Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional systems

Wang, Zhi Jian; Zhang, Laiping; Liu, Yen

doi:10.1016/j.jcp.2003.09.012

Cited by 276 publications

(235 citation statements)

References 24 publications

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“…A monotone higher order scheme must necessarily be nonlinear, which usually takes the form of limiter functions that depend on the current solution. For many high-order schemes, such as the DG method [6] and the SV method [28][29][30] limiters based on the minmod function have proved quite successful, and have led to convergence proofs in the total-variation-diminishing (TVD) [9] or total-variation-bounded (TVB) [20] framework. We use a similar general approach, which is outlined in the following for one and two space dimensions.…”

Section: Discontinuity Capturing and Data Limitingmentioning

confidence: 99%

“…Two triangular grids were generated with 133,480, and 533,920 triangular cells, respectively. All the simulations were carried until t = 0.2 using the CUSP flux and TVB limiter similar to the one developed for the SV method [30]. Figure 9 shows the density contours computed with the third-order SD scheme on the coarse, and fine grids.…”

Section: Double Mach Reflectionmentioning

confidence: 99%

“…In the present study, the SD method is further extended to the Euler equations. The primary motivation for developing another numerical method is to seek a simpler to implement and more efficient method than the current state of the art-the discontinuous Galerkin (DG) method [2,5,6], and the spectral volume (SV) method [17,[27][28][29][30], to name just a few high-order methods for conservation laws on unstructured grids. As a matter of fact, the DG, SV, and SD methods are similar in that they share the same the solution space, i.e., the space of piece-wise discontinuous polynomials, and some Riemann solvers [11,18,19] are used at the element interfaces to provide solution coupling between the discontinuous elements and appropriate numerical dissipation necessary to achieve stability.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2006

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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.

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Section: Discontinuity Capturing and Data Limitingmentioning

confidence: 99%

Section: Double Mach Reflectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2006

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“…Then the method was extended to analyze some nonlinear convection diffusion problems in [23]. In [20,21], such approach is adopted to construct more general higher order schemes on unstructured grids, but no theoretical analysis was presented. By use of Gauss integration and approximation replacement of the weak forms, analysis of a discrete biquadratic FVE scheme on rectangular meshes was presented for elliptic problems in [22].…”

Section: Introductionmentioning

confidence: 99%

A second-order finite volume element method on quadrilateral meshes for elliptic equations

Yang¹

2006

ESAIM: M2AN

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Abstract. In this paper, by use of affine biquadratic elements, we construct and analyze a finite volume element scheme for elliptic equations on quadrilateral meshes. The scheme is shown to be of second-order in H 1 -norm, provided that each quadrilateral in partition is almost a parallelogram. Numerical experiments are presented to confirm the usefulness and efficiency of the method.Mathematics Subject Classification. 65N30, 65N15.

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“…Structured meshes make p-refinement easier by allowing straightforward stencil size adjustments. However, general unstructured grids are more complex to handle [66,67,68]. Another drawback of larger stencils is the order reduction in the primal and dual solutions around the boundaries of the domain [69], where the stencil cannot be extended to allow sufficient accuracy (except in the case of periodic boundary conditions).…”

Section: Adjoint Interpolation and Restriction Operators For The Finimentioning

confidence: 99%

Space–time adaptive solution of inverse problems with the discrete adjoint method

Alexe

Sandu

2014

Journal of Computational Physics

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Abstract. Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms.In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, h/p-refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem. Space-time adaptive solution of inverse problems with the discrete adjoint method2

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Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional systems

Cited by 276 publications

References 24 publications

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

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

A second-order finite volume element method on quadrilateral meshes for elliptic equations

Space–time adaptive solution of inverse problems with the discrete adjoint method

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