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We derive in this paper a time stable seventh-order dissipative compact finite difference scheme with simultaneous approximation terms (SATs) for solving two-dimensional Euler equations. To stabilize the scheme, the choice of penalty coefficients for SATs is studied in detail. It is demonstrated that the derived scheme is quite suitable for multi-block problems with different spacial steps. The implementation of the scheme for the case with curvilinear grids is also discussed. Numerical experiments show that the proposed scheme is stable and achieves the design seventh-order convergence rate.
High-order finite difference methods are well suited for simulations of complex physics as they admit high resolution properties and save large amount of computational resources. Typical examples can be found in fluid dynamics.[1, 2] Although the derivation of high-order finite difference schemes in the interior of the computational domain is quite straightforward, boundary closures often need special investigation[3–5] to avoid accuracy and stability issues. However, it is not an easy task to construct suitable high-order boundary closures to ensure stability.[3]
In our previous work,[6] we showed that a globally seventh-order scheme is not time stable when boundary conditions are imposed directly with the traditional injection method. To rectify this issue, we employed simultaneous approximation terms (SATs)[7] to impose boundary conditions weakly for a dissipative compact finite difference scheme, resulting in a time stable method with global accuracy of the seventh order. The method was demonstrated to be effective for solving one-dimensional (1D) problems, including linear advection equations and Euler equations. The aim of this paper is to extend the algorithm to solve two-dimensional (2D) Euler equations.
To this end, we need to make some modifications to the scheme since SATs involve some subtle issues in the 2D case. As will be seen in section
We consider in section
Since curvilinear grids are often used in practice, we also discuss in section
Consider the following two-dimensional Euler equations
To extend the seventh-order scheme considered in Ref.[6] (some details can also be found in Appendix A) to solve Eq. (
Here
For a computational domain
Intuitively, one may simply choose the penalty coefficients to be same values as the one-dimensional case, i.e.,
To show how to improve the scheme, we take as an example the solution point
Using the scheme of Eqs. (
In this section we intend to show that the developed scheme with SATs is well suited for multi-block problems, which are often used for complex configurations.
For simplicity, the computational domain
To verify the scheme, we calculate the vortex convection problem of Eqs. (
Next we demonstrate that by exchanging information through penalty terms, the interface technique can naturally handle grid partition with junction points. This time we divide the computational domain into four subdomains with two interfaces situated at
Once again we consider the vortex convection problem of Eqs. (
In this section, we study how to implement the numerical scheme for problems with curvilinear grids, which are often needed in practice for complex configurations. In this case, we need to consider the problem in the computational coordinates
The first example is a model governed by the following Euler equations with a source term
The numerical test for the stationary model of Eqs. (
The curvilinear grid[21] considered here (see Fig.
Consider a channel flow[22, 23] governed by the Euler equations (
The initial flow field is set to be the free stream with
We run the calculation on a grid with 61 × 41 solution points until the residue reaches the machine zero. It can be observed from Fig.
In addition, to compare the results obtained by the modified scheme and the scheme with the same penalty coefficients as Ref.[6], we show in Fig.
In this paper, we generalized the globally seventh-order dissipative compact scheme with SATs[6] to the 2D Euler equations. The choice of penalty coefficients for SATs was reconsidered to stabilize the scheme. It was shown that the scheme with SATs is very convenient for dealing with multi-block problems with conformal grids. In addition, the implementation of the scheme for the case with curvilinear grids was also discussed, including the slip-wall boundary condition. Various numerical experiments were performed to verify the proposed scheme. The extension to Navier–Stokes equations may be considered in further work.
Here we revisit briefly the spacial schemes presented in Ref.[6]. For the one-dimensional conservation law
While the values Fj involved in the above difference scheme can be computed directly by
The Jacobian matrices of
Diagonalizing
Similarly, we have
Here we introduce for the two-dimensional Euler equations the slip-wall boundary condition imposed through SATs proposed in Ref.[24]. Consider a Cartesian grid and suppose that we are considering a point
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