A robust high-order residual distribution type scheme for steady Euler equations on unstructured grids
Introduction
One of the major research areas of computational aerodynamics is the numerical simulation of the complex flow field with practical configurations. Many numerical methodologies have been developed for the simulations based on the finite difference method, the finite volume method and the finite element method. All these methods can be used to discretize the Euler or the Navier–Stokes equations, which may result in highly nonlinear algebraic systems. Developing robust and efficient solvers for the nonlinear systems is a main challenge in the numerical simulation. Many recent solvers have been developed to enhance the numerical efficiency, see, e.g. [6], [30], [41], [22] and reference therein. These solvers have used some modern numerical techniques, including multigrid acceleration [2], [7], [8], [18], post-residual smoothing [14], [15], [20] and enthalpy damping [17]. With these acceleration techniques, the nonlinear systems can be solved with several multigrid iterations (a few seconds) to machine accuracy. An overview of these techniques can be found in [29]. However, most of known solvers have only second-order accuracy, while the experience on the structured grids suggests that higher-order accurate approximations can significantly improve the quality of numerical solutions [3]. Recently many efforts have been made for developing high-order methods on the unstructured grids.
On unstructured grids, the situation is more complex than that on the structured grids. In spite of this, many high-order methods have been presented on unstructured grids. Barth [3] developed the concept of k-exact reconstruction scheme for the median-dual finite volume scheme. Such technique was extended to the cell-centered finite volume scheme by Mitchell and Walters [32]. Based on the idea of selecting the locally smoothest stencil, ENO/WENO schemes [21], [25], [35] are proposed for the hyperbolic system of conservation laws. Recently, discontinuous Galerkin (DG) schemes [11] and spectral volume (SV) methods [36], [40] have been developed; both employ the information inside the control volume. All these numerical schemes demonstrated excellent high-order behaviors.
In this paper, we present a robust high-order numerical algorithm for two-dimensional steady Euler equations on unstructured grids, based on the work done by Li et al. [24]. In [24], only linear reconstruction is used, so the numerical solution has just second-order accuracy (for smooth solutions). To obtain higher order numerical accuracy, we extend the reconstruction to the quadratic case, and use hierarchical reconstruction strategy of [27], [26], [42] to avoid the non-physical oscillations. The idea of hierarchical reconstruction is simple. After the initial reconstruction, the information on the centroid of each cell such as the function values, gradients and second-order derivatives is known. What the hierarchical reconstruction does is to recompute all the information level by level from the highest order terms to the lowest order terms, with certain non-oscillatory method. For the quadratic reconstruction which gives a quadratic polynomial in each cell, for example, coefficients of second-order terms are recomputed using constant term and gradients in different reconstruction patch, and several candidates for these coefficients are generated. Then the MUSCL scheme or ENO/WENO schemes are used to determine the limited coefficients for the second-order terms. For the linear terms, a similar way is used to generate limited coefficients with constant term and the updated second-order terms of the polynomial. In this paper, we adopt the WENO type hierarchical reconstruction to limit the approximate polynomial in each cell. The numerical results in the final section show that such limiting strategy works very well: the high-order numerical accuracy is kept in the smooth region, and at the mean time, the spurious oscillations nearby the shock region are removed or reduced significantly.
Different from the standard k-exact reconstruction which the approximate polynomial on each cell are generated based on cell averages, the approximate polynomial in this paper is given based on the functional value on the centroid of each cell. Then the polynomial is used directly to describe the steady-state of Euler equations in an integrated form. In fact, the scheme proposed in this paper fall into the category of the so-called Residual Distribution schemes, which have been extensively discussed in the literature [12], [1], [10]. Residual distribution schemes start with point values of the solution, then an interpolation to obtain a polynomial approximation of the solution, then an approximation of the residual resulting from an integrated version of the steady-state PDE (which, by the divergence theorem, involves only integrals of the flux along the cell boundaries), then a pseudo-time or another method to compute the resulting nonlinear system. The scheme in this paper differs from traditional residual distribution schemes only in the last step, namely the nonlinear system is solved directly by the Newton type method rather than through residual distribution and pseudo-time marching. We point out that our algorithm has almost the same efficiency as finite volume schemes on the unstructured mesh case, since for both schemes, a least square system (5 × 5 for our scheme and 6 × 6 for finite volume schemes) is solved in every cell in each Newton-iteration step.
The numerical algorithm to be presented in this paper is as follows. On unstructured grids, we first discretize the steady Euler equations to obtain the nonlinear algebraic system using residual distribution schemes, which is then linearized by using the standard Newton-iterations. In the steady case, this algebraic system is actually singular without the time evolution term and require to be regularized. In the standard methods, the system is regularized by adding a local artificial time relaxation. In our algorithm we use the term to regularize the system instead of using the artificial time relaxation, where RHS is the residual of the linearized system on each cell, and is a positive constant.
Since the Newton-iteration is employed to linearize the system, we may need the differential forms for the flux to obtain the local Jacobian matrix. Note that the flux itself may be very complex, such as HLLC [37], CUPS [19], etc., so it is difficult to obtain the differential forms of these fluxes and to use them correctly in the implementations. In this paper, we use numerical differentiation rather than the exact formulas. From our numerical experience, the numerical differentiation can simplify the implementation significantly. We just use the exact forms of the flux and one simple C++ function to get the desired results. In the last section, a number of numerical experiments demonstrate the reliability of the numerical differentiation method.
The resulting regularized linear system for the Newton-iteration is solved by a linear multigrid method in which the block LU-SGS proposed by Wang [38] is utilized as the smoother. In implementing the multigrid methods, a sequence of coarse meshes is generated following the idea of [6]. The coarse meshes are generated to construct the projection operator from fine mesh to coarse mesh, applied on the sparse matrix and the right hand side of the regularized linear system. The standard V-cycle type multigrid iteration is used in our implementation.
The rest of this paper is organized as follows. In the next section, the numerical discretization for the Euler equations will be described. In Sections 3 High order reconstruction, 4 The WENO hierarchical limiting strategy, the high-order reconstruction technique and the limiting strategy will be introduced. Then the Newton-iteration and the multigrid method for solving the corresponding nonlinear system will be discussed in Sections 5 Newton-iteration, 6 The geometric multigrid solver, respectively. Numerical experiments will be carried out in Section 7.
Section snippets
Numerical discretization for the Euler equations
The compressible inviscid Euler equations can be written aswhere U is the vectors of the conservative variables and F is the inviscid flux. We consider the ideal flows with U and F defined aswhere is the density, is the velocity, p is the pressure, and E is the total energy. Finally we close the system by using the equation of statewhere is the ratio of specific heats for air.
For stationary solutions,
High order reconstruction
In this paper, we use the quadratic reconstruction as an example to describe the procedure of reconstruction. Note that the reconstruction is performed on the primitive variables.
Expanding the solution at the centroid of control volume by using Taylor expansion, and truncating after the quadratic term, we get the following approximate polynomial solution:Note that is the function value at the centroid of cell,
The WENO hierarchical limiting strategy
In the hierarchical limiting strategy, several candidates for certain term in the polynomial will be generated. In [26], [27], the non-oscillatory MUSCL method or second-order ENO method is adopted to determine the last term from these candidates. Since the inherent non-differentiability of the ENO schemes, convergence of the solution to steady-state is impossible [31]. In this paper, we focus on the steady-state of the Euler equations, so we follow [42] where the WENO method is used in the
Newton-iteration
We use Newton-iteration to linearize the nonlinear system (5):where . The is updated bywhere is a relaxation parameter on .
The calculation of the derivatives of the numerical flux to the conservative variables has been discussed extensively in the references. Often the derivatives are obtained by using chain rule, which may lead to long and
The geometric multigrid solver
In step 2 of Algorithm 1, we use the geometric multigrid method to solve the linear system. There are two main components of the multigrid method: projection operator and smoother. For constructing project operator, we need to build the so-called element patches. First, the seed cells are chosen by using the following principles: any two seed cells have no common vertexes or sides; and according to [6], it is advised that each element patch contains at least four (for 2D) cells. In the
Numerical results
In this section, we mainly show two aspects of our algorithm. One is that with the WENO hierarchical limiting strategy; the high-order numerical accuracy is obtained in the smooth region, while the numerical oscillations nearby the shock are removed or reduced significantly. The other one is about robustness; our algorithm can work well for several test examples with one fixed set of parameters.
The two aspects above are demonstrated in the following subsections. In the simulations, we always
Conclusion remarks
In this work, we developed a high-order and robust algorithm for steady two-dimensional Euler equations on unstructured grids, which uses the Newton-iteration and the multigrid method to solve the linearized Jacobian matrix. The block LU-SGS iteration was adopted as the smoother of the multigrid algorithm. The WENO hierarchical limiting strategy was presented, which can reduce the numerical oscillations significantly and can also keep the high numerical accuracy generated by the quadratic
Acknowledgments
The research of Hu is supported by a studentship from Hong Kong Baptist University. The research of Li was supported in part by the National Basic Research Program of China under the Grant 2005CB321701 and the National Science Foundation of China under the Grant 10731060. The research of Tang was supported in part by Hong Kong Research Grants Council, the FRG grants of Hong Kong Baptist University and an Cheung Kong Chair Professorship of the Chinese Ministry of Education through Beijing
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