Godunov Method Research Articles

Von Neumann stability analysis of globally constraint-preserving DGTD and PNPM schemes for the Maxwell equations using multidimensional Riemann solvers

The time-dependent equations of computational electrodynamics (CED) are evolved consistent with the divergence constraints on the electric displacement and magnetic induction vector fields. Respecting these constraints has proved to be very useful in the classic finite-difference time-domain (FDTD) schemes. As a result, there has been a recent effort to design finite volume time domain (FVTD) and discontinuous Galerkin time domain (DGTD) schemes that satisfy the same constraints and, nevertheless, draw on recent advances in higher order Godunov methods. This paper catalogues the first step in the design of globally constraint-preserving DGTD schemes. The algorithms presented here are based on a novel DG-like method that is applied to a Yee-type staggering of the electromagnetic field variables in the faces of the mesh. The other two novel building blocks of the method include constraint-preserving reconstruction of the electromagnetic fields and multidimensional Riemann solvers; both of which have been developed in recent years by the first author.The resulting DGTD scheme is linear, at least when limiters are not applied to the DG scheme. As a result, it is possible to carry out a von Neumann stability analysis of the entire suite of DGTD schemes for CED at orders of accuracy ranging from second to fourth. The analysis requires some simplifications in order to make it analytically tractable, however, it proves to be extremely instructive. A von Neumann stability analysis is a necessary precursor to the design of a full DGTD scheme for CED. It gives us the maximal CFL numbers that can be sustained by the DGTD schemes presented here at all orders. It also enables us to understand the wave propagation characteristics of the schemes in various directions on a Cartesian mesh. We find that constraint-preserving DGTD schemes permit CFL numbers that are competitive with conventional DG schemes. However, like conventional DG schemes, the CFL of DGTD schemes decreases with increasing order. To counteract that, we also present constraint-preserving PNPM schemes for CED. We find that the third and fourth order constraint-preserving DGTD and P1PM schemes have some extremely attractive properties when it comes to low-dispersion, low-dissipation propagation of electromagnetic waves in multidimensions. Numerical accuracy tests are also provided to support the von Neumann stability analysis. We expect these methods to play a role in those problems of engineering CED where exceptional precision must be achieved at any cost.

A new high-order accuracy numerical method for numerical modeling of supernovae explosions

The paper presents a new numerical method of high accuracy order for simulation of supernova explosion. The numerical method is based on a combination of the method of separation of operators, the Godunov method, the HLL scheme, and the piecewise-parabolic method on local stencil. The results of the verification of the new method on shock tube problems, the Sedov problem, and the collapse of the cloud are presented. The results of modeling a non-central supernova explosion characteristic of SNIa are presented.

Mathematical modeling of wide-range compressible two-phase flows

The paper considers flow modeling of two-phase heterogeneous medium. Each phase of the medium is considered as continuum, which is described by the compressible Euler equations. The phases are separated by the contact surface (interface) and are not mixed on the molecular level. Examples of such medium are: mixtures of solid particles and gas (in dense or dilute concentration of particles), liquids with small bubbles, solid porous materials filled with gas or liquid. The phase can be of connected structure (dense particles, porous solid, gas between dilute particles, etc.) or of non-connected structure (separated inclusions as gas bubbles, dilute particles, closed pores in solid, etc.). The connectivity of the phase is closely related to the propagation of acoustic perturbations. An attempt was made to consider all the cases of the phase connectivity in the framework of a unique approach. The paper presents an approach that couples the models designed for different cases of phase connectivity to the generalized hyperbolic and thermodynamically consistent form. The proposed model is applicable for simulating flows with change of the phase connectivity, e.g. dense-to-dilute two-phase flows. The model is verified on several problems of gas–solid granular medium flows. In numerical simulations the Godunov method with the HLLEM flux approximation on arbitrary moving Euler grids is used.

Efficiency of ENO-schemes for computation of linear waves in an elastic body

The efficiency of the UNO- and TVD-modifications of the Godunov method in computing one and two-dimensional linear waves in an elastic body is numerically studied. The body is simulated by an isotropic linear-elastic semi-space. Riemann problem of cylindrical discontinuity in hydrostatic pressure in a body and a problem of propagation of waves caused by the impulsive action on the free surface of a metal body are considered. It is shown that in all the cases both modifications resolve the waves in the body significantly better than the classic Godunov method. At that, in the case of presence of the pronounced extrema in the solution, the UNO- modification appears more preferable because the TVD- modification “cuts” the extrema to implement the TVD-property.

An Unstructured-Grid Based Morphodynamic Model for Sandbar Simulation in the Modaomen Estuary, China

The Modaomen Estuary is the most important passageway in discharging flood and sediment of the Pearl River Delta, which is one of the most complex estuarine systems in China. Due to the coupling effect among tidal currents, waves, and sediments, an immense sandbar area evolved in the outer subaqueous delta, impeding the flood releasing during wet season, as well as salinity intrusion during the dry season. In this work, an unstructured-grid based morphodynamic model was proposed to simulate the sandbar evolution process in the Modaomen Estuary. The proposed model was constructed by using the two-dimensional shallow water equations for tidal flow, the advection-diffusion equations for salinity and suspended sediment transport, and the non-equilibrium formulation of the Exner equation for bed evolution. To simulate the wave-induced longshore currents in the Modaomen Estuary, an adaptive time-stepping approach was proposed to couple the unstructured-grid based Simulating WAves Nearshore (SWAN) model and the shallow flow model. An integrated solver is proposed for computing flow, salinity, and sediment transport fluxes simultaneously, and then the shallow water equations and advection-diffusion equations are jointly solved by a high-resolution, unstructured-grid Godunov method. Application of the model to the Modaomen Estuary, using calibrated parameter values, gives results comparable to the measured data. The butterfly shape of the sandbar in the Modaomen Estuary is considerably well simulated by the proposed model, which matches well with the measured topography.

Numerical Simulation of the Interaction of an Air Shock Wave with a Surface Gas–Dust Layer

Within the framework of the one-velocity and multivelocity models of a dust-laden gas with the use of the Godunov method with a linearized Riemann solver, the problem of the interaction of a shock wave with a dust-laden gas layer located along a solid plane surface has been studied.

The efficiency of TVD-modification of the Godunov method in computation of waves in AN elastic-plastic body

The efficiency of TVD-modification of the Godunov method in computation of waves in AN elastic-plastic body

Sensitivity Analysis and Numerical Diffusion Effects for Hyperbolic PDE Systems with Discontinuous Solutions. The Case of Barotropic Euler Equations in Lagrangian Coordinates

Sensitivity analysis (SA) is the study of how the output of a mathematical model is affected by changes in the inputs. SA is widely studied, due to its many applications: uncertainty quantifi-cation, quick evaluation of close solutions, and optimization, to name but a few. In this work we show that the classic SA techniques, in particular the continuous sensitivity equation (CSE) method, cannot be used if the mathematical model is a system of hyperbolic partial differential equations (PDEs) with discontinuous solutions. The problem arises from the fact that the CSE method requires the differentiation of the state variable: if the latter is discontinuous, this in turns generates Dirac delta functions in the sensitivity. The focus of the first part of this work is to define a system of sensitivity equations valid also in case of discontinuous state: in order to do that, we add a correction term based on the Rankine-Hugoniot conditions. In the second part of this work we illustrate with some numerical tests how some classical finite volume schemes (an exact Godunov method and a Roe-type method) do not converge to the analytical solution for the sensitivity, due to the numerical diffusion: for this reason, we present an anti-diffusive numerical scheme, which provides the correct results for the sensitivity. In this work we carry out the computation in detail for the barotropic Euler equations in Lagrangian coordinates, but the approach is general and can be applied to any hyperbolic system with discontinuous solutions.

Numerical Modeling of the Gas Dynamics of the Safety Valve

Рассматриваются газодинамические процессы в полости предохранительного клапана с определением силового воздействия газового потока на диск с учетом нестационарности и сжимаемости потока. Предохранительный клапан представляет собой механизм, автоматически открывающийся при повышении давления выше рабочего и закрывающийся при снижении давления до рабочего или несколько ниже него. Задачей расчета предохранительных клапанов является определение пропускной способности, подбор характеристик пружины к ним, динамических усилий, возникающих при срабатывании предохранительного клапана. Известные математические модели работы клапанов, описывающие движения диска под действием сил со стороны газа и пружины, основанные либо на применении эмпирических коэффициентов, либо на сложно реализуемом решении задачи о газодинамических процессах, осуществляемых в трехмерной постановке с использованием пакета ANSYS. В данной статье обосновывается возможность численного моделирования основных газодинамических характеристик клапана на основе решения осесимметричных уравнений газовой динамики. Для численного решения системы газодинамических уравнений с осевой симметрией применяется метод контрольного объема. Параметры газа на границах контрольных объемов определяются по методу С. К. Годунова с использованием автомодельного решения задачи о распаде произвольного разрыва. В расчете применяется структурированная ортогональная разностная сетка, построенная комплексным методом граничных элементов и отображающая криволинейную расчетную область на прямоугольник. Записана математическая модель движения диска клапана с учетом нестационарного изменения давления в емкости, в которой регулируется давление. Получены результаты численного моделирования нестационарного процесса открытия и закрытия клапана. Проведенное численное моделирование газодинамических процессов в предохранительном пружинном клапане прямого действия показало, что течение в области, расположенной выше по потоку от диска, имеет осевую симметрию. Звуковая линия, «запирающая» эту область от возмущений ниже по потоку, преобладающее время соприкасается с краем диска. Результаты эксперимента подтверждают динамику открытия клапана, рассчитанную по рассматриваемому методу.

Diffraction of a Shock Wave on a Wedge in a Dusty Gas

Within the framework of one- and multivelocity dusty-gas models, the author has investigated, on a curvilinear grid, flow in reflection of a shock wave from the wedge-shaped surface in an air–droplet mixture using the Godunov method with a linearized Riemannian solver.

Multiple-correction hybrid k-exact schemes for high-order compressible RANS-LES simulations on fully unstructured grids

A Godunov's type unstructured finite volume method suitable for highly compressible turbulent scale-resolving simulations around complex geometries is constructed by using a successive correction technique. First, a family of k-exact Godunov schemes is developed by recursively correcting the truncation error of the piecewise polynomial representation of the primitive variables. The keystone of the proposed approach is a quasi-Green gradient operator which ensures consistency on general meshes. In addition, a high-order single-point quadrature formula, based on high-order approximations of the successive derivatives of the solution, is developed for flux integration along cell faces. The proposed family of schemes is compact in the algorithmic sense, since it only involves communications between direct neighbors of the mesh cells. The numerical properties of the schemes up to fifth-order are investigated, with focus on their resolvability in terms of number of mesh points required to resolve a given wavelength accurately. Afterwards, in the aim of achieving the best possible trade-off between accuracy, computational cost and robustness in view of industrial flow computations, we focus more specifically on the third-order accurate scheme of the family, and modify locally its numerical flux in order to reduce the amount of numerical dissipation in vortex-dominated regions. This is achieved by switching from the upwind scheme, mostly applied in highly compressible regions, to a fourth-order centered one in vortex-dominated regions. An analytical switch function based on the local grid Reynolds number is adopted in order to warrant numerical stability of the recentering process. Numerical applications demonstrate the accuracy and robustness of the proposed methodology for compressible scale-resolving computations. In particular, supersonic RANS/LES computations of the flow over a cavity are presented to show the capability of the scheme to predict flows with shocks, vortical structures and complex geometries.

Computational electrodynamics in material media with constraint-preservation, multidimensional Riemann solvers and sub-cell resolution – Part I, second-order FVTD schemes

While classic finite-difference time-domain (FDTD) solutions of Maxwell's equations have served the computational electrodynamics (CED) community very well, formulations based on Godunov methodology have begun to show advantages. We argue that the formulations presented so far are such that FDTD schemes and Godunov-based schemes each have their own unique advantages. However, there is currently not a single formulation that systematically integrates the strengths of both these major strains of development. While an early glimpse of such a formulation was offered in Balsara et al. [16], that paper focused on electrodynamics in plasma. Here, we present a synthesis that integrates the strengths of both FDTD and Godunov-based schemes into a robust single formulation for CED in material media.Three advances make this synthesis possible. First, from the FDTD method, we retain (but somewhat modify) a spatial staggering strategy for the primal variables. This provides a beneficial constraint preservation for the electric displacement and magnetic induction vector fields via reconstruction methods that were initially developed in some of the first author's papers for numerical magnetohydrodynamics (MHD). Second, from the Godunov method, we retain the idea of upwinding, except that this idea, too, has to be significantly modified to use the multi-dimensionally upwinded Riemann solvers developed by the first author. Third, we draw upon recent advances in arbitrary derivatives in space and time (ADER) time-stepping by the first author and his colleagues. We use the ADER predictor step to endow our method with sub-cell resolving capabilities so that the method can be stiffly stable and resolve significant sub-cell variation in the material properties within a zone.Overall, in this paper, we report a new scheme for numerically solving Maxwell's equations in material media, with special attention paid to a second-order-accurate formulation. Several numerical examples are presented to show that the proposed technique works. Because of its sub-cell resolving ability, the new method retains second-order accuracy even when material permeability and permittivity vary by an order-of-magnitude over just one or two zones. Furthermore, because the new method is also unconditionally stable in the presence of stiff source terms (i.e., in problems involving giant conductivity variations), it can handle several orders-of-magnitude variation in material conductivity over just one or two zones without any reduction of the time–step. Consequently, the CFL depends only on the propagation speed of light in the medium being studied.

Modelling and simulation of pavement drainage

ABSTRACTThe development of a numerical model to describe pavement surface runoff by a system of non-linear hyperbolic partial differential equations enables the analysis and prediction of safety-relevant road design and construction. Aquaplaning risks could be minimised as well as damages to the road surface geometry prevented. The hydromechanical approach presented in this article by a two-dimensional Pavement Surface Runoff Model (PSRM) based on the depth-averaged Shallow Water Equations – in contrast to empirical models – represents a general simulation. PSRM includes features like the treatment of an irregular topography or the possibility to treat different surface textures. The model enables the calculation of water depths. Discontinuous solutions to the underlying equations are numerically gained by applying Riemann solvers with Godunov methods to compute approximate intercell fluxes over the discretised problem using Finite Volume Methods. The Harten–Lax–Van Leer (HLL) Riemann solver was chosen to calculate the intercell fluxes. Simulation examples show the wide range of possibilities to describe real pavement surface runoff problems, for example junctions and superelevation transitions. Flow resistance is modelled with the Darcy–Weisbach equation by considering the pavements mean texture depth (MTD) as a variable parameter. In order to take into account the pavement surface drainage capabilities due to infiltration processes in porous surface layers, a coupled three-dimensional numerical formulation enlarges the macroscopic PSR Model. The saturation of the porous surface geometry and the infiltration velocities by Darcy velocities under consideration of interface conditions on the interface between free flow on the road surface and drainage into the porous structure are implemented within an emerging hydromechanical numerical model.

Exact and approximate Riemann solvers for compressible two-phase flows

Numerical methods for solving equations of two-phase hydrodynamics, which describe the flow of a dispersed solid and gas mixture are considered. The Godunov method is applied as the main approach to approximate numerical fluxes in solutions of the relevant Riemann problems. The formulations of these problems for the solid and gas phases are given, their exact analytical solution is described, and possible simplified approximate solutions are discussed. The obtained theoretical results are applied to the construction of a discrete model, which results in the generalization of the well-known Godunov-type and Rusanov-type methods to the case of nonequilibrium two-phase media. The numerical results involve the verification of the constructed methods on the analytical solutions of two-phase equations.

Account of Interfractional Heat Transfer in a Hyperbolic Model of a One-Velocity Heterogeneous Mixture

A modified generalized equilibrium model of a one-velocity heterogeneous mixture has been presented in which account is taken of interfractional heat transfer. A characteristic analysis of the model′s equations has been made, and their hyperbolicity has been shown. The Prandtl–Meyer problem and the problem on air–droplet-mixture flow past a wedge has been solved on a curvilinear structured grid using the Godunov method with a linearized Riemannian solver. Results of numerical calculations have been compared with self-similar solutions.

The Godunov method for a 2-phase model

Abstract We consider the Godunov numerical method to the phase-transition trafic model, proposed in [1], by Colombo, Marcellini, and Rascle. Numerical tests are shown to prove the validity of the method. Moreover we highlight the differences between such model and the one proposed in [2], by Blandin, Work, Goatin, Piccoli, and Bayen.

Comparison of the Interface-Capturing Behavior between LSM and MCLS

Numerical simulation of the moving interface is widely investigated in the fields of two-phase flow, crystal growth, molten metal casting and so on. To simulate accurately and quickly motion state of the moving interface, two types of interface-capturing method, LSM (level set method) and MCLS (mass-conserving LSM coupled with VOF (volume of fluid)), were comparatively analysed and studied. Two kinds of moving interface, rotation motion of Zaleska slotted disk interface and shearing deformation of the circle interface, were simulated. The level set advection equation was solved with IATVD (the characteristic integral-averaging finite volume method), RK-WENO (the third order Runge-Kutta method with WENO (weighted essentially non-oscillatory)), or the low order method, respectively. The level set function was re-initialized with modified Godunov method. Computation was implemented in different size mesh. When it is independent with mesh size, results show that the high order method, IATVD and RK-WENO, can get higher computing accuracy in LSM without topological geometry change. But computing time cost of RK-WENO is much greater than that of IATVD. For the low order method, LSM captures the moving interface in bad, however, MCLS’s numerical result is very good. But computing time cost of MCLS is much and much greater than that of LSM. For interface-capturing with topological geometry change, these two high order methods all can not capture the moving interface in good in LSM. Nevertheless, the resulting interfaces all are same as original interface in MCLS. These indicate that an appropriate numerical method and interface-capturing method can be chosen to accurately and quickly capture the related moving interface.

Transient simulation of a two-phase loop thermosyphon with a model out of thermodynamic equilibrium

Numerical investigation of two-phase loop thermosyphon (2PLT) in steady and transient states is addressed. A one-dimensional two-phase flow model describing a liquid-gas mixture in both mechanical and thermal equilibrium but out of thermodynamic equilibrium is developed. The model considers subcooled liquid and over heated vapor as well as phase transition (evaporation and condensation). The flow model is solved with a specific hyperbolic solver based on Godunov method and Harten-Lax-van Leer-Contact (HLLC) Riemann solver. A parametric study on the thermal power at the evaporator is performed in steady and transient states, the aim being to determine the effects of thermal power increase at the evaporator on the loop behavior. The comparison between Goodwin and Stiffened Gas (SG) equation of state (EOS) models shows fair agreement for latent heat of vaporization, specific volume and enthalpy for both liquid and vapor phases. Simulation of four test cases, corresponding to different evaporator thermal loads, is also carried out in transient state showing that loop response is correctly reproduced by this numerical approach, novel in the context of thermosyphon loops.

Numerical solution of the Navier-Stokes equations by discontinuous Galerkin method

Detailed unstructured grids and numerical methods of high accuracy are frequently used in the numerical simulation of gasdynamic flows in areas with complex geometry. Galerkin method with discontinuous basis functions or Discontinuous Galerkin Method (DGM) works well in dealing with such problems. This approach offers a number of advantages inherent to both finite-element and finite-difference approximations. Moreover, the present paper shows that DGM schemes can be viewed as Godunov method extension to piecewise-polynomial functions.As is known, DGM involves significant computational complexity, and this brings up the question of ensuring the most effective use of all the computational capacity available. In order to speed up the calculations, operator programming method has been applied while creating the computational module. This approach makes possible compact encoding of mathematical formulas and facilitates the porting of programs to parallel architectures, such as NVidia CUDA and Intel Xeon Phi.With the software package, based on DGM, numerical simulations of supersonic flow past solid bodies has been carried out. The numerical results are in good agreement with the experimental ones.

New hyperbolic models of dusty gas flow

The paper presents hyperbolic models for dusty gas flow formulated for single- and multivelocity approximations with account for inter-fractional heat transfer. Characteristic analysis of equations for the model was performed. The Godunov method and linearized Riemann solver was applied for a solution on curvilinear mesh: the Prandtl−Meyer problem for air-droplet mixture was solved. The simulation results were compared with the self-similar solution.