+ ϵ =, (9a) t (qn+ ϵ q n ϵ ) + (ρ n ϵ u n ϵ u n ϵ + αp(ρ n ϵ )) + ϵ 2 ( αϵ2 ) p(ρ n+ ϵ ) =. (9b) Reformulating the equation (9b) into ( ) q n+ ϵ = q n ϵ t ρ n ϵ u n ϵ u n ϵ + αp(ρ n ϵ ) t ϵ 2 ( αϵ2 ) p(ρ n+ ϵ ) () and substituting it into the equation (9a), one gets ρ n+ ϵ t2 ϵ 2 ( αϵ2 ) p(ρ n+ ϵ ) =ρ n ϵ t q n ϵ + t 2 ( ρ n ϵ u n ϵ u n ϵ + αp(ρ n ϵ ) ). This is a scalar elliptic equation that is easy to solve numerically. To reduce the computational cost, we can linearize and approimate p(ρ ϵ ) by ( ) p (ρ n ϵ ) ρ n+ ϵ. Then () becomes a linear elliptic equation for ρ n+ ϵ : ρ n+ ϵ t2 ϵ 2 ( αϵ2 ) (p (ρ n ϵ ) ρ n+ ) ϵ =ρ n ϵ t q n ϵ + t 2 ( ρ n ϵ u n ϵ u n ϵ + αp(ρ n ϵ ) ). When programming, we first solve the linear elliptic equation (2) to get ρ n+ ϵ and then update q ϵ by plugging ρ n+ ϵ into (). This is a linearized semi-discrete framework. One can simplify the calculation of (7); (8) by using the same strategy. That is, for the two sub-steps (7) and (8) we reformulate them into elliptic equations for ρ n+/2 ϵ and ρ n+ ϵ respectively, linearize the implicit pressure term and form two new sub-steps with their corresponding equations. The details are omitted here. AP property of the semi-discrete scheme. To investigate the AP property of (7), (8) we only consider periodic boundary conditions, while etensions to other boundary conditions are similar and straight forward. By introducing the epansions ρ n ϵ = ρ n () + ϵ2 ρ n (2) +, un ϵ = u n () + ϵ2 u n (2) +, (3) we are going to show that in the limit ϵ, (7), (8) becomes a semi-discretization of (5). Let us start from (6). The AP proof in [5] is for (), (9b), i.e. the semi-discrete scheme before linearization. Here we give the AP proof for (2) (9b). Assuming that ρ() n is independent of space and un () =, we look at ρn+ () and u n+ (). By substituting the epansions (3) into (2) and equating the O(/ϵ 2 ) terms, one gets (p (ρ n () ) ρn+ () ) =. That ρn () is a constant in space gives ρn+ () =. Together with the periodic boundary conditions, ρ n+ () is independent of space. Moreover, the conservation of indicates () (2) ρ n+ () = ρ n (). (4a)

6 6 MIN TANG The O() terms of (2) are ρ n+ () t 2 (p (ρ n () ) ρn+ (2) + p (ρ n (2) ) ρn+ () ) = ρ n () t (ρn () un () ) + t2 (ρ n () un () un () + αp(ρn () )). Thus from (4a) and u n () =, we get = p (ρ n () ) ρn+ (2) + (ρ n () un () un () + αp(ρn ())). (4b) The O() terms of (9b) are t (ρn+ () un+ () ρ n () un () ) + (ρn () un () un () + αp(ρn () )) + p (ρ n+ () ) ρn+ (2) =, (4c) where we have used p(ρ n+ ϵ ) = ( p ( ρ n+ () + ϵ 2 ρ n+ (2) + o(ϵ 2 ) )) = p(ρ n+ () ) + ϵ2 ( p (ρ n+ ) () )ρn+ (2) + o(ϵ 2 ) = ϵ 2 p (ρ n+ () ) ρn+ (2) + o(ϵ 2 ). Comparing the divergence of (4c) and (4b) and using (4a), we have u n+ () = t (un+ () ) () u n () ) + (un () un () ) + p (ρ n+ ρ n+ () ρ n+ (2) =, which is a discretization of (5) where u n (), p (ρ n () ) ρ ρ n n (2) are respectively approimations of u, p 2 at time t n. () Let us go back to the second order scheme (7), (8). We can do the same linearization simplification for (7) as in (2) but with a time step t/2. Therefore, when ϵ, ρ n+ 2 ϵ, u n+ 2 ϵ satisfy ρ n+ 2 () = ρ n (), u n+ 2 () =, 2 2 () u n () ) + (u () u () ) n + p n+ 2 (2) =, t (un+ Similarly, the second step (8) can be reformulated into (5) with ρ n+ ϵ t2 4ϵ 2 ( αϵ2 ) (p (ρ n ϵ ) ρ n+ ) ϵ = ϕ(ρ n ϵ, q n ϵ, ρ n+ 2 ϵ, q n+ 2 ϵ ), (6) ϕ(ρ n ϵ, q n ϵ, ρ n+ 2 ϵ, q n+ 2 ϵ ) =ρ n ϵ t q n ϵ + t2 2 (ρ ϵu ϵ u ϵ + αp(ρ ϵ )) n+ 2 + t 2 4ϵ 2 ( αϵ2 ) p(ρ n ϵ ). By substituting the epansions (3) into (6) and equating the terms of O(/ϵ 2 ), O() in (6) and O() in the equation of (8) one gets u n+ () =, t (un+ () u n () ) + (u () u () ) n ( p n (2) + ) (7) pn+ (2) =.

7 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS 7 p n+ 2 (2) and 2 ( p n (2) + ) pn+ (2) can be viewed as the lagrange multiplier for the incompressible constraints. Thus (5), (7) is a second order Rouge-Kutta (RK) discretization of (5) and our scheme (7), (8) is AP. This new scheme maintains the AP property, as well as the second order convergence for both finite and low Mach numbers. 3. Second order Space discretization. Model (3) is a nonlinear hyperbolic system. Due to the nonlinearity, even if the initial conditions are smooth, shocks develop as time goes on [23]. We need to capture the shocks as well as maintain the AP property. In this section we use (9a), (9b) to discuss the strategy of space discretizations. The etension to (7); (8) is straight forward. Numerical diffusion is needed to stabilize the system when there are discontinuities [23]. If we do a Taylor epansion, a wide range of numerical schemes are designed to approimate (9) by t (ρn+ ϵ ρ n ϵ ) + q n+ ϵ = β Dρ n ϵ, t (qn+ ϵ q n ϵ ) + ( ρ n ϵ u n ϵ u n ϵ + αp(ρ n ϵ ) ) + ϵ 2 ( αϵ2 ) p(ρ n+ ϵ ) = β 2 Dq n ϵ, where Dρ ϵ, Dq ϵ represent the diffusion terms and the diffusion coefficients β, β 2 should be small to maintain consistency. Here Dq ϵ = (Dq ϵ, Dq ϵ2 ) T in two dimensions and β, β 2 depend on the specific hyperbolic solver we consider. For instance, the first order Rusanov scheme used in [5] satisfies β, β 2 = O( ) and, for the second order KT scheme, β, β 2 = O( 2 ) [9, 2]. The inefficiency of standard techniques in the low Mach number regime is due to the fast wave speed, which requires large β, β 2. There are two antagonist demands: we need β, β 2 to be big enough to stabilize the scheme; at the same time, for better accuracy and CFL condition, we want them to be small. The balance is tuned by introducing a new parameter α. Consider (8) as two subsystems: t (ρn+ ϵ ρ n ϵ ) + q n ϵ = β Dρ n ϵ, t (qn+ ϵ q n ϵ ) + ( ρ n ϵ u n ϵ u n ϵ + αp(ρ n ϵ ) ) = β 2 Dq n ϵ, t (ρn+ t (qn+ ϵ ρ n ϵ ) + q n+ ϵ =, ϵ q n ϵ ) + ϵ 2 ( αϵ2 ) p(ρ n+ ϵ ) =, The first subsystem is hyperbolic while the second one is a nonlinear wave equation for ρ. When calculating β, β 2 according to (9), i.e. determining the shock speeds in the shock capturing methods by the eigenvalues of the Jacobian of (9), it is likely that both subsystem are stable. At the same time, β, β 2 remain small that are independent of ϵ. The two antagonist demands are fulfilled simultaneously. However, the nonphysical oscillations that are introduced by the discontinuity in the wave equation (2) can break down the stability as well as destroy the accuracy. Proper choices of α can diminish the effect of nonphysical oscillations, and keep the demands fulfilled. (8) (9) (2)

8 8 MIN TANG Taking into account the numerical diffusions, the right hand side of (2) becomes ϕ(ρ n ϵ, u n ϵ ) = ρ n ϵ + t 2 ( (ρ n ϵ u n ϵ u n ϵ +αp(ρ n ϵ )) β 2 Dq n ϵ ) t ( (ρϵ u ϵ ) n β Dρ n ϵ ). (2) To design higher order scheme, we only need to use the flues of any high order shock capturing method for (9) get the approimations of ϕ(ρ n ϵ, u n ϵ ) according to (2), and then, discretize the left hand side of (2) by high order finite difference method. Specifically, we use the second order KT scheme [9, 2] for the flues and central finite differences for the elliptic operator. Moreover, to avoid alternating between two staggered grids, regular, non-staggered meshes are used according to [22]. 3.. One dimensional scheme. In one dimension, assume that the computational domain is Ω = [, ], = /N and the grid points are j = j, j =,,, N. Let ρ ϵj, q ϵj be the approimations of ρ ϵ ( j ), q ϵ ( j ) respectively and r represent either q ϵ or ρ ϵ. Assuming that g is a function of ρ ϵ, q ϵ, we define the numerical flues by ( Fr g(ρ ϵ, q ϵ ) ) = ( ) (g(ρ R j+ 2 2 ϵ,j+, q R 2 ϵ,j+ ) + g(ρ L 2 ϵ,j+, q L 2 ϵ,j+ )) λ 2 j+ (rr 2 j+ r L 2 j+ ), 2 with j =,, N. Here λ j+ = ma { qϵj n 2 + αp (ρ n j ), qϵ,j+ n + αp (ρ n j+ )} ρ n ϵj ρ n ϵ,j+ is the discrete local propagation speed. This speed is given by the absolute value of the eigenvalue of the Jacobian of the flu at the discrete level. Moreover, rj+/2 R = r j+ 2 (r ) j+, rj+/2 L = r j + 2 (r ) j, with ( (r ) j = minmod (r j r j ), (r j+ r j ) minmod(a, b) = [sgn(a) + sgn(b)] min( a, b ). 2 The full discretization of the scheme (2), () is as follows: where ρ n+ ϵj t2 ϵ 2 ( αϵ2 )D j (p (ρ n ϵ )D j ρ n+ ϵ ) = ϕ(ρ n ϵ, q n ϵ ), (22a) t (qn+ ϵj q n ϵj) + D j F qϵ ( ρ n ϵ (u n ϵ ) 2 + αp(ρ n ϵ ) ) ), + 2ϵ 2 ( αϵ2 ) ( D j+ + D ) 2 j p(ρ n+ ϵ ) =. (22b) 2 ϕ(ρ ϵ, q n ϵ ) = ρ n ϵ + t 2 D j D j ( ρ n ϵ (u n ϵ ) 2 + αp(ρ n ϵ ) ) td j F ρϵ (q n ϵ ) Dj r = (r j+/2 r j /2 ), D j+ r = 2 (r j+ r j ), D j r = 2 (r j r j ). and r j+/2 = ) rj+ + r j. 2(

9 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS 9 Here, we define D r because D j+ j D j = (D D ). 2 j+ 2 j 2 When ϵ = O(), (22a), (22b) is second order in space. Moreover, it is easy to check that 22a), (22b) is conservative with respect to both ρ ϵ and q ϵ. AP property of fully discrete scheme. The proof of the AP property is to show that in the limit ϵ, (22a), (22b) becomes a discretization of the incompressible limit (5). In one dimension with periodic boundary conditions, the solution of (5) is that ρ, u are constants in both space and time. Let α be O(). We prove the AP property by inserting the asymptotic epansions ρ n ϵj = ρ n () + ϵ2 ρ n (2)j +, qn ϵj = q n () + ϵ2 q n (2)j + (23) into (22a), (22b) and equate the same order of ϵ. We use ρ n ()c, qn ()c to indicate that the O() terms of the epansions are independent of space. The proof is divided into three parts: ρ n+ ()j is independent of j. The O(/ϵ2 ) terms of (22a) are Dj ( p (ρ n () )D j ρ n+ ) () =. (24) From the form of p(ρ ϵ ) in (4) and the fact that ρ n () is independent of space, (24) is equivalent to Dj D j ρn+ () =. Then, the periodic boundary condition gives that ρ n+ ()j is independent of j. Therefore, we can use ρn+ ()c to represent for all j. ρ n+ ()j ρ n+ ()c is independent of n. By summing up (22a) at all grid points and using the periodic boundary conditions, one gets the conservation of mass such that j ρ n+ ϵj = j ρ n ϵj. The O() terms are j ρn+ ()c = j ρn ()c, so that ρ n+ ()c = ρn ()c. (25) q n+ () is independent of space. Equating the O() terms of (22a), since u n = qϵ n /ρ n ϵ, using that ρ n ()c, qn ()c are constants and that ρn+ ()c = ρn ()c, we have Dj ( p (ρ n ()c )D j ρ n+ ) (2) =. Then from the periodic boundary conditions, we get ρ n+ (2)j = ρn+ (2)c, that is ρ n+ (2) is also independent of space. Now, we substitute the epansion (23) into the equation (22b) and equate the O() terms. Then the O() term of the is constant such that: q n+ ()j = qn ()j = qn ()c. (26) In summary, with the periodic boundary conditions, when ϵ in (22a), (22b) both and are independent of j and n. (25), (26) is a discretization of (5) in one space dimension and the fully discrete scheme is AP.

10 MIN TANG 3.2. Two dimensional scheme. For the two dimensional case, we assume that the computational domain is Ω = [, ] [, ] and the mesh grids are Let ( i, y j ) = (i, j y), i =,,, M; j =,,, N. U = (ρ ϵ, q ϵ, q ϵ2 ) T, Q = ϵ 2 ( αϵ2 )(, p(ρ ϵ ), y p(ρ ϵ )) T q ϵ q ϵ2 G (U) = ρ ϵ u 2 ϵ() + αp(ρ ϵ), G 2 (U) = ρ ϵ u ϵ u ϵ2. (27) ρ ϵ u ϵ u ϵ2 ρ ϵ u 2 ϵ2 + αp(ρ ϵ ) System (3) can be written as t U + G (U) + y G 2 (U) + QU =. Thus the discretization in two dimensions is as follows ( ) ρ n+ ϵi,j t2 ϵ 2 ( αϵ2 ) Di,j(p (ρ n ϵ )Di,jρ n+ ϵ ) + D y i,j (p (ρ n ϵ )D y i,j ρn+ ϵ ) = ϕ(ρ n ϵ, qϵ, n qϵ2), n (28a) t (qn+ ϵi,j qn ϵi,j) + Di,jF q ( ϵ ρ n ϵ (u n ϵ) 2 + αp(ρ n ϵ ) ) + D y i,j F q y ( ϵ ρ n ϵ u n ϵu n ) ϵ2 + ( αϵ 2 )( 2ϵ 2 Di+/2,j + ) D i /2,j p(ρ n+ ϵ ) =, (28b) t (qn+ ϵ2i,j qn ϵ2i,j) + Di,jF q ( ϵ2 ρ n ϵ u n ϵu n ) ϵ2 + D y i,j F q y ( ϵ2 ρ n ϵ (u n ϵ2) 2 Here +αp(ρ n ϵ ) ) + 2ϵ 2 ( αϵ 2 )( D y i,j+/2 + Dy i,j /2 ϕ(ρ n ϵ, qϵ, n qϵ2) n =ρ n ϵi,j t ( Di,jF ρ ϵ (qϵ) n + D y i,j F ρ y ϵ (qϵ2) n ) + t 2( Di,jD i,j( ρ n ϵ (u n ϵ) 2 + αp(ρ n ϵ ) ) + Di,jD i,j( y ρ n ϵ u n ϵu n ) ϵ2 + D y i,j D i,j(ρ n ϵ u n ϵu n ϵ2) + D y i,j Dy i,j( ρ n ϵ (u n ϵ2) 2 + αp(ρ n ϵ ) )), ) p(ρ n+ ϵ ) =, (28c) D i,ju = (u i+/2,j u i /2,j ), D y i,j u = y (u i,j+/2 u i,j /2 ), Di+/2,j u = (u i+,j u i,j ), D y i,j+/2 u = y (u i,j+ u i,j ), and the other operators are similar as in the one dimensional discretization (22a), (22b) such that ( ) Fr g(ρ ϵ, q ϵ, q ϵ2 ) = 2 (g(ρ R ϵ, qϵ R ) + g(ρ L ϵ, qϵ L )) λ (r R r L ), ( ) Fr y g(ρ ϵ, q ϵ, q ϵ2 ) = 2 (g(ρ B ϵ, qϵ B ) + g(ρ T ϵ, qϵ T )) λ y (r T r B ), (29) with u R i+/2,j = u i+,j 2 (u ) i+,j, u L i+/2,j = u i,j + 2 (u ) i,j, u T i,j+/2 = u i,j+ y 2 (u y) i,j+, u B i,j+/2 = u i,j + y 2 (u y) i,j, (u ) i,j = minmod ( (u i,j u i,j ), (u i+,j u i,j ) ), (u y ) i,j = minmod ( y (u i,j u i,j ), y (u i,j+ u i,j ) ).

11 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS The local propagation speed now is λ i+/2,j = ma { u n ϵi,j + αp (ρ n i,j ), un ϵi+,j + αp (ρ n i+,j )}, λ y i,j+/2 = ma { u n ϵ2i,j + αp (ρ n i,j ), un ϵ2i,j+ + αp (ρ n i,j+ )}. The proof of the AP property is similar to that of the first order scheme in [5]. The key idea is that, based on the epansions as in (23), we equate the same order of ϵ in (28a), (28b), (28c) and obtain a discretization of (5). The details are omitted here. Remark. Numerically, it is important to discretize ϕ as in (29) and use 2 (D i+/2,j + Di /2,j ) and 2 (D i,j+/2 +D i,j /2 ) in the discretization in (28b), (28c). This is motivated by full discretization for the incompressible Euler equations. The incompressible Euler equation itself possesses some numerical difficulties, for instance, some numerical techniques require artificial viscosity for the elimination of stagnation fluctuations, whereas numerical viscosity is not needed in the incompressible regime [2, 3]. We choose this specific space discretization to avoid the zigzag pattern and maintain the convergence order. 4. Nonphysical oscillations. The AP property guarantees that, for well prepared initial conditions, i.e. with (3) satisfied initially, ρ n () a constant in space and u () =, our scheme can capture the incompressible limit with unresolved meshes. Because strong shocks are not easy to generate in weakly compressible flow, we can focus ourselves on well prepared initial conditions when the Mach number is small. However, in this section, we are interested in the scheme performances for non-well prepared initial data. When shocks develop in the low Mach number regime, numerically, some nonphysical oscillations will appear [5, ]. The differences between our scheme and the standard KT scheme are that: )it has smaller diffusion coefficients β, β 2 (recall (8)); 2) the stiff terms are treated implicitly. These are the two possible sources of the nonphysical oscillations. If we choose α as in (32), the nonphysical oscillations can be diminished by using smaller t. The effect of implicitness. For the first order time discretization, the implicit parts behave like diffusion and can introduce additional numerical viscosity to suppress the oscillations [5]. However, this is not the case for the second order time discretization. In one dimension, assume t ρ n ϵ + q n ϵ =, t q n ϵ + ( ρ n ϵ u n2 ϵ + ϵ 2 p(ρn ϵ ) ) =. When ρ k ϵ, qϵ k, k = n, n + 2, n + in (6) are smooth enough, using Taylor epansions, the semi-discrete time discretization (8) can be written as t ρ n ϵ + qϵ n t2 2 ( t ρ n ϵ u n2 ϵ + ϵ 2 p(ρn ϵ ) ) + O( t 3 ) =, t q n ϵ + ( ρ n ϵ u n2 ϵ + ϵ 2 p(ρn ϵ ) ) t2 24 ( tt ρ n ϵ u n2 ϵ 2 3αϵ2 ϵ 2 p(ρ n ϵ ) ) + O( t 3 ) =. Similarly to La-Wendroff and Beam-Warming method [23], these O( t 2 ) third order derivative terms cause dispersion, which leads to an oscillatory wave train at (3)

12 2 MIN TANG the discontinuity. When ϵ t, the amplitude of these oscillations can be reduced by using smaller time step. Whereas when ϵ t or in the transition regime t = O(ϵ), thanks to the uniform convergence with respect to ϵ that was proved in [8] for the neutron transport equation, the accuracy for well prepared initial data is guaranteed in the incompressible regime. It is reasonable to only consider well prepared initial conditions for the incompressible flow, since shocks can hardly form. However, in the subsequent shock tests, in order to see the effect of dispersion, we choose non-well prepared step functions initially. The choice of α. We need to choose α carefully to stabilize the scheme, as well as get better accuracy. Two demands should be met simultaneously. On the one hand, when ϵ = O() and shocks appear, the numerical viscosity should stabilize the scheme. On the other hand, when ϵ is small, it is not easy to generate shocks. Then the smooth or weakly discontinuous solutions require much less numerical viscosity. As discussed in [5], α is chosen to depend on t and varies with time. Though the analysis in [5] highly depends on the first order Rusanov scheme, the CFL conditions are the same for both first and second order methods. We can etend the methodology of choosing α to our present scheme. Heuristically, bigger α not only introduces more numerical viscosity, i.e. larger diffusion coefficients β, β 2 (recall (8) ), but also reduces the effect of dispersion. This is because numerical viscosity smoothes the solution and decreases the range of frequencies. Therefore, larger α is preferred as far as the scheme remains stable. Similarly to [5], we use the largest possible α that can guarantee stability. Due to the CFL condition, the time step should satisfy t σ ma Ω { u ϵ + αp (ρ ϵ ) }. (3) with Ω the computational domain and σ a constant. σ is the Courant number that is less than theoretically, and we will specify its value in the numerical tests. Then the stability can be guaranteed when t σ ma Ω { u ϵ } + α ma Ω { p (ρ ϵ )}. Thus the corresponding α should satisfy ( σ t α ma Ω{ u ϵ } ma Ω { p (ρ ϵ )} and correspondingly we choose α to be { ( σ α = min ϵ 2, t ma } Ω{ u ϵ } ) 2 ma Ω {. (32) p (ρ ϵ )} This choice of α is suitable in the following four aspects: i) It guarantees the numerical stability. ii) When ϵ = O(), α = ϵ, the scheme becomes eplicitness. As discussed in the 2 effect of implicity, this eplicit scheme is the same as the standard KT scheme ecept for some additional O( t 2 ) third order derivative terms. Moreover, it is conservative and can capture the shocks. In the incompressible regime, when ϵ is small, ( σ t α = ma Ω{ u ϵ } ) 2 ma Ω { p (ρ ϵ )} ) 2

13 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS 3 remains O(). The proofs of AP properties in section 2 and 3 guarantee the reliability of our scheme with unresolved meshes for well prepared initial conditions. In summary, by choosing α as in (32), the scheme meets the two demands in the introduction. iii) When ϵ is small and we start from non-well prepared initial data, (32) can no longer guarantee that the nonphysical oscillations will disappear. When t is fied but not small enough, the oscillations cannot be suppressed by choosing proper α: either there are oscillations when α is small, or the stability of the scheme breaks down when α is big. However, the oscillations can be diminished by using smaller t. As long as ( σ t α = ma Ω{ u ϵ } ) 2 ma Ω { < /ϵ 2, p (ρ ϵ )} smaller t gives bigger α, bigger β, β 2 (see (8)), more diffusion, and additionally from (3) smaller dispersion. All these effects can reduce the oscillations. iv) In particular, no matter if we are in the compressible or incompressible regime, when t is so small that α = /ϵ 2, the scheme becomes eplicit. This choice of α is natural, because our goal is to bypass the severe CFL constraint of the original hyperbolic system. But if t is small enough, the special semi-implicit treatment is not needed any more. Numerical tests for shocks. In the subsequent numerical shock tests for all range of Mach numbers, we confirm the choice of α and show the influence of smaller diffusion coefficients β, β 2 and the dispersive effects brought by the implicitness. Two shock tests are given to verify the previous analytical observations. For simplicity, let p(ρ ϵ ) = ρ ϵ. The two tests are ) La shock tube problem; 2) Sod shock tube problem, for which the incompressible limit cannot be maintained. The referred eact solutions are calculated by second order Runge-Kutta method (RK2) coupled with eplicit KT scheme, using a very fine mesh = /6, t = ϵ /6. In this part, each figure composed by four subplots depicts the and as function of : top left: ; top right: ; bottom left: amplification of the top left subplot for a better look at the oscillations; bottom right: amplification of the top right subplot for a better look at the oscillations. The initial conditions of the La shock tube are ρ L =.445, u L =.698, [, ] (33) ρ R =, u R =, (, ]. Firstly, for fied α =, we verify that the Courant number is approimately σ = numerically and the results are displayed in Table. Let σ =.8 and α be chosen as in (32). Since the computational domain is [, ] and the shock velocity is O(/ϵ), to keep the shocks away from the boundaries, the computational time should be O(ϵ). Therefore, taking ϵ =.3 and T =.5 as an illustrative instance, we show the scheme performance in the compressible regime. In Figure, the initial and and numerical results at T =.5 are displayed as function of. Comparing the results of = /2, t = /3 and = /2, t = /6, we can see that although there are oscillations at the shock location, the numerical

14 4 MIN TANG ϵ ma{ u ϵ + } t ma{ u ϵ + } t.3.69 /4 / /8 / /6 / /4 / /8 / /6 /45.5 Table. La tube problem. When we fi α =, for both ϵ =.3 at T =. and ϵ =.3 at T =., to guarantee stability, the required t for different are displayed, so are the corresponding Courant number. simulations are stable. These wave like oscillations are due to dispersion, which can be diminished when t becomes small. To look at the numerical results with unresolved meshes for non-well prepared initial conditions in the incompressible regime, we test a very small ϵ = 4. Since the shock velocity is O( 4 ) due to the small ϵ, the computational domain [, ] is too small for unresolved time steps. We consider a bigger domain [, 8] with similar initial conditions ρ L =.445, u L =.698, [, 4] ρ R =, u R =, (4, 8]. The numerical results at T = 2 4 with = /, t = 5 6 and = /, t = 2 6 are displayed in Figure 2. Even if there are a lot of oscillations, the shock can yet be captured. Numerically does not necessarily resolve ϵ, while for non-well prepared initial conditions, t has to be at the scale of ϵ to get non-oscillating solutions. The initial and velocity of the Sod tube problem are ρ L =., u L =, [, ], ρ R =.25, u R =, (, ]. (34) The numerical results are presented in Figure 3 with α determined by (32) and Courant number σ =.8. For both ϵ =.3 and.3, heads of the rarefaction wave are moving to the left and shock fronts are moving to the right, which is in accordance with the full Euler simulations. For fied α, σ is found numerically in the La tube problem. The question is whether σ depends on α, or in other words, if there eists a uniform Courant number that guarantees the stability independently of α. In this test, we try to find numerically this minimum σ for different α. Firstly, we choose different t that satisfy t < σ ma u ϵ, where σ is the Courant number when α = which can be found numerically as in the La tube problem. Then for each t, we search the biggest possible α (less than /ϵ 2 ) that allows the scheme to be stable. For this biggest possible α, we determine σ α by the formula: t = σ α ma u ϵ + α. (35)

15 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS a) b) Figure. The La shock tube problem for ϵ =.3. a): The initial (top) and (bottom); b): When T =.5, the (left) and (right) calculated by RK2CN with α as in (32). The bottom left and right subplots are respectively the zoom out of the top left and top right subplots. Here the dashed, dotted and solid lines represent the numerical results of = /2, t = /3 and = /2, t = /6 and the eact solution respectively. Thus the corresponding σ α can be determined by t and α. In Table 2, for fied = /2 and different t, we find the biggest α that can stabilize the scheme and then σ α according to (35). For both ϵ =.3 and.3, different α give almost the same σ α. Therefore, it seems that there eists a uniform

16 6 MIN TANG Figure 2. La shock tube problem with bigger computational domain when ϵ = 4. The bottom left and right subplots are respectively the zoom out of the top left and top right subplots. The and at T = 2 4, calculated with unresolved mesh = /, t = 5 6 and = /, t = 2 6 are shown by dashed and dash dotted lines respectively, while the solid line is the eact solution given by eplicit KT with very fine mesh. Courant number which is approimately numerically and is the same as for the eplicit hyperbolic solver. ϵ ma,t { u ϵ } t α σ / / / / /4 /ϵ / / / / /4 /ϵ 2 Table 2. Sod tube problem. At T =.5, for ϵ =.3 and at T =.5 when ϵ =.3, to guarantee stability, the required σ for different t are presented. Here = /2 for all the calculations and when t = /4 for ϵ =.3 and t = /4 for.3, the allowed α becomes /ϵ 2 and so σ is no longer determined by α

17 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS a) b) Figure 3. RK2CN for Sod shock tube with α as in (32). The eact solutions are given by solid lines. a): When T =.5, ϵ =.3, the and calculated with = /2, t = / and = /2, t = /2 are represented by dashed and dash dotted lines respectively; b): The numerical results of T =.5, ϵ =.3 calculated with = /2, t = / and = /2, t = /2 are displayed by dashed and dash dotted lines respectively. For fied α, it is easy to see that our proposed scheme is second order in both space and time. Moreover, in the same way as for the eplicit KT [9, 2], its L convergence order reduces to when shocks appear. The question is when α depends on t and flow velocity u ϵ as in (32), and consequently, α

18 8 MIN TANG varies with time, does the scheme maintain the same convergence order? The answer is yes and we can see this from Table 3, in which the numerical errors calculated with different, t are given. Here we use σ =.8 and the discrete L norms of the relative errors defined by M j E(R) L = R j r( j ) M e j r(, j) are presented. R is the numerical solution and r is the eact solution obtained on a very fine mesh. M, M e are respectively the numbers of grid points of the numerical and eact solutions. In Table 3, because ϵ =.3 and.3 correspond to different values of CFL conditions, the same requires different t. Additionally, we can check numerically that when is fied, simply decreasing t can suppress the oscillations but not improve the accuracy. ϵ t E(ρ ϵ ) E(q ϵ ).3 /5 / / / /2 / /4 / /5 / / / /2 / /4 / Table 3. Sod tube problem. With α being chosen as in (32), the L error of the ρ ϵ and q ϵ are displayed. The errors are measured at T =.5 for ϵ =.3 and at T =.5 for ϵ =.3. When is fied, decreasing t cannot improve the accuracy and it is obvious that the convergence order here is. Here and for the subsequent part of this paper, as far as the CFL condition is satisfied, we show the convergence order for fied t/. This is due to the two numerical observations: ) If is fied, simply reducing t cannot improve the accuracy, 2) the CFL condition indicates that t must be at O( ), i.e. the scheme is not stable when using small but big t. 5. Numerical Results. In this section, we present the performances of our second order all speed scheme. Four eamples are given. The first two ones are one dimensional tests, one for comparison of first and second order time discretizations when shocks evolve, while the other one is a smooth case of collision of two acoustic waves. Then two, two dimensional eamples are also presented. One is to show that when ϵ, the scheme can capture the incompressible Euler limit, and the other one is a more realistic simulation of a Gresho vorte [6, 7]. In all these calculations, α is determined by (32) with σ =.8. Eample : In this eample, the convergences of the first and second order time discretizations (9a), (9b); (7), (8) (coupled with the same second order space

19 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS 9 First Order t = O( ) ϵ t E ρ E q.3 /5. / /. / /2. / /4. / /5. / /. / /2. / /4. / Second Order t=o( ) ϵ t 2 Eρ 2 Eq 2.3 /5.*/ /.*/ /2.*/ /4.*/ /5.*/ /.*/ /2.*/ /4.*/ Table 4. Eample. When T =., for both ϵ =.3 and.3, the L norms of the numerical errors calculated by the second order time discretization. We can see the first order convergence when ϵ =.3. When ϵ =.3, all errors are smaller than. discretizations) are investigated. Let p(ρ ϵ ) = ρ ϵ and the initial conditions be: ρ ϵ (, ) =, p ϵ (, ) = ϵ 2 /2 [,.2] [.8, ]; ρ ϵ (, ) = + ϵ 2, p ϵ (, ) = (.2,.3]; ρ ϵ (, ) =, p ϵ (, ) = + ϵ 2 /2 (.3,.7] ρ ϵ (, ) = ϵ 2, p ϵ (, ) = (.7,.8]. To compare the first and second order time discretizations, with α as in (32), the numerical results at T =. calculated with =.5, t =.5. are presented in Figure 5. We can see that in the compressible regime ϵ =.3, both methods give similar results, while when ϵ =.3, the first order time discretization introduces more diffusion and RK2CN has severe oscillations. This verifies the analytical observation that the implicitness in the first order time discretization introduces diffusion while in second order one, it introduces dispersion. The numerical errors of the first order time discretization and the RK2CN scheme with t =. are given in Table 4, from which we can see that when ϵ =.3,, t resolve ϵ, the convergence orders in both space and time are from Figure 4. When ϵ =.3, with mesh size of order O(ϵ), the convergence becomes slower but the errors are already smaller than O( ). Though Figure 5 shows that the RK2CN scheme introduces more oscillations than the first order method, it has better accuracy for both ϵ =.3 and.3 from Table 4. If the time step satisfies the CFL condition and is O( ), when shocks evolve, it is enough to couple the first order time discretization with the second order space discretization. This is

20 2 MIN TANG first order second order rho3 q rho3 3.5 q3 Figure 4. Eample. The loglog plot of the numerical errors in Table 4 with respect to. what we do in [2] However, we will see in the net eample that for the smooth case, the RK2CN scheme is more accurate. Eample 2: In a similar way as in [5, 27], we simulate the evolution of two colliding acoustic waves. Because there is no shock in this problem, α can be set to. However, to test the performance of a scheme that can apply to both shock and non-shock cases, we use α as in (32). Well prepared initial conditions Let p(ρ ϵ ) and the initial conditions be p(ρ ϵ ) = ρ.4 ϵ, for [, ], ρ ϵ (, ) = ϵ 2( cos(2π) ), u ϵ (, ) = ϵ 2 sign().4 ( cos(2π) ). Periodic boundary conditions are employed here. These two acoustic pulses, one right-running and one left-running, will collide and superpose and then separate again. No shock will form during this whole procedure. When T =.5, ϵ =., Table 5 gives the L norms of the numerical errors. The referred eact solution is calculated by the eplicit KT scheme with very fine mesh = /6, t = /6 2. We can see that the convergence orders of both time and space are second order from Figure 6. Note that the initial conditions are well prepared, that is when ϵ goes to, the and are consistent with the incompressible limit. To show the advantage of the AP scheme, Figure 7 gives the numerical results for ϵ =. at T =.5, calculated with the AP scheme. With the same, t, our AP scheme can get the right solution, whereas the eplicit KT scheme is no longer stable. We can see that for unresolved, as far as the scheme is stable, the errors are O( ). Non-well prepared initial conditions: Let p(ρ ϵ ) and the initial ρ ϵ, u ϵ be p(ρ ϵ ) = ρ.4 ϵ, for [, ], ρ ϵ (, ) = ϵ ( cos(2π) ), u ϵ (, ) = sign().4 ( cos(2π) ).

21 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS a) b) Figure 5. Eample. When T =., the and calculated with =.5, t =.5. using the first and second order time discretization are shown by the dashed and dash dotted lines respectively. Here the solid lines are the eact solutions. a): ϵ =.3; b): ϵ =.3. Each of the figures are composed of four subplots, top left: the ; top right: the ; bottom left: the amplification of the top left subplot to get a better look at the oscillations; bottom right: the amplification of the top right subplot to get a better look at the oscillations. Here non-well prepared indicates that, though ϵ is small, the and are not close to solutions of the incompressible equations. We can see from Figure 8 that the pulses collide at T =.5 and generate the maimum. When T =.8, the two pulses separate again. The fronts of the two separated pulses are steeper than the initial pulses, which is due to the weakly non-linear effect as discussed in [25]. Eample 3: A two dimensional eample is illustrated. As discussed in [5], the first order method introduces too much diffusion, no matter if ϵ is big or small.

22 22 MIN TANG t E ρ E q /25 / /5 / / / /2 / Table 5. Eample 2. When T =.5, ϵ =., the L norm of the numerical errors calculated by our second order AP scheme with different, t =. are presented second order rho q Figure 6. Eample 2. The loglog plot of the numerical errors in Table 5 with respect to. We use the same initial conditions as Eample 3 in [5] to show the improvement brought by the second order scheme. Let p(ρ ϵ ) = ρ 2 ϵ and the computational domain be (, y) Ω = [, ] [, ]. The initial data are ρ(, y, ) = + ϵ 2 sin 2 (2π( + y)), q (, y, ) = sin(2π( y)) + ϵ 2 sin(2π( + y)), q 2 (, y, ) = sin(2π( y)) + ϵ 2 cos(2π( + y)). Numerically, the CFL condition in two dimensions is similar to (3). Thus we keep using α determined by (32) and choose σ = in the numerical simulations. When ϵ =.8, the numerical results at T = calculated with our AP scheme, using = y = /6, t = /4 and the eplicit KT scheme [2], using = y = /6, t = /4 and = t = /52, t = / are displayed in Figure 9. As soon as the oscillations are suppressed, there is no need to use smaller time steps, especially when no shock forms. We can see that, besides the advantage of bypassing the CFL constraint, the AP scheme gives better approimation than the standard KT scheme when the same space mesh is used. This is because less diffusion is induced by the AP scheme. This suggests that the advantage of our AP method is not limited to the incompressible regime. Comparing with the standard KT scheme, our scheme has less strict stability requirement, better accuracy and less diffusion.

23 SECOND ORDER ALL SPEED METHOD FOR THE ISENTROPIC EULER EQUATIONS Figure 7. Eample 2. When ϵ =., T =.5, the eact solution and the numerical results of our AP scheme calculated with = /25, t = / and = /25, t = /2 are given by solid, dashed and dashed dotted lines respectively. Here the eplicit KT scheme is not stable with the same, t Figure 8. Eample 2. When ϵ =., the numerical results of the AP scheme calculated with = /3, t = / are displayed. The dashed lines are the initial conditions and solid lines are the results at time T. Top: T =.5; Bottom: T =.8 The scheme performance in the incompressible regime with under-resolved meshes is illustrated in Figure, where the numerical solutions for ϵ = 4 are displayed. Here we compare the numerical results of our AP method, using = y = /32, t = /2, and solutions of the limiting incompressible Euler equations calculated

24 24 MIN TANG y a) y q p y b) y q2 p y c) y Figure 9. Eample 3. When ϵ =.8, T =, the numerical results calculated by our AP method with = y = /6, t = /5 (left), by the standard KT scheme with = y = /6, t = /4 (middle) and the eact solution calculated by the KT scheme with = y = /52, t = / (right) are represented. To see the results clearer, we use contour plot for the eact solution, where the appearance of shocks in ρ is observed. a) ρ ϵ ; b) q ϵ ; c) q 2ϵ. with a staggered difference method [3]. The staggered difference method was first introduced by F. H. Harlow and J. E. Welch to simulate incompressible viscous flows and its stable pressure-velocity coupling can give solutions without artificial viscosity. The discrete L norm of the difference between the numerical results of under-resolved meshes and the limiting incompressible Euler equation can be diminished when we reduce. This confirms the AP property of the two dimensional scheme. According to (32), smaller time steps increase α and more diffusion is induced. As long as the scheme is stable, bigger time steps are preferred. To find the convergence order of the two dimensional scheme, numerical errors for different mesh sizes are presented in Table 6. Here when ϵ =.8, the referrence solution is calculated by the eplicit KT scheme using = /52, t = /. The eact solution for ϵ =. is given by the limiting incompressible Euler equations using the staggered difference method with = y = /28, t = /6 [3]. E ρ, E q are the discrete L norms of the relative errors as in one dimension. We can see from Figure due to shock formation, numerical errors for ϵ =.8 give first order convergence. By contrast, thanks to the smoothness of the