Upwind Flux Research Articles

Superconvergence of Discontinuous Galerkin Methods for Two-Dimensional Hyperbolic Equations

This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the (2k+1)th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree k are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of (k+1)th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of (k+2)th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp.

Implicit Runge--Kutta Methods and Discontinuous Galerkin Discretizations for Linear Maxwell's Equations

In this paper we consider implicit Runge--Kutta methods for the time integration of linear Maxwell's equations. We first present error bounds for the abstract Cauchy problem which respect the unboundedness of the differential operators using energy techniques. The error bounds hold for algebraically stable and coercive methods such as Gauss and Radau collocation methods. The results for the abstract evolution equation are then combined with a discontinuous Galerkin discretization in space using upwind fluxes. For the case that permeability and permittivity are piecewise constant functions, we show error bounds for the full discretization, where the constants do not deteriorate if the spatial mesh width tends to zero.

Efficient time integration for discontinuous Galerkin approximations of linear wave equations

We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro‐magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first‐order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization. For the time integration we discuss advantages and disadvantages of explicit and implicit Runge–Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro‐magnetic and elastic waves.

Energy conserving discontinuous Galerkin spectral element method for the Vlasov–Poisson system

We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov–Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [4], with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss–Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge–Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).

A new technique for freestream preservation of finite-difference WENO on curvilinear grid

A new technique for a finite-difference weighted essentially nonoscillatory scheme (WENO) on curvilinear grids to preserve freestream is introduced. This technique first divides the standard finite-difference WENO into two parts: (1) a consistent central difference part and (2) a numerical dissipation part. For the consistent central difference part, the conservative metric technique is directly adopted. For the numerical dissipation part, it is proposed that the metric term should be frozen for constructing the upwinding flux. This treatment only affects the numerical dissipation part, and the order of accuracy is maintained. With this technique, the freestream is perfectly preserved, and the flow fields are better resolved on wavy and random grids.

A Discontinuous Galerkin Time Domain Framework for Periodic Structures Subject to Oblique Excitation

A nodal Discontinuous Galerkin (DG) method is derived for the analysis of time-domain (TD) scattering from doubly periodic PEC/dielectric structures under oblique interrogation. Field transformations are employed to elaborate a formalism that is free from any issues with causality that are common when applying spatial periodic boundary conditions simultaneously with incident fields at arbitrary angles of incidence. An upwind numerical flux is derived for the transformed variables, which retains the same form as it does in the original Maxwell problem for domains without explicitly imposed periodicity. This, in conjunction with the amenability of the DG framework to non-conformal meshes, provides a natural means of accurately solving the first order TD Maxwell equations for a number of periodic systems of engineering interest. Results are presented that substantiate the accuracy and utility of our method.

Changes in fluxes of heat, H2O, and CO2 caused by a large wind farm

The Crop Wind-Energy Experiment (CWEX) provides a platform to investigate the effect of wind turbines and large wind farms on surface fluxes of momentum, heat, moisture, and carbon dioxide (CO2). In 2010 and 2011, eddy covariance flux stations were installed between two lines of turbines at the southwest edge of a large Iowa wind farm from late June to early September. We report changes in fluxes of momentum, sensible heat, latent heat, and CO2 above a corn canopy after surface air had passed through a single line of turbines. In 2010, our flux stations were placed within a field with homogeneous land management practices (same tillage, cultivar, chemical treatments). We stratify the data according to wind direction, diurnal condition, and turbine operational status. Within these categories, the downwind–upwind flux differences quantify turbine influences at the crop surface. Flux differences were negligible in both westerly wind conditions and when the turbines were non operational. When the flow is perpendicular (southerly) or slightly oblique (southwesterly) to the row of turbines during the day, fluxes of CO2 and water (H2O) are enhanced by a factor of five in the lee of the turbines (from three to five turbine diameter distances downwind from the tower) as compared to a west wind. However, we observe a smaller CO2 flux increase of 30–40% for these same wind directions when the turbines are off. In the nighttime, there is strong statistical significance that turbine wakes enhance upward CO2 fluxes and entrain sensible heat toward the crop. The direction of the scalar flux perturbation seems closely associated to the differences in canopy friction velocity. Spectra and co-spectra of momentum components and co-spectra of heat also demonstrate nighttime influence of the wind turbine turbulence at the downwind station.

Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper, we study superconvergence properties of the discontinuous Galerkin (DG) method for one-dimensional linear hyperbolic equations when upwind fluxes are used. We prove, for any polynomial degree $k$, the $2k+1$th (or $2k+1/2$th) superconvergence rate of the DG approximation at the downwind points and for the domain average under quasi-uniform meshes and some suitable initial discretization. Moreover, we prove that the derivative approximation of the DG solution is superconvergent with a rate $k+1$ at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.

Multidimensional HLLC Riemann solver for unstructured meshes – With application to Euler and MHD flows

The goal of this paper is to formulate genuinely multidimensional HLL and HLLC Riemann solvers for unstructured meshes by extending our prior papers on the same topic for logically rectangular meshes Balsara (2010, 2012) [4,5]. Such Riemann solvers operate at each vertex of a mesh and accept as an input the set of states that come together at that vertex. The mesh geometry around that vertex is also one of the inputs of the Riemann solver. The outputs are the resolved state and multidimensionally upwinded fluxes in both directions.A formulation which respects the detailed geometry of the unstructured mesh is presented. Closed-form expressions are provided for all the integrals, making it particularly easy to implement the present multidimensional Riemann solvers in existing numerical codes. While it is visually demonstrated for three states coming together at a vertex, our formulation is general enough to treat multiple states (or zones with arbitrary geometry) coming together at a vertex. The present formulation is very useful for two-dimensional and three-dimensional unstructured mesh calculations of conservation laws. It has been demonstrated to work with second to fourth order finite volume schemes on two-dimensional unstructured meshes. On general triangular grids an arbitrary number of states might come together at a vertex of the primal mesh, while for calculations on the dual mesh usually three states come together at a grid vertex.We apply the multidimensional Riemann solvers to hydrodynamics and magnetohydrodynamics (MHD) on unstructured meshes. The Riemann solver is shown to operate well for traditional second order accurate total variation diminishing (TVD) schemes as well as for weighted essentially non-oscillatory (WENO) schemes with ADER (Arbitrary DERivatives in space and time) time-stepping. Several stringent applications for compressible gasdynamics and magnetohydrodynamics are presented, showing that the method performs very well and reaches high order of accuracy in both space and time. The present multidimensional Riemann solver is cost-competitive with traditional, one-dimensional Riemann solvers. It offers the twin advantages of isotropic propagation of flow features and a larger CFL number.Please see http://www.nd.edu/~dbalsara/Numerical-PDE-Course for a video introduction to multidimensional Riemann solvers.

Numerical simulation of two-phase reactive flow with moving boundary

Purpose – In computational fluid dynamics for two-phase reactive flow of interior ballistic, the conventional schemes (MacCormack method, etc.) are known to introduce unphysical oscillations in the region where the gradient is high. This paper aims to improve the ability to capture the complex shock wave during the interior ballistic cycle. Design/methodology/approach – A two-phase flow model is established to describe the complex physical process based on a modified two-fluid theory. The solution of model is obtained including the following key methods: an approximate Riemann solver to construct upwind fluxes, the MUSCL extension to achieve high-order accuracy, a splitting approach to solve source terms, a self-adapting method to expand the computational domain for projectile motion and a control volume conservation method for the moving boundary. Findings – The paper is devoted to applying a high-resolution numerical method to simulate a transient two-phase reactive flow with moving boundary in guns. Several verification tests demonstrate the accuracy and reliability of this approach. Simulation of two-phase reaction flow with a projectile motion in a large-caliber gun shows an excellent agreement between numerical simulation and experimental measurements. Practical implications – This paper has implications for improving the ability to capture the complex physics phenomena of two-phase flow during interior ballistic cycle and predict the combustion details, such as the flame spreading, the formation of pressure waves and so on. Originality/value – This approach is reliable as a prediction tool for the understanding of the physical phenomenon and can therefore be used as an assessment tool for future interior ballistics studies.

Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg–de Vries equation

We study the superconvergence property of the local discontinuous Galerkin (LDG) method for solving the linearized Korteweg–de Vries (KdV) equation. We prove that, if the piecewise Pk polynomials with k≥1 are used, the LDG solution converges to a particular projection of the exact solution with the order k+3/2, when the upwind flux is used for the convection term and the alternating flux is used for the dispersive term. Numerical examples are provided at the end to support the theoretical results.

Behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations

This article reports the behaviour of the numerical entropy production of the one-and-a-half-dimensional shallow water equations. The one-and-a-half-dimensional shallow water equations are the one-dimensional shallow water equations with a passive tracer or transverse velocity. The studied behaviour is with respect to the choice of numerical fluxes to evolve the mass, momentum, tracer-mass (transverse momentum), and entropy. When solving the one-and-a-half-dimensional shallow water equations using a finite volume method, we recommend the use of a double sided stencil flux for the mass and momentum, and in addition, a single sided stencil (upwind) flux for the tracer-mass. Having this recommended combination of fluxes, we use a double sided stencil entropy flux to compute the numerical entropy production, but this flux generates positive overshoots of the numerical entropy production. Positive overshoots of the numerical entropy production are avoided by use of a modified entropy flux, which satisfies a discrete numerical entropy inequality. References F. Bouchut. Efficient numerical finite volume schemes for shallow water models. In V. Zeitlin (editor), Nonlinear dynamics of rotating shallow water: methods and advances, Volume 2 of Edited series on advances in nonlinear science and complexity, pages 189--256. Elsevier, Amsterdam, 2007. http://dx.doi.org/10.1016/S1574-6909(06)02004-1 R. J. LeVeque. Finite-volume methods for hyperbolic problems. Cambridge University Press, Cambridge, 2002. http://dx.doi.org/10.1017/CBO9780511791253 S. Mungkasi and S. G. Roberts. Numerical entropy production for shallow water flows. ANZIAM Journal, 52(CTAC2010):C1--C17, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3786/1410 G. Puppo and M. Semplice. Numerical entropy and adaptivity for finite volume schemes. Communications in Computational Physics, 10(5):1132--1160, 2011. http://dx.doi.org/10.4208/cicp.250909.210111a J. J. Stoker. Water Waves: The Mathematical Theory with Application. Interscience Publishers, New York, 1957. http://onlinelibrary.wiley.com/book/10.1002/9781118033159

A convergent FEM-DG method for the compressible Navier–Stokes equations

This paper presents a new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix–Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax–Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.

An unstructured finite‐volume method for coupled models of suspended sediment and bed load transport in shallow‐water flows

SUMMARYThe aim of this work is to develop a well‐balanced finite‐volume method for the accurate numerical solution of the equations governing suspended sediment and bed load transport in two‐dimensional shallow‐water flows. The modelling system consists of three coupled model components: (i) the shallow‐water equations for the hydrodynamical model; (ii) a transport equation for the dispersion of suspended sediments; and (iii) an Exner equation for the morphodynamics. These coupled models form a hyperbolic system of conservation laws with source terms. The proposed finite‐volume method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe's scheme using the sign of the Jacobian matrix in the coupled system. A well‐balanced discretization is used for the treatment of source terms. In this paper, we also employ an adaptive procedure in the finite‐volume method by monitoring the concentration of suspended sediments in the computational domain during its transport process. The method uses unstructured meshes and incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep sediment concentrations and bed load gradients that may form in the approximate solutions. Details are given on the implementation of the method, and numerical results are presented for two idealized test cases, which demonstrate the accuracy and robustness of the method and its applicability in predicting dam‐break flows over erodible sediment beds. The method is also applied to a sediment transport problem in the Nador lagoon.Copyright © 2013 John Wiley & Sons, Ltd.

A study of convective flux schemes for aerospace flows

This paper presents the effects of some convective flux computation schemes on boundary layer and shocked flow solutions. Second-order accurate centered and upwind convective flux computation schemes are discussed. The centered Jameson scheme, plus explicitly added artificial dissipation terms are considered. Three artificial dissipation models, namely a scalar and a matrix version of a switched model, and the CUSP scheme are available. Some implementation options regarding these methods are proposed and addressed in the paper. For the upwind option, the Roe flux-difference splitting scheme is used. The CUSP and Roe schemes require property reconstructions to achieve second-order accuracy in space. A multidimensional limited MUSCL interpolation method is used to perform property reconstruction. Extended multidimensional limiter formulation and implementation are here proposed and verified. Theoretical flow solutions are used in order to provide a representative testbed for the current study. It is observed that explicitly added artificial dissipation terms of the centered scheme may nonphysically modify the numerical solution, whereas upwind schemes seem to better represent the flow structure.

A conservative orbital advection scheme for simulations of magnetized shear flows with the PLUTO code

Explicit numerical computations of super-fast differentially rotating disks are subject to the time-step constraint imposed by the Courant condition. When the bulk orbital velocity largely exceeds any other wave speed the time step is considerably reduced and a large number of steps may be necessary to complete the computation. We present a robust numerical scheme to overcome the Courant limitation by extending the algorithm previously known as FARGO (Fast Advection in Rotating Gaseous Objects) to the equations of magnetohydrodynamics (MHD). The proposed scheme conserves total angular momentum and energy to machine precision and works in Cartesian, cylindrical, or spherical coordinates. The algorithm is implemented in the PLUTO code for astrophysical gasdynamics and is suitable for local or global simulations of accretion or proto-planetary disk models. By decomposing the total velocity into an average azimuthal contribution and a residual term, the algorithm solves the MHD equations through a linear transport step in the orbital direction and a standard nonlinear solver applied to the MHD equations written in terms of the residual velocity. Since the former step is not subject to any stability restriction, the Courant condition is computed only in terms of the residual velocity, leading to substantially larger time steps. The magnetic field is advanced in time using the constrained transport method in order to preserve the divergence-free condition. Conservation of total energy and angular momentum is enforced at the discrete level by properly expressing the source terms in terms of upwind fluxes available during the standard solver. Our results show that applications of the proposed orbital-advection scheme to problems of astrophysical relevance provides, at reduced numerical cost, equally accurate and less dissipative results than standard time-marching schemes.

A nodal discontinuous Galerkin method for computational aeroacoustics and comparison with finite difference schemes

A nodal discontinuous Galerkin formulation, which is based on Lagrange polynomials basis, is used to directly simulate the acoustic wave propagation. Two test problems of wave propagation with initial disturbance consisting of a Gaussian profile or rectangular pulse are investigated. We evaluate the performance of the schemes in short, intermediate, and long waves. Moreover, the comparisons of numerical results between the nodal discontinuous Galerkin method and finite difference type schemes are performed, which indicate that the numerical solution obtained using nodal discontinuous Galerkin method with a pure central flux has obvious high frequency oscillations for initial disturbance consisting of rectangular pulse, which is the same as those obtained using finite difference type schemes without artificial selective damping. If an upwind flux is adopted, spurious waves are eliminated effectively except for the location of discontinuities. When a limiter is used, obviously the spurious short waves are almost completely removed. The quality of the computed solution has greatly improved.

A Comparative Study of Three Finite Element-Based Explicit Numerical Schemes for Solving Maxwell's Equations

Three finite element-based explicit numerical algorithms, named the dual-field domain decomposition at the element level (DFDD-ELD), the discontinuous Galerkin time-domain method with upwind fluxes (DGTD-Upwind), and the discontinuous Galerkin time-domain method with central fluxes (DGTD-Central), are investigated and compared in terms of accuracy and efficiency. All three algorithms can perform an efficient domain decomposition and avoid the inversion of a global system matrix, yet the study shows that they differ from each other in terms of accuracy and efficiency. Hybrid implicit-explicit schemes, which can relax the restriction on the time step size imposed by the smallest elements in the computational domain, are also investigated for both DFDD and DGTD and compared in terms of efficiency.

A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows

In this paper we present a genuinely two-dimensional HLLC Riemann solver. On logically rectangular meshes, it accepts four input states that come together at an edge and outputs the multi-dimensionally upwinded fluxes in both directions. This work builds on, and improves, our prior work on two-dimensional HLL Riemann solvers. The HLL Riemann solver presented here achieves its stabilization by introducing a constant state in the region of strong interaction, where four one-dimensional Riemann problems interact vigorously with one another. A robust version of the HLL Riemann solver is presented here along with a strategy for introducing sub-structure in the strongly-interacting state. Introducing sub-structure turns the two-dimensional HLL Riemann solver into a two-dimensional HLLC Riemann solver. The sub-structure that we introduce represents a contact discontinuity which can be oriented in any direction relative to the mesh. [Display omitted] The Riemann solver presented here is general and can work with any system of conservation laws. We also present a second order accurate Godunov scheme that works in three dimensions and is entirely based on the present multidimensional HLLC Riemann solver technology. The methods presented are cost-competitive with traditional higher order Godunov schemes.The two-dimensional HLLC Riemann solver is shown to work robustly for Euler and Magnetohydrodynamic (MHD) flows. Several stringent test problems are presented to show that the inclusion of genuinely multidimensional effects into higher order Godunov schemes indeed produces some very compelling advantages. For two dimensional problems, we were routinely able to run simulations with CFL numbers of ∼0.7, with some two-dimensional simulations capable of reaching higher CFL numbers. For three dimensional problems, CFL numbers as high as ∼0.6 were found to be stable. We show that on resolution-starved meshes, the scheme presented here outperforms unsplit second order Godunov schemes that are based on conventional one-dimensional Riemann solver technology. Strong discontinuities are shown to propagate very isotropically using the methods presented here. The present Riemann solver provides an elegant resolution to the problem of obtaining multi-dimensionally upwinded electric fields in MHD without resorting to a doubling of the dissipation in each dimension.

Superconvergence of Discontinuous Galerkin Methods for Scalar Nonlinear Conservation Laws in One Space Dimension

In this paper, an analysis of the superconvergence property of the semidiscrete discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $\left(k + \frac32\right)$th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.