An Incompressible High-Order Solver with Thermal Coupling

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

Mass conservation of the unified continuous and discontinuous element-based Galerkin methods on dynamically adaptive grids with application to atmospheric simulations

On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

We propose a new technique for computing highly accurate approximations to linear functionals in terms of Galerkin approximations. We illustrate the technique on a simple model problem, namely, that of the approximation of J(u), where $$J(\cdot )$$ is a very smooth functional and u is the solution of a Poisson problem; we assume that the solution u and the solution of the adjoint problem are both very smooth. It is known that, if $$u_h$$ is the approximation given by the continuous Galerkin method with piecewise polynomials of degree $$k>0$$ , then, as a direct consequence of its property of Galerkin orthogonality, the functional $$J(u_h)$$ converges to J(u) with a rate of order $$h^{2k}$$ . We show how to define approximations to J(u), with a computational effort about twice of that of computing $$J(u_h)$$ , which converge with a rate of order $$h^{4k}$$ . The new technique combines the adjoint-recovery method for providing precise approximate functionals by Pierce and Giles (SIAM Rev 42(2):247–264, 2000), which was devised specifically for numerical approximations without a Galerkin orthogonality property, and the accuracy-enhancing convolution technique of Bramble and Schatz (Math Comput 31(137):94–111, 1977), which was devised specifically for numerical methods satisfying a Galerkin orthogonality property, that is, for finite element methods like, for example, continuous Galerkin, mixed, discontinuous Galerkin and the so-called hybridizable discontinuous Galerkin methods. For the latter methods, we present numerical experiments, for $$k=1,2,3$$ in one-space dimension and for $$k=1,2$$ in two-space dimensions, which show that $$J(u_h)$$ converges to J(u) with order $$h^{2k+1}$$ and that the new approximations converges with order $$h^{4k}$$ . The numerical experiments also indicate, for the p-version of the method, that the rate of exponential convergence of the new approximations is about twice that of $$J(u_h)$$ .

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

High-order methods are quickly becoming popular for turbulent flows as the amount of computer processing power increases. The flux reconstruction (FR) method presents a unifying framework for a wide class of high-order methods including discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV). It offers a simple, efficient, and easy way to implement nodal-based methods that are derived via the differential form of the governing equations. Whereas high-order methods have enjoyed recent success, they have been known to introduce numerical instabilities due to polynomial aliasing when applied to under-resolved nonlinear problems. Aliasing errors have been extensively studied in reference to DG methods; however, their study regarding FR methods has mostly been limited to the selection of the nodal points used within each cell. Here, we extend some of the de-aliasing techniques used for DG methods, primarily over-integration, to the FR framework. Our results show that over-integration does remove aliasing errors but may not remove all instabilities caused by insufficient resolution (for FR as well as DG).

SummaryIn this paper, a numerical method, which is about the coupling of continuous and discontinuous Galerkin method based on the splitting scheme, is presented for the calculation of viscoelastic flows of the Oldroyd‐B fluid. The momentum equation is discretized in time by using the Adams‐Bashforth second‐order algorithm, and then decoupled via the splitting approach. Considering the Oldroyd‐B constitutive equation, the second‐order Runge‐Kutta approach is selected to complete the temporal discretization. As for the spatial discretizations, the fundamental purpose is to make the best of finite element method (FEM) and discontinuous Galerkin (DG) method to handle different types of equations. Specifically speaking, for the subequations, FEM is chosen to treat the Poisson and Helmholtz equations, and DG is employed to deal with the nonlinear convective term. In addition, because of the hyperbolic nature, DG is also utilized to discretize the Oldroyd‐B constitutive equation spatially. This coupled method avoids resorting to extra stabilization technique occurred in standard FEM framework even for moderately high values of Weissenberg number and also reduces the complexity compared with unified DG scheme. The Oldroyd‐B model is applied to investigate several typical and challenging benchmarks, such as the 4:1 planar contraction flow and the lid‐driven cavity flow, with a wide range of Weissenberg number to illustrate the feasibility, robustness, and validity of our coupled method.

Discretely nonlinearly stable weight-adjusted flux reconstruction high-order method for compressible flows on curvilinear grids

A comparison of continuous and discontinuous Galerkin methods for hydrodynamic transport

Locally adapted meshes and polynomial degrees can greatly improve spectral element accuracy and applicability. A balancing domain decomposition by constraints (BDDC) preconditioner is constructed and analysed for both continuous (CG) and discontinuous (DG) Galerkin discretizations of scalar elliptic problems, built by nodal spectral elements with variable polynomial degrees. The DG case is reduced to the CG case via the auxiliary space method. The proposed BDDC preconditioner is proved to be scalable in the number of subdomains and quasi-optimal in both the ratio of local polynomial degrees and element sizes and the ratio of subdomain and element sizes. Several numerical experiments in the plane confirm the obtained theoretical convergence rate estimates, and illustrate the preconditioner performance for both CG and DG discretizations. Different configurations with locally adapted polynomial degrees are studied, as well as the preconditioner robustness with respect to discontinuities of the elliptic coefficients across subdomain boundaries. These results apply also to other dual-primal preconditioners defined by the same set of primal constraints, such as FETI-DP preconditioners.

High-Order Implicit Large Eddy Simulation using Entropically Damped Artificial Compressibility

A bulk metal oxide semiconductor field-effect transistor (MOSFET) is modelled with symmetric interior penalty discontinuous Galerkin (DG) method. The proposed framework is based on the drift-diffusion (DD) model and the Poisson equation. The Gummel algorithm is employed to decouple relevant equations. The potential and carrier distribution under a given bias are therefore obtained to demonstrate the feasibility of the proposed framework. Compared with the traditional continuous Galerkin (CG) method, this method is proven to achieve better numerical stability.

Ensuring provable nonlinear stability provides bounds on the discrete solution and guarantees that the numerical scheme remains convergent. In the context of compressible flows, nonlinear stability is achieved by enforcing a secondary conservation principle—the second law of thermodynamics. For high-order numerical methods, such as discontinuous Galerkin (DG), discrete nonlinear and entropy stability have been effectively realized. These stability guarantees are typically derived using properties of the L2-norm. We have developed a nonlinearly stable flux reconstruction (NSFR) scheme for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to continue to benefit from the use of larger time-steps than DG due to the ESFR correction functions while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We employ a modified mass matrix in a weight-adjusted form that reduces the computational cost in curvilinear coordinates through a precomputed projection operator that approximates the dense matrix inversion and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum factorization. A novel, fully-discrete, nonlinearly stable flux reconstruction (FD-NSFR) framework is introduced, which guarantees entropy stability in both spatial and temporal domains for high-order methods via the relaxation Runge-Kutta scheme. We developed an FD-NSFR scheme that prevents a temporal numerical entropy change in the broken Sobolev norm if the governing equations admit a convex entropy function that can be expressed in inner-product form. Through the use of a bound-preserving limiter, positivity of thermodynamic quantities is preserved and enables the extension of this scheme to hyperbolic conservation laws. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy-conserving NSFR scheme and find that our proposed entropy-conserving NSFR scheme is computationally competitive with the conservative DG scheme.

Coupling local discontinuous and continuous galerkin methods for flow problems

SUMMARYTo evaluate the computational performance of high‐order elements, a comparison based on operation count is proposed instead of runtime comparisons. More specifically, linear versus high‐order approximations are analyzed for implicit solver under a standard set of hypotheses for the mesh and the solution. Continuous and discontinuous Galerkin methods are considered in two‐dimensional and three‐dimensional domains for simplices and parallelotopes. Moreover, both element‐wise and global operations arising from different Galerkin approaches are studied. The operation count estimates show, that for implicit solvers, high‐order methods are more efficient than linear ones. Copyright © 2013 John Wiley & Sons, Ltd.