Yen Liu

Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations

An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high-computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.

Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems

In this paper, the fifth in a series, the high-order spectral finite-volume, or spectral volume (SV) method for unstructured grids is extended to three dimensions. Limitations of conventional structured and unstructured methods are first reviewed. The spectral finite-volume method for generalized conservation laws is then described. It is shown that if all grid cells are partitioned into structured sub-cells in a similar manner, the discretizations become universal, and are reduced to the same weighted sum of unknowns involving just a few simple adds and multiplies. Important aspects of the data structure and its effects on communication and the optimum use of cache memory are discussed. Previously defined one-parameter partitions of the SV in 2D are extended to multiple parameters and then used to construct 3D partitions. Numerical solutions of plane electromagnetic waves incident on perfectly conducting circular cylinders and spheres are presented and compared with the exact solution to demonstrate the capability of the method. Excellent agreement has been found. Computation timings show that the new method is more efficient than conventional structured and unstructured methods.

Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids III: One Dimensional Systems and Partition Optimization

In this paper, the third in a series, the Spectral Volume (SV) method is extended to one-dimensional systems—the quasi-1D Euler equations. In addition, several new partitions are identified which optimize a certain form of the Lebesgue constant, and the performance of these partitions is assessed with the linear wave equation. A major focus of this paper is to verify that the SV method is capable of achieving high-order accuracy for hyperbolic systems of conservation laws. Both steady state and time accurate problems are used to demonstrate the overall capability of the SV method.

Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids

A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is developed. The concept of discontinuous and high-order local representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG)[1] and the Spectral Volume (SV)[2] methods, but while these methods are based on the integrated forms of the equations, the new method is based on the differential form to attain a simpler formulation and higher efficiency. Conventional unstructured finite-difference (FD)[3] and finite-volume (FV)[4] methods require data reconstruction based on the least-squares formulation using neighboring point or cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every point or cell at each time step, or store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In addition, the finite-difference method does not satisfy the integral conservation in general. By contrast, the DG and SV methods employ a local, universal reconstruction of a given order of accuracy in each cell in terms of internally defined conservative unknowns. Since the solution is discontinuous across cell boundaries, a Riemann solver is necessary to evaluate boundary flux terms and maintain conservation. In the DG method, a Galerkin finite-element method is employed to update the nodal unknowns within each cell. This requires the inversion of a mass matrix, and the use of quadratures of twice the order of accuracy of the reconstruction to evaluate the surface integrals and additional volume integrals for non-linear flux functions. In the SV method, the integral conservation law is used to update volume averages over subcells defined by a geometrically similar partition of each grid cell. As the order of accuracy increases, the partitioning for 3D requires the introduction of a large number of parameters, whose optimization to achieve convergence becomes increasingly more difficult. Also, the number of interior facets required to subdivide non-planar faces, and the additional increase in the number of quadrature points for each facet, increases the computational cost greatly.

Nonequilibrium flow computations. I. an analysis of numerical formulations of conversation laws

Modern numerical techniques employing properties of flux Jacobian matrices are extended to general, nonequilibrium flows. Generalizations of the Beam-Warming scheme, Steger-Warming and van Leer Flux-vector splittings, and Roes approximate Riemann solver are presented for 3-D, time-varying grids. The analysis is based on a thermodynamic model that includes the most general thermal and chemical nonequilibrium flow of an arbitrary gas. Various special cases are also discussed.

Efficient quadrature-free high-order spectral volume method on unstructured grids

An efficient implementation of the high-order spectral volume (SV) method is presented for multi-dimensional conservation laws on unstructured grids. In the SV method, each simplex cell is called a spectral volume (SV), and the SV is further subdivided into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions. In the traditional implementation, Gauss quadrature formulas are used to approximate the flux integrals on all faces. In the new approach, a nodal set is selected and used to reconstruct a high-order polynomial approximation for the flux vector, and then the flux integrals on the internal faces are computed analytically, without the need for Gauss quadrature formulas. This gives a significant advantage over the traditional SV method in efficiency and ease of implementation. For SV interfaces, a quadrature-free approach is compared with the Gauss quadrature approach to further evaluate the accuracy and efficiency. A simplified treatment of curved boundaries is also presented that avoids the need to store a separate reconstruction for each boundary cell. Fundamental properties of the new SV implementation are studied and high-order accuracy is demonstrated for linear and non-linear advection equations, and the Euler equations. Several well known inviscid flow test cases are utilized to show the effectiveness of the simplified curved boundary representation.

The Spectral Difference Method for the 2D Euler Equations on Unstructured Grids

An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite- difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, a monotonicity limiter is implemented, and tested for a double Mach reflection problem. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.

An Implicit Space-Time Spectral Difference Method for Discontinuity Capturing Using Adaptive Polynomials

*† ‡ A new implicit high-order space-time spectral difference method for capturing discontinuities is developed. The proposed method has the following new improvements over the conventional spectral difference method: (1) The method treats time and space in the same fashion and hence allows conservation laws to be satisfied in the coupled space and time coordinates. (2) The method is fully implicit and can be any order of accuracy and therefore it allows a much larger time step to be taken without losing accuracy. (3) The method can be implemented to obtain the solution by using adaptive polynomial refinements. An approach employing a self-adaptive polynomial refinement will be discussed. (4) A new approach for discontinuity-capturing will be discussed. This approach does not require a limiter and unlike the conventional shock capturing schemes, the proposed method resolves discontinuities by polynomial refinements.

On the accuracy and efficiency of discontinuous Galerkin, spectral difference and correction procedure via reconstruction methods

Numerical accuracy and efficiency of several discontinuous high-order methods, including the quadrature-based discontinuous Galerkin (QDG), nodal discontinuous Galerkin (NDG), spectral difference (SD) and flux reconstruction/correction procedure via reconstruction (FR/CPR), for the conservation laws are analyzed and compared on both linear and curved quadrilateral elements. On linear elements, all the above schemes are one-dimensional in each natural coordinate direction. However, on curved elements, not all schemes can be reduced to a one-dimensional form, although the SD and CPR formulations remain one-dimensional by design. The efficiency and accuracy of various formulations are compared on highly skewed curved elements. Several benchmark problems are simulated to further evaluate the performance of these schemes.

General multi-group macroscopic modeling for thermo-chemical non-equilibrium gas mixtures

This paper opens a new door to macroscopic modeling for thermal and chemical non-equilibrium. In a game-changing approach, we discard conventional theories and practices stemming from the separation of internal energy modes and the Landau-Teller relaxation equation. Instead, we solve the fundamental microscopic equations in their moment forms but seek only optimum representations for the microscopic state distribution function that provides converged and time accurate solutions for certain macroscopic quantities at all times. The modeling makes no ad hoc assumptions or simplifications at the microscopic level and includes all possible collisional and radiative processes; it therefore retains all non-equilibrium fluid physics. We formulate the thermal and chemical non-equilibrium macroscopic equations and rate coefficients in a coupled and unified fashion for gases undergoing completely general transitions. All collisional partners can have internal structures and can change their internal energy states after transitions. The model is based on the reconstruction of the state distribution function. The internal energy space is subdivided into multiple groups in order to better describe non-equilibrium state distributions. The logarithm of the distribution function in each group is expressed as a power series in internal energy based on the maximum entropy principle. The method of weighted residuals is applied to the microscopic equations to obtain macroscopic moment equations and rate coefficients succinctly to any order. The models accuracy depends only on the assumed expression of the state distribution function and the number of groups used and can be self-checked for accuracy and convergence. We show that the macroscopic internal energy transfer, similar to mass and momentum transfers, occurs through nonlinear collisional processes and is not a simple relaxation process described by, e.g., the Landau-Teller equation. Unlike the classical vibrational energy relaxation model, which can only be applied to molecules, the new model is applicable to atoms, molecules, ions, and their mixtures. Numerical examples and model validations are carried out with two gas mixtures using the maximum entropy linear model: one mixture consists of nitrogen molecules undergoing internal excitation and dissociation and the other consists of nitrogen atoms undergoing internal excitation and ionization. Results show that the original hundreds to thousands of microscopic equations can be reduced to two macroscopic equations with almost perfect agreement for the total number density and total internal energy using only one or two groups. We also obtain good prediction of the microscopic state populations using 5-10 groups in the macroscopic equations.