Spectral Volume Method Research Articles

Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws

AbstractIn this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.

Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients

Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients

Unified Analysis of Any Order Spectral Volume Methods for Diffusion Equations

Unified Analysis of Any Order Spectral Volume Methods for Diffusion Equations

Any order spectral volume methods for diffusion equations using the local discontinuous Galerkin formulation

In this paper, we present and study two spectral volume (SV) schemes of arbitrary order for diffusion equations by using the local discontinuous Galerkin formulation to discretize the viscous flux. The basic idea of the scheme is to rewrite the diffusion equation into an equivalent first-order system first, and then use the SV method to solve the system. The SV scheme is designed with control volumes constructed by using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as LSV and RSV schemes, respectively. The stability analysis for the linear diffusion equations based on alternating fluxes are provided, and optimal error estimates are established for both the exact solution and the auxiliary variable. Furthermore, a rigorous mathematical proof are given to demonstrate that the proposed RSV method is identical to the standard LDG method when applied to constant diffusion problems. Numerical experiments are presented to demonstrate the stability, accuracy and performance of the two SV schemes for both linear and nonlinear diffusion equations.

A new well-balanced spectral volume method for solving shallow water equations over variable bed topography with wetting and drying

A new well-balanced spectral volume method for solving shallow water equations over variable bed topography with wetting and drying

Analysis of Spectral Volume Methods for 1D Linear Scalar Hyperbolic Equations

Analysis of Spectral Volume Methods for 1D Linear Scalar Hyperbolic Equations

Numerical simulation of 1-D oil and water displacements in petroleum reservoirs using the correction procedure via reconstruction (CPR) method

The focus of this work is to investigate and to apply, for the first time, a very high-resolution correction procedure via reconstruction (CPR) numerical discretization technique for the hyperbolic saturation equation that describes 1-D oil-water displacement through heterogeneous porous media. The CPR method can achieve high-order accuracy via a compact stencil consisting of the current cell and its immediate neighbors; in addition, the CPR recovers simplified versions of nodal discontinuous Galerkin (NDG), spectral volume (SV), and spectral difference (SD) methods using an adequate polynomial reconstruction function, whose coefficients are preprocessed and stored. Indeed the CPR versions of NDG and SV/SD are highly efficient. In order to suppress numerical oscillations near shocks, which are typical from higher-order schemes, a hierarchical MLP (multi-dimensional limiting process) is used in the reconstruction stage. The integration in time is carried out using a third-order RK (Runge-Kutta) method. A number of 1-D two-phase flow benchmark problems are solved, to verify the accuracy, efficiency, and shock-capturing capability of the adopted methodology.

A new WENO based Chebyshev Spectral Volume method for solving one- and two-dimensional conservation laws

High-order methods have attracted considerable attention in the CFD fields because of their potential in achieving higher accuracy with a lower cost than low order methods. In this paper, a new class of high order spectral volume method based on Chebyshev polynomials as an approximation function called Chebyshev Spectral Volume (CSV) method is presented for hyperbolic conservation laws in quadrilateral meshes. As we know, the hyperbolic conservation law may contain discontinuities even if the initial conditions are smooth. To capture these discontinuities, we propose a new limiter to reconstruct the numerical approximation on the target cell, which is marked as a troubled cell. It is based on the Chebyshev polynomials in the target and its immediate neighboring cells to achieve a high order of accuracy and non-oscillatory properties. Also, a new indicator is introduced to detect troubled cells. This indicator depends neither on the numerical order of accuracy nor on a problem-dependent parameter. A series of 1D and 2D scalar and system conservation laws are presented to evaluate the performance of the CSV method coupled with the new limiter and indicator.

Grid Convergence Study of Spectral Volume Method for Hybrid Unstructured Mesh

Grid convergence characteristics of the spectral volume (SV) method is explored for hybrid unstructured meshes. In this study, the SV method originally developed for tetrahedral computational cells is extended to use prismatic and hexahedral cells. The extended SV method (hereafter SV+) for hybrid unstructured mesh is first applied to solve the linear advection and diffusion problems. The formal spatial accuracy is rigorously kept in both problems. Then, the SV+ is applied to solve the turbulent boundary layer flow over a flat plate. It is shown that the SV+ can obtain grid convergence even for using tetrahedral cells in the turbulent boundary layer, where the mesh aspect ratio becomes as large as 20,000. Convergence toward steady solutions using prismatic layers with the same number of DOFs becomes about two times faster than that using tetrahedral cells in the turbulent boundary layer. Finally, the SV+ is applied to compute the flowfield around the NASA-CRM. It is shown that the present SV+ gives mesh converged solutions which quantitatively agree with the corresponding experimental data regardless of mesh type.

Study of Fast Prediction Method for Transonic Flutter Boundary Using Radial Basis Function

A spectral volume method for hybrid unstructured mesh developed earlier is built in an aeroelastic code to determine transonic flutter boundary of AGARD 445.6 wing. It is shown that the computed transonic dip of flutter boundary reasonably agrees with the experimental data. In the present study, we focus on to develop a fast numerical method for evaluating the bottom of transonic dip. In order to reduce the number of computational cases to determine the flutter boundary, an interpolation method using the radial basis function is examined. The initial sampling points are sparsely distributed over the entire parameter plane to roughly determine the flutter boundary by interpolating damping ratio of wing oscillation. Then, based on the consideration of load distribution over the wing, some additional sampling points are introduced in the region where the transonic dip likely appears. It is demonstrated that the present approach allows us to evaluate the bottom of transonic dip with less computational efforts for the present problem. Discussions are included to examine the possible efforts of the arrangement of initial sampling points and interpolation method on the shape of flutter boundary near the dip.

Extension of Spectral Volume Method to Hexahedral Mesh and Application to Flowfield Calculation around NASA-CRM

Extension of Spectral Volume Method to Hexahedral Mesh and Application to Flowfield Calculation around NASA-CRM

Further Development and Application of High-Order Spectral Volume Methods for Compressible Flows

The present work investigates the high-order spectral finite volume (SFV) method with emphasis on applicability aspects for compressible flows. The intent is to improve the understanding and implementation of numerical techniques related to high-order unstructured grid schemes. In that regard, a hierarchical moment limiter and high-order mesh capability are developed for a two-dimensional Euler SFV solver. The limiter formulation and geometry interpreter for high-order mesh creation are new contributions for the SFV method. Literature test cases are evaluated to assess the interaction of curved mesh, limiter and spatial reconstruction features of the SFV scheme. An order of accuracy study is presented along with steady and unsteady problems with strong shock waves and other discontinuities typical of compressible flows. Moreover, second, third and fourth-order spatial resolution analyses are explored and the SFV results are compared with results from different numerical methods.

High order sub-cell finite volume schemes for solving hyperbolic conservation laws I: basic formulation and one-dimensional analysis

In this paper, a family of sub-cell finite volume schemes for solving the hyperbolic conservation laws is proposed and analyzed in one-dimensional cases. The basic idea of this method is to subdivide a control volume (main cell) into several sub-cells and the finite volume discretization is applied to each of the sub-cells. The averaged values on the sub-cells of current and face neighboring main cells are used to reconstruct the polynomial distributions of the dependent variables. This method can achieve arbitrarily high order of accuracy using a compact stencil. It is similar to the spectral volume method incorporating with $\mathrm{P_NP_M}$ technique but with fundamental differences. An elaborate utilization of these differences overcomes some shortcomings of the spectral volume method and results in a family of accurate and robust schemes for solving the hyperbolic conservation laws. In this paper, the basic formulation of the proposed method is presented. The Fourier analysis is performed to study the properties of the one-dimensional schemes. A WENO limiter based on the secondary reconstruction is constructed.

Summation-by-parts operators for correction procedure via reconstruction

The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. Linearly stable schemes were presented by Vincent et al. (2011, 2015), but proofs of non-linear (entropy) stability in this framework have not been published yet (to the knowledge of the authors). We reformulate CPR methods using summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), a framework popular for finite difference methods, extending the results obtained by Gassner (2013) for a special discontinuous Galerkin spectral element method. This reformulation leads to proofs of conservation and stability in discrete norms associated with the method, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Additionally, extending the skew-symmetric formulation of conservation laws by additional correction terms, entropy stability for Burgers' equation is proved for general SBP CPR methods not including boundary nodes.

Hybrid spectral difference/embedded finite volume method for conservation laws

Recently, interests have been increasing towards applying the high-order methods to various engineering applications with complex geometries [30]. As a result, a family of discontinuous high-order methods, such as Discontinuous Galerkin (DG), Spectral Volume (SV) and Spectral Difference (SD) methods, is under active development. These methods provide high-order accurate solutions and are highly parallelizable due to the local solution reconstruction within each element. But, these methods suffer from the Gibbs phenomena when discontinuities are present in the flow fields. Various types of limiters [43–45] and artificial viscosity [46,48] have been employed to overcome this problem.A novel hybrid spectral difference/embedded finite volume method is introduced in order to apply a discontinuous high-order method for large scale engineering applications involving discontinuities in the flows with complex geometries. In the proposed hybrid approach, the finite volume (FV) element, consisting of structured FV subcells, is embedded in the base hexahedral element containing discontinuity, and an FV based high-order shock-capturing scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is captured at the resolution of FV subcells within an embedded FV element. In the smooth flow region, the SD element is used in the base hexahedral element. Then, the governing equations are solved by the SD method. The SD method is chosen for its low numerical dissipation and computational efficiency preserving high-order accurate solutions. The coupling between the SD element and the FV element is achieved by the globally conserved mortar method [56]. In this paper, the 5th-order WENO scheme with the characteristic decomposition is employed as the shock-capturing scheme in the embedded FV element, and the 5th-order SD method is used in the smooth flow field.The order of accuracy study and various 1D and 2D test cases are carried out, which involve the discontinuities and vortex flows. Overall, it is shown that the proposed hybrid method results in comparable or better simulation results compared with the standalone WENO scheme when the same number of solution DOF is considered in both SD and FV elements.

Hybrid格子を用いたSpectral Volume法の検討

Hybrid格子を用いたSpectral Volume法の検討

High-order methods for computational fluid dynamics: A brief review of compact differential formulations on unstructured grids

Popular high-order schemes with compact stencils for Computational Fluid Dynamics (CFD) include Discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV) methods. The recently proposed Flux Reconstruction (FR) approach or Correction Procedure using Reconstruction (CPR) is based on a differential formulation and provides a unifying framework for these high-order schemes. Here we present a brief review of recent progress in FR/CPR research as well as some pacing items and future challenges.

Reviews of high-order methods on unstructured and hybrid grid

While 2nd order methods are dominant in most compressible flow simulations, many types of problems, such as computational aeroacoustics (CAA), vortex-dominant flows and large eddy simulation (LES) of turbulent flows, call for higher order accuracy (third order and more). The main deficiency of widely available, second-order methods for the accurate simulations of the above-mentioned flows is the numerical diffusion and dissipation of vorticity to unacceptable level. Applications of high-order accu- rate, low-diffusion and low dissipation numerical methods can significantly alleviate this deficiency of the traditional second order methods, improve predictions of vortical and other complex, separated, unsteady flows. On the other hand, for complex configurations, the structured/unstructured hybrid grid technique presents the trend of grid generation technique. Therefore the high-order methods on unstructured and hybrid (or mixed) grids are paid much more attention in recent years. In this paper, the high-order methods on unstructured and hybrid grids are reviewed comprehensively, including the k-exact finite volume (FV) methods, the FV methods based on ENO and WENO reconstruction, the discontinuous Galerkin (DG) finite element method, the finite spectral volume (SV) methods, the finite spectral dif- ference (SD) method, DG/FV hybrid methods, the unified method based on correction procedure via reconstruction (CPR). The main ideas of these high-order methods are introduced, and the advantages and disadvantages of these methods are discussed. In addition, some important issues for simulations of complex geometry are discussed, including the treatment of curved boundary, the detector of discontinu-ity and high-order limiters, implicit iteration methods, and hp-multigrid approaches. We believe that the high-order methods on unstructured and hybrid grids will play more and more important role on more accurate simulations of realistic airspace vehicles and study of complex flow mechanism in the future.

An implicit LU-SGS spectral volume method for the moment models in device simulations III: accuracy enhancement using the LDG2 flux formulation for non-uniform grids

The LDG2 viscous flux formulation was recently developed by Kannan and Wang for the spectral volume setting. The LDG2 scheme is a variant of the more popular LDG formulation and retains the attractive features of the latter. In addition, it also displays higher degree of symmetry and accuracy than the LDG approach and has a milder stability constraint than the original formulation. In this paper, the third in a series, the accuracy of the high-order spectral volume method for the moment models in device simulations is enhanced by using the LDG2 formulation for discretizing the second derivative diffusive fluxes. More specifically, the emphasis is on showing the efficacy of the new formulation, when used in a domain discretized with highly non-uniform grids. Unlike the penalty formulations such as the interior penalty and the Bassi and Rebay 2 (BR2) schemes, the LDG2 formulation requires no length-based penalizing terms and is compact. This eliminates problems associated with obtaining a right length scale when the grid sizes are varying rapidly. A n+–n–n+ diode was assumed for simulation purposes. The results obtained by solving the steady state hydrodynamic models and the energy transport models are compared with the existing LDG, penalty, and the BR2 results. The new formulation always yields more accurate solutions than the ones that used the LDG formulation. It also displays (i) higher fidelity than the BR2 formulations and (ii) comparable fidelity to the penalty formulation when the grids are highly non-uniform. The LDG2 formulation is compact and hence can be easily parallelized. In general, the numerical results are very promising and indicate that the new diffusive flux formulation is suited better than the existing formulations when the grids are non-uniform. Copyright © 2012 John Wiley & Sons, Ltd.