Article

A performance comparison of nodal discontinuous Galerkin methods on triangles and quadrilaterals

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Abstract

This work presents a study on the performance of nodal bases on triangles and on quadrilaterals for discontinuous Galerkin solutions of hyperbolic conservation laws. A nodal basis on triangles and two tensor product nodal bases on quadrilaterals are considered. The quadrilateral element bases are constructed from the Lagrange interpolating polynomials associated with the Legendre–Gauss–Lobatto points and from those associated with the classical Legendre–Gauss points. Settings of interest concern the situation in which a mesh of triangular elements is obtained by dividing each quadrilateral element into two triangular elements or vice versa, the mesh of quadrilateral elements is obtained by merging two adjacent triangular elements. To assess performance, we use a linear advecting rotating plume transport problem as a test case. For cases where the order of the basis is low to moderate, the computing time used to reach a given final time for the quadrilateral elements is shorter than that for the triangular elements. The numerical results also show that the quadrilateral elements yield higher computational efficiency in terms of cost to achieve similar accuracy. Copyright © 2010 John Wiley & Sons, Ltd.

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... From this observation, the use of quadrilateral elements appears to be an appealing means to potentially improve the computational efficiency of DG schemes. In our previous work [13], we conducted a study of the performance of the numerical solution of a linear two-dimensional transport problem using DG methods based on Lagrange nodal bases on triangles and tensor-product nodal bases on quadrilaterals. Indeed, numerical results shown there clearly demonstrated the benefit of using the nodal tensor-product bases on quadrilaterals wherein, for low to moderate interpolation order, the computing time used in the simulation is shorter than that of the simulation using the nodal bases on triangles. ...
... set with the classical two-dimensional Legendre-Gauss distribution was also considered in [13]). Figure 4 depicts the nodal set for p = 5. ...
... elemental mass and stiffness matrices) of each element can no longer be obtained by scaling the elemental matrices associated with the master element. While they can be computed accurately and subsequently stored element-by-element, we adopt a less accurate but more memory-economical approach in approximating such matrices [13]. Such an approach, owing to the use of a (fixed order) classical two-dimensional Gauss quadrature, defines the approximate elemental matrices as a multiplication of the matrices defined on the master element and the matrices involved with the coordinate mapping. ...
Article
We present a comprehensive assessment of nodal and hybrid modal/nodal discontinuous Galerkin (DG) finite element solutions on a range of unstructured meshes to nonlinear shallow water flow with smooth solutions. The nodal DG methods on triangles and a tensor-product nodal basis on quadrilaterals are considered. The hybrid modal/nodal DG methods utilize two different synergistic polynomial bases on polygons in realizing the DG discretization; orthogonal basis functions constructed by the Gram–Schmidt process are used as trial and test functions in a DG weak formulation; and a nodal basis is used as an efficient means for area integration. These are implemented on triangular, quadrilateral, and polygonal elements. In addition, we discuss aspects to be considered in order to achieve the so-called well-balanced property that preserves steady state at rest with a spatially varying bed. The performance in terms of accuracy and computational cost is demonstrated using h and p convergence studies on a nonlinear problem with a manufactured solution and the nonlinear Stommel problem with flat and non-flat beds. To assess the performance of quadrilateral and polygonal elements in comparison to triangular elements, we consider a setting in which a quadrilateral mesh, a mixed triangular–quadrilateral mesh, and polygonal mesh are derived from a given triangular mesh and vice versa. The tests conducted reveal the merit of using the quadrilateral elements in terms of computational cost per accuracy and computing time. More importantly, the numerical results clearly show that high order schemes significantly improve the cost performance for a given level of accuracy, with cubic or bi-cubic interpolants particularly achieving dramatic improvements in accuracy as compared to linear and quadratic interpolants, with diminishing benefit as p>3p>3.
... Though using the high order DG scheme can obtain more accurate results, it also possesses more degrees of freedom and requires a higher computational cost. With the aim of enhancing the computational efficiency, the nodal DG method has begun to attract much attention and numerous research works have been published [5,[7][8][9][10][11]. Hesthaven and Warburton [7,8] applied the nodal DG method into solving the Maxwell and Navier-Stokes equations, along with the quadrature-free approach to get rid of the full-order integration, which is a major part of the computational cost. ...
... , which consists of polynomials in element Ω e with degree at most p and allows discontinuities at element boundaries. In this study, we use the nodal approach from Hesthaven and Warburton [8], where the φ i represents the i th Lagrange interpolation function, and implement the nodal sets with Legendre-Gauss-Lobatto points on the edges of the triangles and quadrilaterals [8][9][10]. As the main aspect of this study is to focus on the construction of the multi-dimensional limiter for nodal DG methods, for the representation of the bases function constructions please refer to [7][8][9]11,17] [29,30]. ...
... A C C E P T E D M A N U S C R I P T is twice than that in the quadrilateral meshes. This is partially because the nodal basis functions on triangular and quadrilateral elements are quite different, and the basis on the quadrilateral elements contains more polynomial terms than the triangular basis [10,11], which is also the main advantage of utilizing the quadrilateral elements with the nodal DG methods. ...
Article
The nodal discontinuous Galerkin (DG) methods possess many good properties that make them very attractive for numerically solving the shallow water equations, but it is necessary to maintain numerical monotonicity by applying a slope-limiting approach to eliminate spurious oscillations. In this study, a new vertex-based slope limiter is developed for the nodal DG method on arbitrary unstructured meshes. This new limiting approach modifies the vertex-based limiter by a weighted reconstruction function, which satisfies the maximum principle and is totally free of tuning parameters. Three different weighting functions are proposed by modifying the stencil reconstruction methods from other limiters, and their performance and accuracy are compared through two shock flow problems and a laboratory-scale tsunami problem on triangular and quadrilateral meshes. The results indicate that the proposed limiter can eliminate the nonphysical oscillations efficiently and provide accurate and robust results on both triangular and quadrilateral unstructured meshes.
... Note that successful SWE CG-based solvers are mostly based on linear interpolation. Numerical investigations in [17,18], which employ test problems with smooth solutions, demonstrate a remarkable advantage of the high-order DG methods (p > 1) in terms of costper-accuracy performance. For test problems used therein, the cost required for a moderate level of error in DG solutions with quadratic or bi-quadratic interpolants are typically two orders of magnitude lower than that of the linear-element DG methods. ...
... Curved solid physical boundary Figure 3: Curvature-boundary-condition approach for two integration points By following [21], the CBC approach for realizing (16) in the DG scheme is carried out by setting the fictitious exterior state at an integration point to (17) where T denotes the unit tangential vector (T · N = 0 and T × N = k ). The velocity of the exterior state given above is the reflection of the interior velocity with respect to T. Note that (17) amounts to specifying the velocity components of the exterior state as ...
... In the above discussion, the resolution of the computation is changed by refining the mesh sizes. The resolution can be also be refined by changing the order of DG polynomials, p, which yields great benefit for a problem with a smooth solution (see for example [17], [18] for a performance study of high-order DG methods). Figures 12 and 13 show the plots of velocity magnitude at t = 2 days under p-refinement on the h-mesh. ...
Article
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In this work, we consider the application of Discontinuous Galerkin (DG) solutions to open channel flow problems, governed by two-dimensional shallow water equations (SWE), with solid curved wall boundaries on which the no-normal flow boundary conditions are prescribed. A commonly used approach consists of straightforwardly imposing the no-normal flow condition on the linear approximation of curved walls. Numerical solutions indicate clearly that this approach could lead to unfavorable results and that a proper treatment of the no-normal flow condition on curved walls is crucial for an accurate DG solution to the SWE. In the test case used, errors introduced through the commonly used approach result in artificial boundary layers of one-grid-size thickness in the velocity field and a corresponding over-prediction of the surface elevation in the upstream direction. These significant inaccuracies, which render the coarse mesh solution unreliable, appear in all DG schemes employed including those using linear, quadratic, and cubic DG polynomials. The issue can be alleviated by either using an approach accounting for errors introduced by the geometric approximation or an approach that accurately represents the geometry.
... High order methods are appealing in a large number of scientific contexts, being well-suited for maximizing performance to accuracy ratios in specific applications [18,22,23,48]. As is frequently the case in nonlinear models, the nonlinear partial differential equations that govern the solution behavior, usually as presented in the weak formulation, are well-known mathematically to exhibit discontinuous solutions and singular behaviors. ...
... In the case of DG methods, the arguments against them are often aimed at the increased degrees of freedom relative to standard finite element methods. However, careful study has shown that these arguments are, at least at present, largely misleading, as DG methods are unusually well-suited for modern supercomputer architectures that emphasize arithmetic intensity over memory access [24], and also since modal enrichment in p is known to scale more efficiently than spatial refinement in h [48]. In fact, among those who use DG methods, the most concerning feature of the method is not its computational expense or scaling, but rather the fact that DG methods tend to demonstrate remarkable instabilities, with relatively frequent blow-up (or blow-down) behavior. ...
Article
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Nonlinear systems of equations demonstrate complicated regularity features that are often obfuscated by overly diffuse numerical methods. Using a discontinuous Galerkin finite element method, we study a nonlinear system of advection-diffusion-reaction equations and aspects of its regularity. For numerical regularization, we present a family of solutions consisting of: (1) a sharp, computationally efficient slope limiter, known as the BDS limiter, (2) a standard spectral filter, and (3) a novel artificial diffusion algorithm with a solution-dependent entropy sensor. We analyze these three numerical regularization methods on a classical test in order to test the strengths and weaknesses of each, and then benchmark the methods against a large application model.
... Concerning Maxwell's equations in time-domain, the DGM has been studied in particular in [7,8,9,10]. The latter two make use of hexahedral meshes, which allow for a computationally more efficient implementation [11]. ...
... The surface integrals in (10) and (11) represent interelement fluxes, the volume integrals are referred to as the mass and stiffness terms according to standard FE nomenclature. In the following the dependence of the spatial and temporal variable is not written down explicitly. ...
Article
An hp-adaptive Discontinuous Galerkin Method for electromagnetic wave propagation phenomena in the time-domain is proposed. The method is highly efficient and allows for the first time the adaptive full-wave simulation of transient problems in three-dimensional space. Refinement is performed anisotropically in the approximation order, p, and the mesh step size, h, regardless of the resulting level of hanging nodes. For guiding the adaptation process a variant of the concept of reference solutions with largely reduced computational costs is proposed. The computational mesh is adapted such that a given error tolerance is respected throughout the entire time-domain simulation.
... The work described in this thesis is restricted to simplical elements, i.e. triangles and tetrahedra. A recent study [147] suggests that quadrilateral meshes yield a better computational efficiency than triangular meshes. In Ref. [147], the lower number of elements and edges in quadrilateral meshes, as well as the richer polynomial basis, are credited for this superior efficiency. ...
... A recent study [147] suggests that quadrilateral meshes yield a better computational efficiency than triangular meshes. In Ref. [147], the lower number of elements and edges in quadrilateral meshes, as well as the richer polynomial basis, are credited for this superior efficiency. It can also be noted that the ratio of the number of DoF's representing the solution on the edges of an element to the total number of DoF's in the element is lower for quadrilaterals than for triangles. ...
... Due to their local nature, DG methods can be extended to achieve high-order accuracy on unstructured meshes through p-refinement in a straightforward manner compared to FV methods where the determination of an appropriate stencil spanning numerous adjacent finite-volume cells is non-trivial [11]. Through the use of high-order approximations, DG methods are able to outperform low-order methods on a cost per accuracy basis [27]. DG methods are also highly parallelizable due to their local stencil and use of explicit time stepping, making them attractive from a high performance computing perspective [3]. ...
Article
Full-text available
This paper presents a computational framework developed to improve both the serial and parallel performance of two dimensional, unstructured, discontinuous Galerkin (DG) solutions to hyperbolic conservation laws. The coding techniques employed factor in advancements trending in HPC technologies. They are designed to maximize loop vectorization, efficiently utilize cache, facilitate straightforward shared memory parallelization, reduce message passing volume, and increase the overlap between computation and communication. With today’s CPU technology and HPC networks rapidly evolving, it is important to quantitatively assess and compare these methodologies with standard paradigms in order to maximize current computational resources. In our benchmark studies, we specifically investigate the shallow water equations to show that the refactored algorithm implementation is able to provide a significant performance increase over the conventional elemental DG code structure in terms of both CPU time and parallel scalability. Our results show that the serial optimizations result in a 28–38 % performance increase. For parallel computations our improvements give rise to a 1.5–2.0 speedup factor for local problem sizes between 10 and 2000 elements per core, regardless of the overall problem size. The computational benchmarks were performed on the Lonestar and Stampede supercomputers at the Texas Advanced Computing Center.
... The unstructured models also have a better implementation of bottom friction and additional physics built in into the model such as tides and nonlinear advection that contribute to overall water levels and currents during storms. The transition from structured to unstructured represents the current state of evolution of coastal and ocean modeling, but it is expected that these models can be significantly improved in their efficiency with higher order methods [Wirasaet et al., 2010]. ...
Article
A Gulf of Mexico performance evaluation and comparison of coastal circulation and wave models was executed through harmonic analyses of tidal simulations, hindcasts of Hurricane Ike (2008) and Rita (2005), and a benchmarking study. Three unstructured coastal circulation models (ADCIRC, FVCOM, and SELFE) validated with similar skill on a new common Gulf scale mesh (ULLR) with identical frictional parameterization and forcing for the tidal validation and hurricane hindcasts. Coupled circulation and wave models, SWAN+ADCIRC and WWMII+SELFE, along with FVCOM loosely coupled with SWAN, also validated with similar skill. NOAA's official operational forecast storm surge model (SLOSH) was implemented on local and Gulf scale meshes with the same wind stress and pressure forcing used by the unstructured models for hindcasts of Ike and Rita. SLOSH's local meshes failed to capture regional processes such as Ike's forerunner and the results from the Gulf scale mesh further suggest shortcomings may be due to a combination of poor mesh resolution, missing internal physics such as tides and nonlinear advection, and SLOSH's internal frictional parameterization. In addition, these models were benchmarked to assess and compare execution speed and scalability for a prototypical operational simulation. It was apparent that a higher number of computational cores are needed for the unstructured models to meet similar operational implementation requirements to SLOSH, and that some of them could benefit from improved parallelization and faster execution speed.
... They demonstrate high order convergence rates [2,4], are well-established as candidates for computationally optimal adaptive technologies (e.g. hp-adaptivity and r-adaptivity) [21,52], are extremely scalable (especially on state-of-the-art architectures with high thread parallel arithmetic intensity, such as GPUs and coprocessors) [1,34,73,75], and are often noted as being remarkably flexible for accurately modeling large categories of coupled systems of initial-boundary value problems with strongly nonlinear forcings. This aspect of DG methods makes them particularly appealing for studying scrape-off layer dynamics, as the system of PDEs in question in the scrape-off layer is often highly variable, requires great flexibility in representation, possess weak regularity features, and may involve complicated geometries (e.g. in the presence of magnetic chaos) with nonlinear boundary forcings that are befitting to finite element methods in general. ...
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A new parallel discontinuous Galerkin solver, called ArcOn, is developed to describe the intermittent turbulent transport of filamentary blobs in the scrape-off layer (SOL) of fusion plasma. The model is comprised of an elliptic subsystem coupled to two convection-dominated reaction–diffusion–convection equations. Upwinding is used for a class of numerical fluxes developed to accommodate cross product driven convection, and the elliptic solver uses SIPG, NIPG, IIPG, Brezzi, and Bassi–Rebay fluxes to formulate the stiffness matrix. A novel entropy sensor is developed for this system, designed for a space–time varying artificial diffusion/viscosity regularization algorithm. Some numerical experiments are performed to show convergence order on manufactured solutions, regularization of blob/streamer dynamics in the SOL given unstable parameterizations, long-time stability of modon (or dipole drift vortex) solutions arising in simulations of drift-wave turbulence, and finally the formation of edge mode turbulence in the scrape-off layer under turbulent saturation conditions.
... However, because the original map is local in x, using spatially localized e a may be more practical when dealing with generic nonlinear problems. For example, one can partition the x space into small elements of size ∆x and decompose Ψ within each cell using Legendre polynomials up to an order ≥ 0, with larger leading to higher fidelity of the map representation [30]. 3 For simplicity, we limit our consideration to = 0. ...
Preprint
Quantum algorithms for computing classical nonlinear maps are widely known for toy problems but might not suit potential applications to realistic physics simulations. Here, we propose how to compute a general differentiable invertible nonlinear map on a quantum computer using only linear unitary operations. The price of this universality is that the original map is represented adequately only on a finite number of iterations. More iterations produce spurious echos, which are unavoidable in any finite unitary emulation of generic non-conservative dynamics. Our work is intended as the first survey of these issues and possible ways to overcome them in the future. We propose how to monitor spurious echos via auxiliary measurements, and we illustrate our results with numerical simulations.
... Flux values are thus uniquely defined at each cell interface quadrature points as the solution of a Riemann problem. In the literature, there are many possible DG approaches -modal [50][51][52], nodal [34,53,54], DG Spectral Element Method (DGSEM) [55][56][57][58] -and all depend on the chosen set of basis functions. Basis functions choice will have an impact on the stability, accuracy and efficiency of the DG method. ...
Thesis
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This thesis examines the extension of the Spectral Difference (SD) method on unstructured hybrid grids involving simplex cells (triangles, tetrahedra) and prismatic elements. The Spectral Difference method is part of high-order spectral discontinuous numerical methods. These methods rely on piecewise continuous polynomial approximation to obtain high-order accuracy with a good parallel efficiency. The standard SD scheme is first presented in the one-dimensional case and then for tensor-product elements (quadrangles and hexahedra). The treatment of simplex cells using Raviart-Thomas elements is detailed for triangles (in 2D) and tetrahedra (in 3D), followed by the implementation for prismatic elements. The linear stability of the Spectral Difference method using Raviart-Thomas elements (SDRT) is studied on triangles and tetrahedra. The SDRT scheme stability is strongly dependent on the interior flux points location. On triangles, the SDRT implementation based on interior flux points located at Williams-Shunn-Jameson quadrature points is found stable up to the fourth-order of accuracy but shown as spatially unstable for higher orders. Nevertheless, it is shown that this implementation can be stabilized for fifth- and sixth-order schemes using suitable temporal integration schemes. This approach being submitted to strict conditions, an optimization of the interior flux points location is conducted to determine spatially stable SDRT formulations for orders higher than four. The optimization process leads to spatially stable schemes up to the sixth-order of accuracy. Finally, the stability analysis on tetrahedra proves that the SDRT scheme based on the interior flux points located at Shunn-Ham quadrature points is stable up to the third-order. The SD/SDRT numerical method is validated on several academic cases for first and second-order Partial Differential Equations (linear advection equation, Euler equations, Navier-Stokes equations). Both proposed implementations (based either on Williams-Shunn-Jameson quadrature points or optimization points) are used. Numerical experiments involve grids composed of quadratic triangles, linear tetrahedral elements as well as 2D hybrid meshes.
... The entire procedure for mesh refinement for both LWDG and ALWDG provides sufficient content for a separate article and will be treated in a forthcoming paper. Unlike other finite element methods, RKDG schemes can easily handle arbitrary triangulations (for details see[34][35][36], performance of triangular and quadrilateral grids are compared in[37]). Though the original LWDG method proposed in[1]is only tested there for uniform grids in 1D and uniform Cartesian grids on 2D, it is capable of handling unstructured or non uniform grids and has been studied in[38](e.g., quadrilaterals and triangles). ...
Article
The Lax–Wendroff time discretization is an alternative method to the popular total variation diminishing Runge–Kutta time discretization of discontinuous Galerkin schemes for the numerical solution of hyperbolic conservation laws. The resulting fully discrete schemes are known as LWDG and RKDG methods, respectively. Although LWDG methods are in general more compact and efficient than RKDG methods of comparable order of accuracy, the formulation of LWDG methods involves the successive computation of exact flux derivatives. This procedure allows one to construct schemes of arbitrary formal order of accuracy in space and time. A new approximation procedure avoids the computation of exact flux derivatives. The resulting approximate LWDG schemes, addressed as ALWDG schemes, are easier to implement than their original LWDG versions. In particular, the formulation of the time discretization of the ALWDG approach does not depend on the flux being used. Numerical results for the scalar and system cases in one and two space dimensions indicate that ALWDG methods are more efficient in terms of error reduction per CPU time than LWDG methods of the same order of accuracy. Moreover, increasing the order of accuracy leads to substantial reductions of numerical error and gains in efficiency for solutions that vary smoothly.
... This class of problems is well suited to approximation by the discontinuous GFEM, as demonstrated in numerous studies (e.g., [6,12]). For parabolic problems, there appears to be substantial variety in discontinuous GFEM (and hybrid discontinuous GFEM) approaches that have been considered [9,[13][14][15][16][17][18]. Furthermore, some studies have focused on discontinuous Galerkin time discretization methods and compared them to continuous GFEM or continuous Galerkin-Petrov finite element methods. ...
Article
Full-text available
This article compares the computational cost, stability, and accuracy of continuous and discontinuous Galerkin Finite Element Methods (GFEM) for various parabolic differential equations including the advection–diffusion equation, viscous Burgers’ equation, and Turing pattern formation equation system. The results show that, for implicit time integration, the continuous GFEM is typically 5–20 times less computationally expensive than the discontinuous GFEM using the same finite element mesh and element order. However, the discontinuous GFEM is significantly more stable than the continuous GFEM for advection dominated problems and is able to obtain accurate approximate solutions for cases where the classic, un-stabilized continuous GFEM fails.
... To accelerate the NDGM solver, the graphic processors have been used by Klöckner et al. [40] , Gandham et al. [41] and Modave et al. [42] . The performance of the NDGM based on the triangular and quadrilateral grids has been compared in the work of Wirasaet et al. [43] . Besides of these contexts, the NDGM has been also employed for the nonlinear elastic wave propagation [44] , the wave propagation in the meta-materials [45] , the fluid flow problems using the lattice Boltzmann equations [46] , the elastic wave propagation in an arbitrary heterogeneous media [47] and the fractional diffusion equations [48] . ...
Article
In this work, a high-order nodal discontinuous Galerkin method is applied and assessed for the simulation of compressible non-cavitating and cavitating flows. The one-fluid approach with the thermal effects is used to properly model the cavitation phenomenon. Here, the spatial and temporal derivatives in the system of governing equations are discretized using the nodal discontinuous Galerkin method and the third-order TVD Runge-Kutta method, respectively. Various numerical fluxes such as the Roe, Rusanov, HLL, HLLC and AUSM+-up and two discontinuity capturing methods, namely, the generalized MUSCL limiter and a generalized exponential filter are implemented in the solution algorithm. At first, the sinusoidal density wave problem which has a smooth solution is simulated and the effects of the numerical fluxes on the accuracy and performance of the nodal discontinuous Galerkin method are studied. Two problems, namely, the shock-density interaction (non-cavitating flow) and the two symmetric expansion waves (cavitating flow) are then computed and the effects of the numerical fluxes and the discontinuity capturing methods on the accuracy and computational cost of the solution are investigated. For non-cavitating flows, the high-pressure water-water shock tube and the low-pressure water-water shock tube are also simulated. Then, three cavitating flow problems, namely, the two symmetric expansion waves, the shock-condensation tube and the collapsing cavitation bubble are simulated to assess the accuracy and robustness of the solution algorithm. Results show that the solution methodology based on the high-order NDGM is accurate and robust for simulating the compressible non-cavitating and cavitating flows.
... D'une part, l'utilisation des éléments triangulaires présente l'avantage d'avoir un déterminant constant du jacobien, ce qui réduit notablement le coût de calcul des intégrales de volume. D'autre part, les éléments rectangulaires présentent une meilleure précision (voir[248] ).En substituant u par son approximation (B.4) dans (B.3) et en calculant pour tous les éléments, le système algébrique globale est obtenu : ...
Thesis
Les structures mécaniques utilisées de nos jours ne cessent d’évoluer en utilisant des matériaux composites ou à gradient fonctionnel afin de répondre aux enjeux de résistance accrue, allégement de la structure et amélioration des performances. Ceux-ci nécessitent un contrôle adéquat de leur état de santé afin de s’assurer de l’intégrité de la structure. L’utilisation des ondes guidées ultrasonores fournit un moyen efficace et rapide d’inspection sur de longues distances. Néanmoins, ces ondes présentent certaines caractéristiques complexes qui rendent la tâche très difficile. L’utilisation d’outils d’analyse tels que les modèles numériques constitue un grand atout pour ce type d’application. Dans ce contexte, l’objectif de cette de thèse est le développement d’un outil de modélisation performant, permettant d’étudier la propagation des ondes guidées ultrasonores avec une grande précision et une faible consommation de ressources et de temps de calculs. De ce fait, l’intérêt est porté sur des méthodes numériques d’ordres élevés dont les propriétés de convergence sont beaucoup améliorées que les méthodes classiques. En particulier, la méthode semi-analytique éléments finis de Galerkin discontinue pour la détermination des courbes de dispersion des ondes guidées est développée. La méthode est applicable aux structures planes et cylindriques fabriquées de matériaux isotropes, anisotropes et hétérogènes (à gradient fonctionnel de propriétés). Une étude comparative sur l’analyse des performances de ces méthodes est effectuée. Celle-ci a démontré la capacité de la méthode à modéliser la propagation des ondes guidées ultrasonores dans des guides d’ondes à section arbitraire avec des performances prometteuses par rapport à la méthode des éléments finis classique.
... Numerical comparison with the example given in [22] is in good agreement even though we use a quadrilateral mesh instead of triangles. An interesting point of view about the influence of mesh nature can be found in Wirasaet et al. [24]. We observe that, when mesh is refined with only a difference of refinement level equal to one, discontinuous Galerkin method has good properties with higher numerical errors in the coarser mesh. ...
... A quadrilateral-dominant mesh was adopted to save the computing time [31] . Refined grids were created near the evaporation Table 1 Experimental observations of relative humidity in [ 6 , 27 , 28 ]. ...
Article
Solar stills are one of the most promising and eco-friendly solutions for providing fresh water in arid regions. This study used computational fluid dynamics (CFD) modelling to investigate the effects of operating pressure pop and geometrical parameters on the performance of a Tubular Solar Still (TSS), with an emphasis on vapor flow in the enclosure. The simulation results indicated that water vapor has a higher circulation velocity when operated under vacuum compared with atmospheric conditions, resulting that the yield rate increased by more than 50% at pop < 60 kPa. Three distinct flow regimes were observed at different thermal Rayleigh numbers RaT, which were identified as steady, periodic, and chaotic flow. Steady flow was observed at low RaT, while periodic or chaotic flow was observed when RaT exceeded 10⁵ and water depth Dw exceeded 0.4r. The vapor flow in different regimes was validated by visualization experiments. Based on the simulation results, a correlation was obtained to predict the Nusselt number for a particular TSS design, which showed a deviation of -7.8−7.5% when compared with independent experimental datasets.
... The former two apply tetrahedral meshes, which provide flexibility for the generation of meshes also for complicated structures. The latter two make use of hexahedral meshes, which allow for a computationally more efficient implementation [9]. In [3] the authors state that the method can easily deal with meshes with hanging nodes since no inter-element continuity is required, which renders it particularly well suited for hp-adaptivity. ...
Article
A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials (p-adaptation) as well as their combination. The computation of the approximation within locally adapted elements is based on projections between finite element spaces (FES), which are shown to preserve an upper limit of the electromagnetic energy. The formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency. Error and smoothness estimates based on interface jumps are presented and applied to the fully hp-adaptive simulation of two examples in one-dimensional space. A full wave simulation of electromagnetic scattering form a radar reflector demonstrates the applicability to large scale problems in three-dimensional space.
Article
We describe a local timestepping (LTS) approach within the Runge-Kutta discontinuous Galerkin (RKDG) method, and the application of this method to the modeling of hurricane storm surge. Modeling storm surge requires the numerical solution of the shallow water equations with wind and atmospheric pressure forcing, over complex domains which include wet and dry regions. The RKDG method is well suited for these applications; however, well-resolved simulations of storm surge can require highly graded meshes, which can lead to severe global CFL constraints. The LTS approach allows for elements to use timesteps which approximately satisfy only local CFL conditions. We describe a fully parallel implementation of the LTS method within an RKDG shallow water simulator, developed by the author and collaborators over a period of several years. We demonstrate that, for a specific hurricane, namely Hurricane Ike, the LTS method can reduce parallel run-times by nearly a factor of two with no degradation in accuracy.
Article
In the modelling of hydrodynamics, the Discontinuous Galerkin (DG) approach constitutes a more complex and modern alternative to the well-established finite volume method. The latter retains some desired practical features for modelling hydrodynamics, such as well-balancing between spatial flux and steep topography gradients, ability to incorporate wetting and drying processes, and computational affordability. In this context, DG methods were originally devised to solve the two-dimensional (2D) Shallow Water Equations (SWE) with irregular topographies and wetting and drying, albeit at reduction in the formulation’s complexity to often being second-order accurate (DG2). The aims of this paper are: (a) to outline a so-called “slope-decoupled” formulation of a standard 2D-DG2-SWE simulator in which theoretical complexity is deliberately reduced; (b) to highlight the capabilities of the proposed slopedecoupled simulator in providing a setting where the simplifying assumptions are verified within the formulation. Both the standard and the slope-decoupled 2D-DG2-SWE models adopt 2D modal basis functions for shaping local planar DG2 solutions on quadrilateral elements, by using an average coefficient and two slope coefficients along the Cartesian coordinates. Over a quadrilateral element, the stencil of the slope-decoupled 2D-DG2 formulation is simplified to remove the interdependence of slope-coefficients for both flow and topography approximations. The fully well-balanced character the slope-decoupled 2D-DG2-SWE planar solutions is theoretically studied. The performance of the latter is compared with the standard 2D-DG2 formulation in classical simulation tests. Other tests are conducted to diagnostically verify the conservative properties of the 2D-DG2-SWE method in scenarios involving sharp topography gradients and wet and/or dry zones. The analyses conducted offer strong evidence that the proposed slope-decoupled 2D-DG2-SWE simulator is very attractive for the development of robust flood models.
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This paper investigates Local Time Stepping (LTS) with the RKDG2 (second-order Runge-Kutta Discontinuous Galerkin) non-uniform solutions of the inhomogeneous SWEs (Shallow Water Equations) with source terms. A LTS algorithm – recently designed for homogenous hyperbolic PDE(s) – is herein reconsidered and improved in combination with the RKDG2 shallow-flow solver (LTS-RKDG2) including topography and friction source terms as well as wetting and drying. Two LTS-RKDG2 schemes that adapt 3 and 4 levels of LTSs are configured on 1D and/or 2D (quadrilateral) non-uniform meshes that, respectively, adopt 3 and 4 scales of spatial discretization. Selected shallow water benchmark tests are used to verify, assess and compare the LTS-RKDG2 schemes relative to their conventional Global Time Step RKDG2 alternatives (GTS-RKDG2) considering several issues of practical relevance to hydraulic modelling. Results show that the LTS-RKDG2 models could offer (depending on both the mesh setting and the features of the flow) comparable accuracy to the associated GTS-RKDG2 models with a savings in runtime of up to a factor of 2.5 in 1D simulations and 1.6 in 2D simulations.
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In this work, a high-order nodal discontinuous Galerkin method (NDGM) is developed and assessed for the simulation of 2D incompressible flows on triangle elements. The governing equations are the 2D incompressible Navier–Stokes equations with the artificial compressibility method. The discretization of the spatial derivatives in the resulting system of equations is made by the NDGM and the time integration is performed by applying the implicit dual-time stepping method. Three numerical fluxes, namely, the local Lax–Friedrich, Roe and AUSM+-up are formulated and applied to assess and compare their accuracy and performance in the simulation of incompressible flows using the NDGM. Several steady and unsteady incompressible flow problems are simulated to examine the accuracy and robustness of the proposed solution methodology and they are the Kovasznay, backward facing step, NACA0012 airfoil, circular cylinder and two side-by-side circular cylinders. Indications are that the NDGM applied for solving the incompressible Navier–Stokes equations with the artificial compressibility approach and the implicit dual-time stepping method is accurate and robust for the simulation of steady and unsteady incompressible flow problems.
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A Godunov-type numerical model, which is based on the local planar Runge-Kutta discontinuous Galerkin (RKDG2) solutions to the two dimensional (2D) shallow water equations (SWEs) on a dynamically adaptive quadrilateral grid system, is developed in this work for shallow water wave simulations, with particular application to flood inundation modeling. To be consistent with the dynamic grid adaptation, the well-balanced RKDG2 framework is reformulated to facilitate realistic flood modeling. Grid adaptation and redistribution of flow data are automated based on simple measures of local flow properties. One analytical and two diagnostic test cases are used to validate the performance of the dynamically adaptive RKDG2 model against an alternative RKDG2 code based on uniform quadrilateral meshes. The adaptive model is then assessed by further applying it to reproduce a laboratory-scale tsunami benchmark case and the historical Malpasset dam-break event. Numerical evidence indicates that the new algorithm is able to resolve the moving wave features adequately at much less computational cost than the refined uniform grid-based counterpart.
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Geophysical flows over complex domains often encompass both coarse and highly resolved regions. Approximating these flows using shock-capturing methods with explicit timestepping gives rise to a Courant–Friedrichs–Lewy (CFL) timestep constraint. This approach can result in small global timesteps often dictated by flows in small regions, vastly increasing computational effort over the whole domain. One approach for coping with this problem is to use locally varying timesteps. In previous work, we formulated a local timestepping (LTS) method within a Runge–Kutta discontinuous Galerkin framework and demonstrated the accuracy and efficiency of this method on serial machines for relatively small-scale shallow water applications. For more realistic models involving large domains and highly complex physics, the LTS method must be parallelized for multi-core parallel computers. Furthermore, additional physics such as strong wind forcing can effect the choice of local timesteps. In this paper, we describe a parallel LTS method, parallelized using domain decomposition and MPI. We demonstrate the method on tidal flows and hurricane storm surge applications in the coastal regions of the Western North Atlantic Ocean.
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A new high-resolution finite element scheme is introduced for solving the two-dimensional (2D) depth-integrated shallow water equations (SWE) via local plane approximations to the unknowns. Bed topography data are locally approximated in the same way as the flow variables to render an instinctive well-balanced scheme. A finite volume (FV) wetting and drying technique that reconstructs the Riemann states by ensuring non-negative water depth and maintaining well-balanced solution is adjusted and implemented in the current finite element framework. Meanwhile, a local slope-limiting process is applied and those troubled-slope-components are restricted by the minmod FV slope limiter. The inter-cell fluxes are upwinded using the HLLC approximate Riemann solver. Friction forces are separately evaluated via stable implicit discretization to the finite element approximating coefficients. Boundary conditions are derived and reported in details. The present model is validated against several test cases including dam-break flows on regular and irregular domains with flooding and drying.
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This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrix-vector operations that should provide for efficient and portable implementations of algorithms for high-performance computers.
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We present a convergent high-order accurate scheme for the solution of linear conservation laws in geometrically complex domains. As our main example we include a detailed development and analysis of a scheme for the time-domain solution of Maxwell&apos;s equations in a three-dimensional domain. The fully unstructured spatial discretization is made possible by the use of a high-order nodal basis, employing multivariate Lagrange polynomials defined on the triangles and tetrahedra, while the equations themselves are satisfied in a discontinuous Galerkin form with the boundary conditions being enforced weakly through a penalty term. Accuracy, stability, and convergence of the semidiscrete approximation to Maxwell&apos;s equations is established rigorously and bounds on the growth of the global divergence error are provided. Concerns related to efficient implementations are discussed in detail. This sets the stage for the presentation of examples, verifying the theoretical results, and illustrating the versatility, flexibility, and robustness when solving two- and three-dimensional benchmark problems in computational electromagnetics. Pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated. (C) 2002 Elsevier Science (USA).
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We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary order for the spatial discretization and a third-order Runge–Kutta scheme to advance in time. The local elements are coupled together by numerical fluxes, evaluated using the HLLC Riemann solver. We apply the model to test cases involving smooth flows and demonstrate the exponentially fast convergence with regard to polynomial order. We also illustrate that even for results of ‘engineering accuracy’ the computational efficiency increases with increasing order of the model and time of integration. The model is found to be robust in the presence of shocks where Gibbs oscillations can be suppressed by slope limiting. Copyright 2004 John Wiley & Sons, Ltd.
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We present a comparative study of two finite element shallow water equation (SWE) models: a generalized wave continuity equation based continuous Galerkin (CG) model—an approach used by several existing SWE models—and a recently developed discontinuous Galerkin (DG) model. While DG methods are known to possess a number of favorable properties, such as local mass conservation, one commonly cited disadvantage is the larger number of degrees of freedom associated with the methods, which naturally translates into a greater computational cost compared to CG methods. However, in a series of numerical tests, we demonstrate that the DG SWE model is generally more efficient than the CG model (i) in terms of achieving a specified error level for a given computational cost and (ii) on large-scale parallel machines because of the inherently local structure of the method. Both models are verified on a series of analytic test cases and validated on a field-scale application.
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This paper presents a review of the role played by trees in the theory of Runge–Kutta methods. The use of trees is in contrast to early publications on numerical methods, in which a deceptively simpler approach was used. This earlier approach is not only non-rigorous, but also incorrect. It is now known, for example, that methods can have different orders when applied to a single equation and when applied to a system of equations; the earlier approach cannot show this. Trees have a central role in the theory of Runge–Kutta methods and they also have applications to more general methods, involving multiple values and multiple stages.
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. The electrostatic interpretation of the Jacobi-Gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex. Moreover, several new estimates, based on extensive numerical studies, for approximation along the line using Jacobi-Gauss-Lobatto quadrature points as the nodal sets are presented. The electrostatic analogy is extended to the two-dimensional case, with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented. The matrices are evaluated by computing the Lebesgue constants and they share the property that the nodes along the edges of the simplex are the Gauss-Lobatto quadrature points of the Chebyshev and Legendre polynomials, respectively. This makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h Gamma p finite element methods. Key words. Polynomial Interpol...
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The characteristics and capabilities of the best codes for solving the initial value problem for ordinary differential equations are studied. The aim is to assist the person who wants to install a high quality code for the initial value problem for ordinary differential equations at his computer center, and to acquaint other interested persons with the characteristics and capabilities of the best software for this problem. Only codes which are readily available, portable, and very efficient are examined. Their sources, distinctive features, documentation, and ease of use are described. Their efficiency is compared with respect to storage, overhead, and derivative evaluations. Their behavior when faced with difficult problems is studied. Some guidelines are developed as to when a particular code is to be preferred. 23 refs.
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A numerical example is presented which shows that the known L 2 error estimate ∥u-u h∥≤ Ch k+ 1/2∥u∥ k+1 for the discontinuous Galerkin method cannot be improved within the class of quasi-uniform meshes, not even for smooth exact solution u.
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In this paper a new approach is developed for analyzing the discontinuous Galerkin method for hyperbolic equations. For a model problem in $R^2$, the method is shown to converge at a rate $O(h^{n + 1})$ when applied with $n$th degree polynomial approximations over a semiuniform triangulation, assuming sufficient regularity in the solution.
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Computer-oriented mesh generators, which serve as pre-processors to finite element programs, have recently been developed by several investigators to alleviate the frustration and to reduce the amount of time involved in the tedious manual subdividing of a complex structure into finite elements. Our purpose here is to describe how the techniques of bivariate ‘blending-function’ interpolation, which were originally developed for, and applied to, geometric problems of computer-aided design and numerically controlled machining of free-form surfaces such as automobile exterior panels, can be adapted and applied to the problems of mesh generation for finite element analyses. We concentrate attention on the problem of curvilinearly co-ordinating simply connected planar domains ℛ by constructing invertible maps of the unit square [0, 1] × [0, 1] onto ℛ. Extensions of the methods described herein to shells in 3-space is straightforward and is illustrated by a practical example taken from the automobile industry. Analogous mesh generators for three-dimensional solids can be developed on the basis of the trivariate ‘blending-function’ formulae found at the end of the second section.
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We discuss an adaptive local refinement finite element method for solving initial-boundary value problems for vector systems of partial differential equations in one space dimension and time. The method uses piecewise bilinear rectangular space-time finite elements. For each time step, grids are automatically added to regions where the local discretization error is estimated as being larger than a prescribed tolerance. We discuss several aspects of our algorithm, including the tree structure that is used to represent the finite element solution and grids, an error estimation technique, and initial and boundary conditions at coarse-fine mesh interfaces. We also present computational results for a simple linear hyperbolic problem, a problem involving Burgers' equation, and a model combustion problem.
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The Finite Element Method: Its Basis and Fundamentals offers a complete introduction to the basis of the finite element method, covering fundamental theory and worked examples in the detail required for readers to apply the knowledge to their own engineering problems and understand more advanced applications. This edition sees a significant rearrangement of the bookâ??s content to enable clearer development of the finite element method, with major new chapters and sections added to cover: Weak forms Variational forms Multi-dimensional field problems Automatic mesh generation Plate bending and shells Developments in meshless techniques Focusing on the core knowledge, mathematical and analytical tools needed for successful application, The Finite Element Method: Its Basis and Fundamentals is the authoritative resource of choice for graduate level students, researchers and professional engineers involved in finite element-based engineering analysis.
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This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDEs, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDEs: Maxwells equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.
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Chapter 1 features an introduction to the theory of partial differential equations (PDEs). Assumed is the knowledge of the elementary functional analysis provided in Appendix A. Discussed is the classification of PDEs, the notion of well-posedness, and selected general existence and uniqueness results for operator equations. The most frequently used types of PDEs — the second-order elliptic, parabolic and hyperbolic equations, are discussed in detail. Covered are their weak formulations, various types of boundary conditions, existence and uniqueness results, maximum principles and various other properties and results. Discussed are linear first-order hyperbolic systems and their solution via the characteristic lines, the Riemann problem, and the creation of discontinuities (shock waves) in nonlinear hyperbolic PDEs.
This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-order accurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on the exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems.
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We consider the approximation of the depth-averaged two-dimensional shallow water equations by both a traditional continuous Galerkin (CG) finite element method as well as two discontinuous Galerkin (DG) approaches. The DG method is locally conservative, flux-continuous on each element edge, and is suitable for both smooth and highly advective flows. A novel technique of coupling a DG method for continuity with a CG method for momentum is developed. This formulation is described in detail and validation via numerical testing is presented. Comparisons between a widely used CG approach, a conventional DG method, and the novel coupled discontinuous–continuous Galerkin method illustrates advantages and disadvantages in accuracy and efficiency. Copyright © 2006 John Wiley & Sons, Ltd.
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This article shows how to obtain multidimensional spectral methods as a warped product of one-dimensional spectral methods, thus generalizing direct (tensor) products. This generalization includes the fast Fourier transform. Applications are given for spectral approximation on a disk and on a triangle. The use of the disk spectral method for simulating the Navier-Stokes equations in a periodic pipe is detailed. The use of the triangle method in a spectral element scheme is discussed. The degree of approximation of the triangle method is computed in a new way, which favorably compares with the classical approximation estimates.
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In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the method to be non-linearly stable regardless of its accuracy, with a finite element space discretization by discontinuous approximations, that incorporates the ideas of numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume schemes. The resulting RKDG methods are stable, high-order accurate, and highly parallelizable schemes that can easily handle complicated geometries and boundary conditions. We review the theoretical and algorithmic aspects of these methods and show several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations.
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We consider the approximation of a simplified model of the depth-averaged two-dimensional shallow water equations by two approaches. In both approaches, a discontinuous Galerkin (DG) method is used to approximate the continuity equation. In the first approach, a continuous Galerkin method is used for the momentum equations. In the second approach a particular DG method, the nonsymmetric interior penalty Galerkin method, is used to approximate momentum. A priori error estimates are derived and numerical results are presented for both approaches.
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Runge–Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge–Kutta time discretizations, and limiters. In most of the RKDG papers in the literature, the Lax–Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. In this paper, we systematically investigate the performance of the RKDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., and second-order TVD fluxes, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.
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This paper proposes a wetting and drying treatment for the piecewise linear Runge–Kutta discontinuous Galerkin approximation to the shallow water equations. The method takes a fixed mesh approach as opposed to mesh adaptation techniques and applies a post-processing operator to ensure the positivity of the mean water depth within each finite element. In addition, special treatments are applied in the numerical flux computation to prevent an instability due to excessive drying. The proposed wetting and drying treatment is verified through comparisons with exact solutions and convergence rates are examined. The obtained orders of convergence are close to or approximately equal to 1 for solutions with discontinuities and are improved for smooth solutions. The combination of the proposed wetting and drying treatment and a TVB slope limiter is also tested and is found to be applicable on condition that they are applied exclusively to an element at the same Runge–Kutta step.
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A discontinuous Galerkin (DG) finite element method is described for the two-dimensional, depth-integrated shallow water equations (SWEs). This method is based on formulating the SWEs as a system of conservation laws, or advection–diffusion equations. A weak formulation is obtained by integrating the equations over a single element, and approximating the unknowns by piecewise, possibly discontinuous, polynomials. Because of its local nature, the DG method easily allows for varying the polynomial order of approximation. It is also “locally conservative”, and incorporates upwinded numerical fluxes for modeling problems with high flow gradients. Numerical results are presented for several test cases, including supercritical flow, river inflow and standard tidal flow in complex domains, and a contaminant transport scenario where we have coupled the shallow water flow equations with a transport equation for a chemical species.
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In this paper, we discuss the development, verification, and application of an hp discontinuous Galerkin (DG) finite element model for solving the shallow water equations (SWE) on unstructured triangular grids. The h and p convergence properties of the method are demonstrated for both linear and highly nonlinear problems with advection dominance. Standard h-refinement for a fixed p leads to p + 1 convergence rates, while exponential convergence is observed for p-refinement for a fixed h. It is also demonstrated that the use of p-refinement is more efficient for problems exhibiting smooth solutions. Additionally, the ability of p-refinement to adequately resolve complex, two-dimensional flow structures is demonstrated in the context of a coastal inlet problem.
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In this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for Runge--Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing (TVD) Runge--Kutta method. Error estimates for the $\mathbb{P}^1$ (piecewise linear) elements are obtained under the usual CFL condition $\tau\leq \gamma h$ for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and $\tau$ are the maximum element lengths and time steps, respectively, and the positive constant $\gamma$ is independent of h and $\tau$. However, error estimates forhigher order $\mathbb{P}^k(k\geq 2)$ elements need a more restrictive time step $\tau\leq \gamma h^{4/3}$. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition $\tau\leq\gamma h$ for the $\mathbb{P}^k$ elements of degree $k\geq 2$. Error estimates of $O(h^{k+1/2}+\tau^2)$ are obtained for general monotone numerical fluxes, and optimal error estimates of $O(h^{k+1}+\tau^2)$ are obtained for upwind numerical fluxes. In this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for Runge--Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing (TVD) Runge--Kutta method. Error estimates for the $\mathbb{P}^1$ (piecewise linear) elements are obtained under the usual CFL condition $\tau\leq \gamma h$ for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and $\tau$ are the maximum element lengths and time steps, respectively, and the positive constant $\gamma$ is independent of h and $\tau$. However, error estimates forhigher order $\mathbb{P}^k(k\geq 2)$ elements need a more restrictive time step $\tau\leq \gamma h^{4/3}$. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition $\tau\leq\gamma h$ for the $\mathbb{P}^k$ elements of degree $k\geq 2$. Error estimates of $O(h^{k+1/2}+\tau^2)$ are obtained for general monotone numerical fluxes, and optimal error estimates of $O(h^{k+1}+\tau^2)$ are obtained for upwind numerical fluxes.
Chapter
Numerical methods for the solution of initial value problems in ordinary differential equations made enormous progress during the 20th century for several reasons. The first reasons lie in the impetus that was given to the subject in the concluding years of the previous century by the seminal papers of Bashforth and Adams for linear multistep methods and Runge for Runge-Kutta methods. Other reasons, which of course apply to numerical analysis in general, are in the invention of electronic computers half way through the century and the needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics. This survey paper follows many of the main strands in the developments of these methods, both for general problems, stiff systems, and for many of the special problem types that have been gaining in significance as the century draws to an end.
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As a tool for large-scale computations in fluid dynamics, spectral methods were prophesized in 1944, born in 1954, virtually buried in the mid-1960's, resurrected in 1969, evangalized in the 1970's, and catholicized in the 1980's. The use of spectral methods for meteorological problems was proposed by Blinova in 1944 and the first numerical computations were conducted by Silberman (1954). By the early 1960's computers had achieved sufficient power to permit calculations with hundreds of degrees of freedom. For problems of this size the traditional way of computing the nonlinear terms in spectral methods was expensive compared with finite-difference methods. Consequently, spectral methods fell out of favor. The expense of computing nonlinear terms remained a severe drawback until Orszag (1969) and Eliasen, Machenauer, and Rasmussen (1970) developed the transform methods that still form the backbone of many large-scale spectral computations. The original proselytes of spectral methods were meteorologists involved in global weather modeling and fluid dynamicists investigating isotropic turbulence. The converts who were inspired by the successes of these pioneers remained, for the most part, confined to these and closely related fields throughout the 1970's. During that decade spectral methods appeared to be well-suited only for problems governed by ordinary diSerential eqllations or by partial differential equations with periodic boundary conditions. And, of course, the solution itself needed to be smooth. Some of the obstacles to wider application of spectral methods were: (1) poor resolution of discontinuous solutions; (2) inefficient implementation of implicit methods; and (3) drastic geometric constraints. All of these barriers have undergone some erosion during the 1980's, particularly the latter two. As a result, the applicability and appeal of spectral methods for computational fluid dynamics has broadened considerably. The motivation for the use of spectral methods in numerical calculations stems from the attractive approximation properties of orthogonal polynomial expansions.
Spectral Methods in Fluid Dynamics. Spring Series in Computational Physics
  • Canuto C My Hussaini
  • A Quarteroni
  • Zang
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Canuto C, Hussaini MY, Quarteroni A, Zang TA. Spectral Methods in Fluid Dynamics. Spring Series in Computational Physics. Springer: Berlin, 1988.
  • Butcher