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High order semiimplicit weighted compact nonlinear scheme for the allMach isentropic Euler system
Advances in Aerodynamics volume 2, Article number: 27 (2020)
Abstract
The computation of compressible flows at all Mach numbers is a very challenging problem. An efficient numerical method for solving this problem needs to have shockcapturing capability in the high Mach number regime, while it can deal with stiffness and accuracy in the low Mach number regime. This paper designs a high order semiimplicit weighted compact nonlinear scheme (WCNS) for the allMach isentropic Euler system of compressible gas dynamics. To avoid severe CourantFriedrichsLevy (CFL) restrictions for low Mach flows, the nonlinear fluxes in the Euler equations are split into stiff and nonstiff components. A thirdorder implicitexplicit (IMEX) method is used for the time discretization of the split components and a fifthorder WCNS is used for the spatial discretization of flux derivatives. The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit. One and twodimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.
1 Introduction
LowMach number flows in fluid dynamics that are slow compared with the speed of sound can be described by the following scaled isentropic Euler equations.
where ρ(x,t)>0,u(x,t)=(u(x,t),v(x,t)) and q=ρu(x,t) are the density, velocity and momentum of the fluid, respectively, and ε>0 is the scaled Mach number that is a measure of compressibility of the fluid. For example, ε=1 corresponds to the fully compressible regime. The operators ∇,∇· and ⊗ are the gradient, divergence and tensor product operators, respectively. The pressure p in Eq. (1) is given by the equation of state
where Λ>0 and γ≥1 are constants. Equation (1) are hyperbolic with wave speeds in direction n
and the sound speed \(c=\sqrt {p'(\rho)}/\varepsilon \).
Klainerman and Majda [1, 2] proved that the solution of Eq. (1) converges to the incompressible counterpart when ε∼0. The asymptotic nature of the solution to Eq. (1) can be studied by pluggingin the ansatz
for variables ρ, p and u. After performing a scale analysis, we obtain an incompressible Euler system for the velocity
based on wellprepared initial data, i.e.,
and appropriate boundary conditions [3], e.g., periodic, wall or homogeneous Neumann boundary conditions. Taking the divergence of the first equation in (5) and using the incompressibility, we obtain
In low Mach number flows, the challenge faced by standard hyperbolic numerical methods for Eq. (1) is that the system becomes stiff. The stiffness is caused by the stringent CourantFriedrichsLevy (CFL) restriction [3–6], i.e. \(\Delta t\le \Delta x/\underset {i}{max}\lambda _{i}=\mathcal {O}(\varepsilon \Delta x)\) with the spatial and temporal grid sizes Δx and Δt. The restriction results in an increasingly large computational time for ε→0. Moreover, the strong stability requirement for ε→0 results in the excessive numerical viscosity of the standard hyperbolic solvers that is proportional to \(\underset {i}{max}\lambda _{i}=\mathcal {O}(\varepsilon ^{1})\). This makes the standard hyperbolic solvers suffer from severe loss of accuracy due to the creation of spurious waves. Therefore, the stability and accuracy of the standard hyperbolic numerical methods highly depend on ε [4].
In the literature, some allspeed schemes [3–16] have been derived for the compressibleincompressible limit problems based on the idea of the asymptoticpreserving (AP) methodology introduced by Jin [17, 18] for relaxation systems. The AP allspeed schemes are uniformly consistent with the lowMach number limit and are uniformly stable. Although a robust AP scheme should allow both explicit and implicit timemarching techniques, the explicit strategies suffer from the CFL stability conditions and thus affect the efficiency of the method for small Mach numbers. The implicit strategies avoid the CFL restriction, but they need to solve highly nonlinear equations and add an excessive numerical dissipation to the slow wave, degrading the accuracy of the method [3]. To overcome the drawbacks, the semiimplicit or implicitexplicit (IMEX) time discretization strategies are widely applied based on the acoustic/advection splitting of the Euler equations into the stiff/nonstiff parts (or fast and slow fluxes) [3–5, 10]. These strategies treat with the stiff part implicitly and the nonstiff part explicitly to solve the isentropic Euler equations effectively and efficiently. The IMEX time discretization combined with the shockcapturing schemes, e.g., the firstorder local LaxFriedrichs (or Rusanov) scheme [4], the secondorder finite volume method [3, 19], the arbitrary high order accurate discontinuous Galerkin method [20] and the fifthorder finite difference weighted essentially nonoscillatory (WENO) scheme [10], has shown its good performances in compressible and incompressible regimes.
Recently a class of weighted compact nonlinear schemes (WCNS’s) as high order accurate shockcapturing schemes have been widely adopted to investigate compressible flows with shock waves and discontinuities [21–27]. The WCNS’s are first proposed by Deng and Zhang [21] based on the cellcentered compact scheme of Lele [28] and are developed in [22–24, 29]. The expression of the classical cellcentered compact scheme is that a linear combination of the unknown approximations of the first derivative of a function at the grid points in a stencil is equal to another linear combination of the function itself at the grid points in the same stencil. Compared to noncompact schemes, e.g., the classical WENO scheme [30], the stencil can be more compact in the compact scheme for the same order of accuracy [22]. WCNS’s that integrate WENO interpolation into the cellcentered compact scheme for the first derivative of a function are variants of the WENO schemes and have the good performance in terms of accuracy, robustness and efficiency for compressible flows. Compared to the classical WENO schemes, the WCNS’s feature a better spectral resolution at the same order of accuracy [21, 22]. They are more flexible in the choice of flux splitting methods, e.g., Roe’s flux difference splitting method, van Leer’s flux vector splitting and Liou’s advection upstream splitting method, and flux derivatives can be computed by different explicit or implicit compact finite difference methods [23, 29]. In this work, in view of the good performance of the IMEX time discretization method and the WCNS’s, we design a high order IMEX WCNS for the allMach isentropic Euler system based on the acoustic/advection splitting strategy [10]. The stiff part governing fast acoustic and the nonstiff part modeling slow nonlinear advection effects are solved by the high order implicit and explicit time discretization methods, respectively. In the zero Mach number limit, the high order IMEX scheme has the AP property and is asymptotically stable under a convective CFL condition independent of the Mach number ε. One and twodimensional numerical examples are given to demonstrate good performance of the designed IMEX WCNS in both compressible and incompressible regimes.
2 WCNS method
In this section, a fifthorder WCNS is considered for the spatial discretization of flux derivatives. The WCNS procedure consists of three components, i.e., midpointtonode differencing, flux evaluation at the midpoint and nodetomidpoint weighted averaging interpolation of fluxes. Here, we briefly describe the framework of WCNS discretization for the first derivative, f_{x}, in the onedimensional scalar case u_{t}+f(u)_{x}=0. For simplicity, we take a uniform grid defined by x_{i}=iΔx=ih,i=0,...,N, where h is the uniform grid spacing and N is the number of cells.
The classical cellcentered compact scheme for the first derivatives [28] has no dissipation, but the dispersive errors dominate the scheme, which may destroy the numerical solutions. A class of hybrid cellnode and celledge WCNS’s proposed in [29] overcome the disadvantages and are written as
which introduce dissipation by approximating the celledge fluxes f_{i±(m−1/2)} based on nonlinear weighted averaging interpolation of fluxes. The coefficients ϱ_{k},η_{l} and ζ_{m} in Eq. (8) can be acquired by matching the Taylor series coefficients of various orders. If the coefficients ϱ_{k}=0, a class of explicit WCNS’s can be derived. The explicit WCNS’s are preferred because of their low cost and simplicity for parallelization and vectorization, compared to the implicit WCNS’s with a certain coefficient \(\phantom {\dot {i}\!}\varrho _{k^{*}}\neq 0\). Here, we consider one of explicit WCNS’s and use a sixthorder explicit centraltype scheme obtained from Eq. (8) to approximate f_{x}, i.e.,
By the Taylor series expansion of (9), we obtain
where the truncation term O(h^{r}) comes from the numerical approximations of the celledge fluxes f_{i±1/2}, denoted by \(\hat {f}_{i\pm 1/2}\). The accuracy of the scheme (9) is min(6,r). Here, the numerical fluxes \(\hat {f}_{i\pm 1/2}\) approximate f_{i±1/2} to the fifth order and thus a fifthorder WCNS is designed for the spatial discretization.
Various flux splitting methods [23] can be used in WCNS to introduce correct upwinding. For simplicity, we use the LaxFriedrichs flux splitting, i.e., the flux is split into positive and negative parts:
where \(\alpha =\underset {u}{max} f'_{u}\) is the maximum characteristic wave speed. The numerical fluxes \(\hat {f}_{i\pm 1/2}\) at the cell centers in (9) are calculated as
A fifthorder WENO interpolation technique is considered to obtain the positive numerical flux \(\hat {f}^{+}_{i+1/2}\). The interpolation process of the negative numerical flux \(\hat {f}^{}_{i+1/2}\) is symmetric with respect to x_{i+1/2} and will not be shown here. For simplicity, the “+” sign in the superscript is removed.
The numerical flux \(\hat {f}_{i+1/2}\) on a cell center can be interpolated locally from the values on a fivepoint upwindbiased stencil S={x_{i±2},x_{i±1},x_{i}}. The stencil S can be divided into three substencils S_{k}={x_{i+k−2},x_{i+k−1},x_{i+k}},k=0,1,2. On each substencil a thirdorder linear interpolation of f at cell center x_{i+1/2} can be derived, i.e.,
where \(\hat {f}_{i + 1/2}^{k}, k=0,1,2,\) are the approximated values at cell center x_{i+1/2} on S_{k},k=0,1,2, respectively. On the stencil S a fifthorder upwindbiased linear interpolation of f at cell center x_{i+1/2} can be derived, i.e.,
which can also be constructed linearly from the three thirdorder interpolations \(\hat {f}_{i + 1/2}^{k}\):
with the linear weights
The centraltype scheme (9) with the upwindbiased linear interpolation (16) or (17) will generate oscillations around shock waves and discontinuities. To capture shock waves and discontinuities, the numerical flux \(\hat {f}_{i + 1/2}\) should be computed by using the nonlinear combination of upwindbiased interpolations, i.e.,
where ω_{k},k=0,1,2, are the upwindbiased nonlinear weights that are associated with the relative smoothness of f on each substencil. In smooth areas the nonlinear weights ω_{k} should be close to the linear weights d_{k} to achieve the designed order and be close to zero near discontinuities to suppress spurious numerical oscillations near discontinuities.
There are several types of nonlinear weights ω_{k} in the literature, e.g., the WENOJS [30], WENOM [31] and WENOZ [32–34] weights. The WENOZ weights place a larger weight to the less smooth substencil, and thus they are less dissipative than the WENOJS weights and more efficient than the WENOM weights. Here, the WENOZ weights ω_{k} proposed by Borges et al. [32] are chosen.
where the secondorder smoothness indicators β_{k},k=0,1,2, derived by Hu [35] are adopted to measure the smoothness of f on the substencils S_{k}.
with
In (20), the global higher order smoothness indicator τ=β_{0}−β_{2} and the parameter δ is a small constant to avoid denominators becoming zero.
Remark 1
The value of δ in (20) will influence the convergence, accuracy and stability of the schemes. Here, a parameterfree algorithm proposed by Zheng et al. [36] is used to calculate the value of δ as follows.
where \(\beta _{ave}=\frac {1}{3}\sum ^{2}_{k=0}\beta _{k},~\beta _{min}=min\{\beta _{0}, \beta _{1},\beta _{2}\},~\beta _{std}=\sqrt {\frac {1}{3}\sum ^{2}_{k=0}(\beta _{ave}\beta _{k})^{2}}\). To avoid denominators becoming zero in (20) and (22), β_{k}+δ in (20) and β_{std} in (22) are replaced with max(β_{k}+δ,err) and max(β_{std},err), respectively. Here, err is the square root of the smallest positive number allowed for a machine.
3 IMEX method
In this section, we briefly describe the framework of IMEX time discretization designed for the numerical integration of the stiff ordinary differential equation (ODE) as follows.
where \(\bar {f}\) and \(\bar {g}\) are the nonstiff and stiff parts of the ODE, respectively. The IMEX time discretization treats the nonstiff part explicitly and the stiff part implicitly in order to overcome the stringent timestep restriction for stability. There are many IMEX methods designed for various problems in the literature [37–41]. Among them, the IMEX RungeKutta (IMEXRK) schemes as combinations of diagonally implicit A or Lstable RK (DIRK) methods and explicit RK methods offer a precise and robust approach to define high order AP schemes. The computational cost per time step is proportional to the number of stages in the IMEXRK schemes and the accuracy is proportional to the timestep size and the order of accuracy.
The IMEXRK p(s,σ,p) scheme is characterized by the s×s matrices \( \mathbf {A}=(a_{ij}), \tilde {\mathbf {A}}=(\tilde {a}_{ij})\), the vectors \(\mathbf {b}, \tilde {\mathbf {b}} \in \mathbb {R}^{s}\), and the vectors \(\mathbf {c}, \tilde {\mathbf {c}} \in \mathbb {R}^{s}\). This scheme with s stages of the implicit scheme for the stiff term and σ(=s+1) stages of the explicit scheme for the nonstiff term has p^{th}order temporal accuracy. Applying this scheme to solve Eq. (23) from t^{n} to t^{n+1} yields
Definition 1
The IMEXRK p(s,σ,p)scheme is of type ARS, i.e., the matrices \( \mathbf {A}, \tilde {\mathbf {A}}\) for the implicit and explicit schemes are described respectively as
with the submatrices \((\tilde {A}_{21})_{(s1) \times 1}, (\tilde {A}_{22})_{(s1) \times (s1)}\) and the invertible submatrix (A_{22})_{(s−1)×(s−1)}.
Definition 2
The IMEXARSp scheme is globally stiffly accurate (GSA) (see [10,40]), i.e., the matrices \( \mathbf {A}, \tilde {\mathbf {A}}\) and the vectors \(\mathbf {b}^{T}, \tilde {\mathbf {b}}^{T}\) satisfy
where e_{s}=(0,...,0,1)^{T}. The IMEXARSp scheme that satisfies the condition \(\mathbf {e}^{T}_{s} \mathbf {A}=\mathbf {b}^{T}\) is called stiffly accurate (SA).
Remark 2
For the GSA IMEXARSp scheme, y^{n+1}=y^{(s)} in Eq. (25).
4 IMEXARSp wCNS
In this section, we design the IMEXARSp WCNS scheme for the isentropic Euler system. Here, the fifthorder WCNS described in the previous section is used for the spatial discretization and the p^{th}order IMEXARSp method is used for the time discretization. We first split the Euler system into two parts based on the acoustic/advection splitting strategy [10] and obtain
where \(\mathbf {U}=(\rho, \mathbf {q}), \mathbf {F}=(0, \nabla \cdot \frac {\mathbf {q} \otimes \mathbf {q}}{\rho })\) is the nonstiff nonlinear term and \(\mathbf {G}=(\nabla \cdot \mathbf {q}, \frac {1}{\varepsilon ^{2}} \nabla p)\) is the stiff linear term.
The p^{th}order IMEXARSp scheme that consists of (24) and (25) is applied to each component of (28). The intermediate stages are described as
where k=1,2,...,s, and the numerical solutions at time t^{n+1} are described as
The intermediate stages (29) and (30) consist of two implicit steps and can be simplified by eliminating q^{(k)} between (29) and (30) [10], and obtain a nonlinear elliptic equation for ρ^{(k)}:
where
Based on (30) and (35), q^{(k)} can be calculated as
Remark 3
To update the solutions from t^{n} to t^{n+1} by the p^{th}order IMEXARSp WCNS, we first solve the elliptic equation (33) to obtain ρ^{(k)} and then calculate explicitly q^{(k)} using (36). Here, (33) is a nonlinear equation for ρ^{(k)}, which is solved by inexact Newtongeneralized minimum residual (NewtonGMRES) algorithms. Refer to [42,43] for the detailed description of the algorithms. Finally, the updates ρ^{n+1} and q^{n+1} are calculated explicitly using (31) and (32) with the intermediate values ρ^{(k)} and q^{(k)}. For the GSA IMEXARSp scheme, ρ^{n+1}=ρ^{(s)} and q^{n+1}=q^{(s)}.
Remark 4
The spatial discretization of both convective terms in (33)(35) and the gradient terms in (34) and (35) are solved by the fifthorder WCNS in the componentbycomponent and dimensionbydimension forms. Here, in the interpolation process of the celledge pressures, the LaxFriedrichs flux with zero numerical viscosity is considered, i.e., \(p=\frac {1}{2}p+\frac {1}{2}p\), otherwise the numerical diffusion will become large when ε→0 [10].
Remark 5
For the explicit time discretization, the CFL stability condition is described as
which leads to unaffordable computational costs in the low Mach number regime. For the IMEXARSp time discretization, the convective CFL stability condition is obtained as
due to semiimplicit nature of the acoustic/advection splitting strategy. In order to avoid an excessively large timestep sizes when the velocity of the fluid is too small, Boscarino et al. [3] recommended the following stability restriction
The stability conditions (38) and (39) of the IMEXARSp method are independent of the Mach number ε and thus are less restrictive contrary to a standard explicit method. Therefore, the IMEXARSp method is more efficient than the explicit method for lowMach number flows.
Remark 6
For onedimensional case, the term △p=p_{xx} in (33) is approximated by a fourthorder centraltype difference scheme:
For twodimensional case, this centraltype difference scheme in a dimensionbydimension form is applied to △p=p_{xx}+p_{yy}.
5 AP property
Definition 3
A scheme for the low Mach number limit of the Euler equations is asymptotic preserving, if it serves as a consistent and stable discretisation, independent of the Mach number ε.
Proposition 1
Consider the system (1) subject to wellprepared initial data (i.e., \(\rho ^{0}_{0}=const, \nabla \cdot \mathbf {u}^{0}_{0}(\mathbf {x})=0\)) and periodic, wall or homogeneous Neumann boundary conditions. The spatial derivatives are assumed to be continuous. The GSA IMEXARSp scheme ((29)(32)) (see Definitions 1 and 2) is used for the time discretization of the system (1) from t=t^{0} to t^{1} with the timestep size Δt=t^{1}−t^{0} given by the stability conditions (38) or (39) independent of ε, and we then have
and
Here, u_{e} is the exact solution of Eq. (5) with initial data \(\mathbf {u}_{e}(\mathbf {x},0)= \mathbf {u}^{0}_{0}(\mathbf {x})\).
Let \(\mathbf {f}=\nabla \cdot \frac {\mathbf {q} \otimes \mathbf {q}}{\rho }, \vec {r}=(r^{(1)},...,r^{(s)})^{T}\) for r=ρ or p, and \(\vec {\mathbf {v}}=(\mathbf {v}^{(1)},...,\mathbf {v}^{(s)})^{T}\) for v=f or q generated at the IMEXARSp intermediate stages t_{k}=t^{0}+c_{k}Δt. Following [10], we give the proof process of Proposition 1.
Proof
Based on (29)(32), we obtain from t=t^{0} to t^{1}
and
where \(\mathbf {e}=(1,...,1)^{T}\in \mathbb {R}^{s}\). □
Taking the divergence of both sides of (44) and substituting \( \nabla \cdot \vec {\mathbf {q}}\) into (43) yields a pressure Poisson equation
In order to study the asymptotic behavior of the solutions, we plug the εasymptotic expansions f(x)=f_{0}(x)+ε^{2}f_{2}(x)+O(ε^{3}) for variables \(\vec {\rho }, \rho ^{1}, \vec {p}, p^{1}\) and g(x)=g_{0}(x)+O(ε^{2}) for variables \(\vec {\mathbf {q}},\mathbf {q}^{1}\) into (43)(47). Note that the term O(ε) does not appear in Eq. (1). At the leading order O(ε^{−2}) terms in (44), we obtain \(\nabla \vec {p}_{0}=\nabla \vec {p}(\vec {\rho }_{0})=0\). Therefore, \(\vec {p}_{0}\) and \(\vec {\rho }_{0}\) are independent of x. Collecting the order O(1) terms in ε generates
where \(\mathbf {q}^{0}_{0}=\rho ^{0}_{0} \mathbf {u}^{0}_{0}\) and \(\vec {\mathbf {f}}_{0}=\vec {\mathbf {q}}_{0} \otimes \vec {\mathbf {q}}_{0}/\vec {\rho }_{0}\).
Integrating (48) over the whole computational domain and applying periodic, wall or homogeneous Neumann boundary conditions yields \(\vec {\rho }_{0}=\rho ^{0}_{0} \mathbf {e}\), i.e., \(\vec {\rho }_{0}\) is independent of t. Since \(\vec {\rho }_{0}=\rho ^{0}_{0} \mathbf {e}\) and \(\nabla \cdot \mathbf {q}^{0}_{0} =\rho ^{0}_{0} \nabla \cdot \mathbf {u}^{0}_{0}=0\), (52) becomes
Based on Definition 1, (53) can be simplified as
where \(\vec {p}'=(p^{(2)},...,p^{(s)})^{T}\) and \(\vec {\mathbf {f}}'_{0}=(\mathbf {f}^{(2)},...,\mathbf {f}^{(s)})^{T}\).
Taking the divergence of both sides of (49) and applying Definition 1 and (54) yields
Based on (50) and (55), we have \(\rho ^{1}_{0}=\rho ^{0}_{0}\), i.e., \(\rho ^{1}(\mathbf {x},t^{1}) \xrightarrow {\varepsilon \to 0} \rho ^{0}_{0}\). Taking the divergence of both sides of (51) and applying Definition 2 generates
By applying Definition 1 and (54), (56) is simplified as \(\nabla \cdot \mathbf {q}^{1}_{0}=\rho ^{1} \nabla \cdot \mathbf {u}^{1}_{0}=0\). Since \(\rho ^{1}_{0}=\rho ^{0}_{0}\), we have \(\nabla \cdot \mathbf {u}^{1}_{0}=0\), i.e., \(\nabla \cdot \mathbf {u}^{1}(\mathbf {x},t^{1}) \xrightarrow {\varepsilon \to 0}0\).
Both sides of (49) and (51) are divided by the constant density \(\rho ^{0}_{0}\) and we obtain the asymptotic limit of scheme for the incompressible Euler system (5):
where \(\vec {p}_{2}\) is given by (53). If the IMEXARSp scheme that has p^{th}order temporal accuracy is adopted to solve the system (5) subject to initial data \( \mathbf {u}_{e}(\mathbf {x},0)= \mathbf {u}^{0}_{0}\), the asymptotic accuracy can be attained, i.e., (42) is proved.
Remark 7
A similar procedure can be adopted when going from t^{n} to t^{n+1} to prove that the limiting IMEXARSp scheme is a consistent discretization of the limiting equation for fixed h and Δt. The IMEXARSp scheme is also asymptotically stable with the timestep restrictions (38) or (39) independent of the Mach number ε.
Remark 8
The AP property of the GSA IMEXARSp scheme is proved based on the assumption that the Euler equations are subject to wellprepared initial data. For the SA IMEXARSp scheme, the AP property can also be proved using a similar procedure under the same assumption. The proof of the AP property of the GSA IMEXARSp scheme combined with the fifthorder WCNS is relatively complex and has not been performed. In the next section, we will verify numerically the AP property of the IMEXARSp WCNS. In addition, the IMEXARSp scheme combined some other shock capturing schemes, e.g., the MUSCL (monotone upstreamcentered scheme for conservation laws) type finite volume scheme proposed in [44], can also maintain numerically the AP property.
6 Numerical examples
The thirdorder GSA IMEXARS3 WCNS is applied to solve the following numerical examples. In some convergence tests, the thirdorder SA IMEXARS3 WCNS is also tested. See Appendix for the characteristics of the GSA and SA IMEXARS3 scheme. In addition, the thirdorder explicit total variation diminishing RungeKutta timestepping method [30] coupled with the fifthorder WCNS (TVDRK3 WCNS) and the thirdorder GAS IMEXARS3 scheme coupled with the thirdorder MUSCL type finite volume scheme (IMEXARS3 MUSCL) are used to solve some examples for comparisons.
The order of temporal accuracy is numerically calculated as
where \(E_{\Delta t_{1}}\) obtained with Δt_{1}=O(h) and \(E_{\Delta t_{2}}\) obtained with Δt_{2}=Δt_{1}/2 are the global errors in the L^{∞} and L^{1} norms, respectively. The spacestep size h deceases with the timestep size Δt. In the convergence tests, to test the temporal order of accuracy the timestep size is taken as Δt=CFL·h (CFL=0.1), while it is set as Δt=CFL·h^{2} (CFL=0.1) to test the spatial order of accuracy. In the other tests, the timestep sizes of the IMEXARS3 WCNS and the IMEXARS3 MUSCL are computed by the stability conditions (38) or (39), while the timestep size of the TVDRK3 WCNS is calculated by the stability condition (37).
Example 1
Convergence test for onedimensional case [3]
This example is adopted to test the temporal order of accuracy of the GSA and SA IMEXARS3 WCNS for onedimensional case. The computational domain is [−2.5,2.5] with periodic boundary conditions. The initial data for onedimensional isentropic Euler equations (1) is given by
where \(\gamma =2,c=2\sqrt {\gamma }/\varepsilon,L=5\) and the final time is fixed to T_{f}=0.1. To measure the temporal convergence of the GSA and SA IMEXARS3 WCNS, the solution computed on a fine mesh with N=2560 cells is used as a reference (“exact”) solution since the exact solution of this problem is unknown. Tables 1 and 2 report the numerical L^{∞} and L^{1}norm errors and orders of accuracy in time for the density obtained with the GSA IMEXARS3 WCNS at different Mach numbers, while Tables 3 and 4 show the corresponding results obtained with the SA IMEXARS3 WCNS. It is observed that both methods show a thirdorder temporal convergence, as expected, for different values of the Mach number, from compressible to incompressible regimes. In the following examples, we only investigate the performance of the GSA IMEXARS scheme. Here, we also test the spatial accuracy of the GSA IMEXARS3 WCNS. The numerical L^{∞} and L^{1}norm errors are computed by comparing the numerical solutions of the density on third consecutive mesh sizes [13]. Tables 5 and 6 report the corresponding results at t=0.05, which demonstrate a fifthorder spatial convergence of the GSA IMEXARS3 WCNS at different Mach numbers.
Example 2
Onedimensional Riemann problem [4,6,10]
This example that consists in several interacting Riemann problems is used to validate the asymptotic stability of the IMEXARS3 WCNS. The computational domain is [0,1] with periodic boundary conditions. The initial data is described as
and the pressure is p(ρ)=ρ^{2}. The timestep size is computed by (38) with CFL=0.3 and the final time is fixed to T_{f}=0.05. The values of the Mach number are chosen as \(\varepsilon =\sqrt {0.99}, 0.3\) and 0.01 corresponding to the compressible, intermediate and incompressible regimes, respectively. The numerical solutions calculated by the GSA IMEXARS3 WCNS and the GSA IMEXARS3 MUSCL with N=200 cells are compared to a reference solution computed by the explicit TVDRK3 WCNS with N=2000 cells. Figure 1 illustrates the solutions for the density and momentum at different Mach numbers. From this figure, we observe the IMEXARS3 WCNS and the TVDRK3 WCNS can capture shocks and contact discontinuities that are stronger when ε is bigger, while the IMEXARS3 MUSCL suffers from excessive numerical diffusion. Spurious oscillations are observed with the TVDRK3 WCNS when \(\varepsilon =\sqrt {0.99}\) (see Fig. 1a and b). The utilization of WENO interpolation based on the characteristic variables instead of the flux functions can effectively suppress spurious oscillations near discontinuities. Compared to the TVDRK3 WCNS, the IMEXARS3 WCNS and the IMEXARS3 MUSCL capture the limit incompressible solution faster when ε is small. When ε approaches 0, the TVDRK3 WCNS can not capture the correct solution, due to the excessive numerical viscosity.
Example 3
Sod shock tube problem [5]
This problem with nonwell prepared initial data is used to investigate the performance of the GSA IMEXARS3 WCNS in the intermediate and incompressible regimes. The computational domain is [0,1] with zero Neumann boundary conditions. The pressure is p(ρ)=ρ. The nonwell prepared initial data including discontinuity is
The timestep size is computed by (39) with CFL=0.1. The values of the Mach number are chosen as ε=0.3 and 0.03 corresponding to the intermediate and incompressible regimes, respectively. For both ε=0.3 and ε=0.03, reference solutions are generated using the GSA IMEXARS3 MUSCL with N=2000 cells. Figure 2 depicts the density and momentum calculated by the GSA IMEXARS3 WCNS with N=200 cells at time T_{f}=0.05 for ε=0.3 and at time T_{f}=0.005 for ε=0.03. From this figure, the density and momentum are not close to the solutions of the incompressible Euler equations. The solutions include the rarefaction waves moving to the left and shock waves moving to the right for both ε=0.3 and ε=0.03. From Fig. 2a and b, it is observed that when ε=0.3, the numerical solution obtained with the IMEXARS3 WCNS has a good agreement with the reference solution. When ε=0.03 in the incompressible regime, nonphysical oscillations arise near discontinuities based on the IMEXARS3 WCNS (see Fig. 2c and d). Nonphysical oscillations are a common phenomenon when shocks develop in the low Mach number regime [4,5] and can be eliminated by decreasing Δt and taking a smaller CFL=0.01.
Example 4
Convergence test for twodimensional case [19]
This example is used to test the temporal order of accuracy of the GSA IMEXARS3 WCNS for twodimensional case. The computational domain is [0,1]×[0,1] with periodic boundary conditions. The pressure is \(p(\rho)=\frac {1}{2}\rho ^{2}\). The final time is T_{f}=0.02. The initial data is given by the following exact solution
where \(r=4\pi \sqrt {(x0.6t0.5)^{2}+(y0.5)^{2}}, k(r)=2\cos (r)+2r\sin (r)+0.125\cos (2r)+0.25r\sin (2r)+0.75r^{2}\) and
Tables 7 and 8 show the numerical L^{∞} and L^{1}norm errors and orders of accuracy in time for the momentum ρu based on different values of the Mach number, from compressible to incompressible regimes, which validates the GSA IMEXARS3 WCNS for twodimensional case can attain the designed temporal accuracy. In addition, the L^{1}norm convergence is better than the L^{∞}norm convergence under the same conditions.
Example 5
Twodimensional Riemann problem [8,10]
This example is used to test the performance of the GSA IMEXARS3 WCNS at a high Mach number for twodimensional case. The computational domain is [0,1]×[0,1] with zero Neumann boundary conditions. The pressure is p(ρ)=ρ^{1.4}. This initial data includes four shock waves and is described as
The timestep size is computed by (38) with CFL=0.4. The values of the Mach number are chosen as ε=1 and 2 to test the robust shockcapturing capability of the IMEXARS3 WCNS in the compressible regime. Figure 3 plots the surfaces of the density profile obtained with the IMEXARS3 WCNS and the IMEXARS3 MUSCL on a mesh 50×50 at T_{f}=0.1. From this figure, we see that the IMEXARS3 WCNS has a higher resolution, compared to the IMEXARS3 MUSCL.
Example 6
Cylindrical explosion problem [6,11]
The computational domain is [−1,1]×[−1,1] with periodic boundary conditions and the pressure is given by p(ρ)=ρ^{2}. The initial density of the fluid is given by
where \(r=\sqrt {x^{2}+y^{2}}\) is the distance to the center (0,0) of the domain. The initial velocity of the fluid is given by
where \(\beta =max(0,1r)e^{16r^{2}}\) and (u,v)=(0,0), if r≤10^{−15}. The computational domain is divided into 50×50 cells. The values of the Mach number are chosen as ε=1 and 0.001. The timestep sizes are computed by (38) with CFL=0.4 for ε=1 and by (39) with CFL=0.03 for ε=0.001, respectively. Figure 4 plots the surfaces of the density profile and the velocity fields for ε=1 obtained with the GSA IMEXARS3 WCNS at different times. Figure 5 shows the density profile and the discrete divergence of the velocity ∇·u at time t=0.05 for a small value of ε=0.001. It is observed that when ε=0.001, the density converges to the constant value 1 and the divergence ∇·u is close to 0.
Example 7
This example is used to verify numerically the AP property of the IMEXARS3 WCNS in the low Mach number limit. The computational domain is [0,2π]×[0,2π] with periodic boundary conditions and the pressure is set to p(ρ)=ρ^{1.4}. Initially, a constant density and an incompressible velocity field are given by ρ(x,0)=π/15 and
The timestep size is computed by (39) with CFL=0.2. The Mach number is ε=10^{−4} and the final time is T_{f}=10. Figure 6 demonstrates the discrete vorticity ω=v_{x}−u_{y} on a mesh 50×50 at different times. From this figure, we see the vorticity quickly develops into rollups with smaller and smaller scales with the time evolution. Figure 7 shows the time history of the L_{1}norm error of the discrete divergence of the velocity ∇·u. It is observed that the L_{1}norm error increases with time, which is in agreement with the phenomenon in [10] and is caused by the appearance of the smaller and smaller scale structures in the shear flow. The maximum error is smaller than that obtained with a finer mesh in [10].
Example 8
KelvinHelmholtz instability problem [10]
This problem is solved in the domain [0,2π]×[0,4π] with periodic boundary conditions. The pressure is p(ρ)=ρ^{2} and the initial data reads
The timestep size is computed by (39) with CFL=0.2. The Mach number is ε=10^{−3} and the final time is T_{f}=40. The discrete vorticity ω=v_{x}−u_{y} on a mesh 50×50 at different times is depicted in Fig. 8, which shows the emergence and subsequent evolution of small scale flow features. The time history of the L_{1}norm error of the discrete divergence of the velocity ∇·u is shown in Fig. 9, from which we also observe that the L_{1}norm error increases with time and the maximum error is smaller than that obtained with a finer mesh in [10].
7 Conclusions
In this work, we have designed a high order IMEX WCNS for the compressibleincompressible limit problems described as scaled isentropic Euler equations. Based on the acoustic/advection splitting strategy, the semiimplicit method combines the thirdorder IMEXARS3 scheme for the time integration with the fifthorder WCNS for the spatial discretization. The GSA IMEXARS3 scheme has been proven to be asymptotic preserving, i.e., it converges to a consistent scheme in the zero Mach number limit, and asymptotically stable under a convective CFL condition. The IMEXARS3 WCNS has been verified numerically that it can achieve a thirdorder temporal accuracy for different values of the Mach number, from compressible to incompressible regimes. Numerical results show that the IMEXARS3 WCNS with the AP property can capture shocks and contact discontinuities with high resolution in the compressible regime, while it can capture incompressible features in the imcompressible regime. The future work will focus on the implementation of the proof of the AP property of the fully discrete scheme and the extension to compressible NavierStokes flows at any Mach number.
8 \thelikesection Appendix
This appendix is used to characterize the GSA and SA IMEXARSp schemes adopted in this paper.
The thirdorder GSA IMEXRK3 scheme is characterized by
and \(\mathbf {b}^{T}=(0, 1.5, 1.5, 0.5, 0.5), \tilde {\mathbf {b}}^{T}=(0.25, 1.75, 0.75, 1.75,0)\). The thirdorder SA IMEXRK3 scheme is characterized by
and \(\mathbf {b}^{T}=\tilde {\mathbf {b}}^{T}=(0,1.208496649,0.644363171,0.4358665215)\).
Availability of data and materials
All data generated or analysed during this study are included in this published article.
Abbreviations
 WCNS:

Weighted compact nonlinear scheme
 IMEX:

Implicitexplicit
 CFL:

CourantFriedrichsLevy
 AP:

Asymptoticpreserving
 WENO:

Weighted essentially nonoscillatory
 ODE:

Ordinary differential equation
 RK:

RungeKutta
 DIRK:

Diagonally implicit RungeKutta
 GSA:

Globally stiffly accurate
 SA:

Stiffly accurate
 NewtonGMRES:

Newtongeneralized minimum residual
 MUSCL:

Monotone upstreamcentered scheme for conservation laws
 TVDRK3:

Thirdorder total variation diminishing RungeKutta
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The present study is supported by the National Numerical Wind Tunnel Project (No. NNW2018ZT4A08), the National Natural Science Foundation of China (Nos. 11872323 and 11971025) and the Natural Science Foundation of Fujian Province (No. 2019J06002)
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Jiang, Y., Chen, X., Zhang, X. et al. High order semiimplicit weighted compact nonlinear scheme for the allMach isentropic Euler system. Adv. Aerodyn. 2, 27 (2020). https://doi.org/10.1186/s42774020000529
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DOI: https://doi.org/10.1186/s42774020000529