Godunov-type Numerical Scheme Research Articles

A Well-balanced and Positivity-preserving Godunov-type Numerical Scheme for a Spray Model

A Well-balanced and Positivity-preserving Godunov-type Numerical Scheme for a Spray Model

On one approach for the numerical solving of hyperbolic initial-boundary problems on an adaptive moving grids

An approach for the quality improvement of numerical modeling of the processes described by nonlinear hyperbolic equations, which have conservative form is proposed. The transition to synchronous moving coordinate system along with Godunov type numerical scheme for the problem on moving mesh is applied. The grid motion law relies heavily on Rankine–Hugoniot relations. In order to prevent the degeneration of moving computational mesh the regularization mechanism is implemented. It allows to almost automatically fit the computational mesh to the admissible width of the solution singularity resolution. The proposed algorithm is tested on several initial-boundary problems for the Hopf equation with well known exact solutions. The hyperbolic Buckley–Leverett equation is also considered as an example of application of the proposed approach.

Development of a Shock-Stable and Contact-Preserving Scheme for Multidimensional Euler Equations

The shock instability, notably the disastrous carbuncle phenomenon, has plagued many low-dissipation Godunov-type numerical schemes in calculations of high Mach flows. In the current work, two popular stability analysis tools and corresponding numerical experiments are used to investigate the root cause of shock instability of the low-dissipation Harten, Lax, and van Leer with contact (HLLC) scheme. The local linear stability analysis reveals that the HLLC flux could attenuate all perturbations in the streamwise direction but not perturbations of density and shear velocity in the transverse direction. Numerical results also indicate that the shock instability of the HLLC scheme is only related to the transverse flux. The viscosity terms of entropy wave and shear wave are introduced into the transverse flux, and the stability analysis shows that with the help of additional viscosities the shock instability of the HLLC scheme is cured. To preserve the capability of resolving contact discontinuities and shear waves, a pressure-based switching function is incorporated into the expressions of viscosity terms so that the additional viscosities are activated only when calculating the transverse flux in the shock layer. A series of numerical experiments give evidence of the robustness and accuracy of the proposed scheme, and the strategy adopted here can be easily applied to cure shock instabilities of other low-dissipation numerical schemes, e.g., the Roe scheme and the AUSM+ scheme.

Modeling of the bubble collapse with water jets and pressure loads using a geometrical volume of fluid based simulation method

The dynamic behavior of bubble collapses, water jets, and pressure loads during the collapse of the bubble near walls and a free surface were numerically investigated via a geometrical volume of fluid (VOF)-based simulation method. The numerical method is based on the compressible Navier–Stokes equations in a conservative form that describe the flow of compressible viscous fluids. The equations are discretized on a general curvilinear grid using an associated Godunov-type numerical scheme, and a reconstruction of a computational finite-volume Riemann solver is introduced for suppression of oscillation near the interface between fluids. The interface was tracked using the VOF reconstruction method. The VOF method is based on a geometrical tool and Lagrangian propagation of the interface reconstructed by a piecewise linear interface calculation (PLIC), resulting in strictly mass-conserving and sharp interface solutions. The numerical procedure was validated for capturing sharp interface and strong shock waves. For the simulation of bubble collapse, grid dependence studies of a spark-generated bubble were studies of both spherical and non-spherical bubbles were conducted. The results showed good agreement between the simulation and experiment of the bubble dynamics during the collapse process. Subsequently, an investigation of a single bubble near a wall with different standoff distances was performed. The pressure loads induced by the jets impacting the walls were calculated and analyzed. Furthermore, a more complex case of bubble collapse near an oblique wall and free surface was simulated. The resulting bubble dynamics with the jets and free surface shape were compared via photographs of the experiments.

Positivity-preserving numerical scheme for hyperbolic systems with $$\delta $$-shock solutions and its convergence analysis

Godunov type numerical schemes for the class of hyperbolic systems, admitting non-classical $\delta-$ shocks are proposed. It is shown that the numerical approximations converge to the solution and preserve the physical properties of the system such as positive density and bounded velocity. The scheme has been extended to positivity preserving and velocity bound preserving second-order accurate scheme by using appropriate slope limiters. The numerical results are compared with the existing the literature and the scheme is shown to capture the solution efficiently. The paper presents a hyperbolic system, for which an entropy satisfying scheme is constructed through an appropriate decoupling of the system into two scalar conservation laws with discontinuous flux.

Nonlocal approaches for multilane traffic models

We present a multilane traffic model based on balance laws, where the nonlocal source term is used to describe the lane changing rate. The modelling framework includes the consideration of local and nonlocal flux functions. Based on a Godunov type numerical scheme, we provide BV estimates and a discrete entropy inequality. Together with the $L^1$-contractivity property, we prove existence and uniqueness of weak solutions. Numerical examples show the nonlocal impact compared to local flux functions and local sources.

Numerical modeling of multiphase compressible flows with the presence of shock waves using an interface-sharpening five-equation model

An accurate shock- and interface-capturing method is introduced for simulations of compressible multiphase flows. First, an associated Godunov-type numerical scheme is established for a five-equation two-phase model obtained from a seven-equation model by assuming a single velocity and a single pressure between two phases. The computational finite-volume Riemann solver using the scheme and computing algorithms is presented. Next, an interface-sharpening technique (IST) is extended for the compressible two-phase model to improve numerical simulations and correct diffusion errors. The modified IST was applied as postprocessing to correct the numerical diffusion error in the solution of the discretization scheme while maintaining a sharp interface with a desired thickness after each time step. A mixture-consistent interface regularization approach of all conservative variables is combined with the IST to obtain consistent thermodynamic laws for the mixture, ensuring the consistency of the variables in the correction process. Several examples of fluid interface simulations including shock-tube, shock-bubble interactions, and underwater explosions were performed to demonstrate the accuracy and capability of the proposed method. Those compressible multiphase flow problems are complicated by the presence of both shock waves and the dynamics of interfaces. Comparisons of the numerical method with theoretical results and experimental data indicate that the present method can simulate interface dynamics with the presence of shock waves and large density differences.

Predicting deep water wave breaking with a non-hydrostatic shock-capturing model

A non-hydrostatic shock-capturing model is developed to address deep water wave evolution including wave breaking. The present model is based on the Euler equations, with a hybrid finite volume/finite difference (FV/FD) method and the high-resolution Godunov-type numerical scheme. Using these numerical schemes, wave breaking can be captured as a shock or a hydraulic jump without adopting additional wave breaking parameters or terms, such as breaking-onset criteria. The parallel method of the Message Passing Interface (MPI) is used to accelerate the simulation. This model was applied to simulate deep-water wave propagation including breaking as a consequence of dispersive focusing or due to the modulational instability. By comparison with existing experiments, the ability of the present model to predict breaking onset and breaking energy dissipation is evaluated. Additionally, a dynamic criterion is applied to provide additional information on wave breaking onset, and the wavelet transform is applied to predict the energy as a function of frequency and time.

Godunov-type Numerical Scheme for the Shallow Water Equations with Horizontal Temperature Gradient

We present a Godunov-type scheme for the shallow water equations with horizontal temperature gradient and variable topography. First, the exact solutions of the Riemann problem in a computational form are given, where algorithms for computing these solutions are described. Second, a Godunov-type scheme is constructed relying on exact solutions of the local Riemann problems. Computing algorithms for the scheme are given. The scheme is shown to be well-balanced and preserve the positivity of the water height. Numerical tests show that the scheme is convergent with a good accuracy, even for the resonant phenomenon, where the exact solutions contain several distinct waves propagating with the same shock speed. Furthermore, the scheme also provides us with good results for the solution of the wave interaction problem.

Non-local multi-class traffic flow models

We prove the existence for small times of weak solutions for a class of non-local systems in one space dimension, arising in traffic modeling. We approximate the problem by a Godunov type numerical scheme and we provide uniform L ∞ and BV estimates for the sequence of approximate solutions. We finally present some numerical simulations illustrating the behavior of different classes of vehicles and we analyze two cost functionals measuring the dependence of congestion on traffic composition.

A Godunov type scheme for a class of LWR traffic flow models with non-local flux

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide $L^\infty$ and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

Building a Godunov-type numerical scheme for a model of two-phase flows

Abstract Our aim in the paper is to build a Godunov-type numerical scheme for an isentropic model of two-phase flows. First, computational Riemann solvers together with computing algorithms in subsonic and supersonic regions are presented. Then, exact solutions of local Riemann problems are employed to build a Godunov-type scheme. The scheme is shown to be well-balanced in the sense that it can capture exactly stationary waves in both phases. Numerical tests show that the Godunov-type scheme possesses a good accuracy in the subsonic region as well as supersonic regions, where approximate solutions are convergent to the exact solution.

The HLLD Riemann solver based on magnetic field decomposition method for the numerical simulation of magneto-hydrodynamics

By splitting magnetic field into two components (internal plus external), we derived an extended formulation of the HLLD Riemann solver for numerical simulation of magnetohydrodynamics (MHD). This new solver is backward compatible with the standard HLLD Riemann solver when the external component of the magnetic field is zero. Moreover, the solver is more robust than the standard HLLD solver in applications to low plasma β (the ratio between thermal and magnetic pressures) cases, where the thermal pressure may become negative from subtracting the kinetic and large magnetic energy from the large total energy density in a Godunov type numerical scheme. Our numerical tests show that the extended HLLD solver works well for the cases of magnetic field decomposition, and maintains high resolution similar to the standard HLLD.

Multiple effects of sediment transport and geomorphic processes within flood events: Modelling and understanding

Flood events can induce considerable sediment transport which in turn influences flow dynamics. This study investigates the multiple effects of sediment transport in floods through modelling a series of hydraulic scenarios, including small-scale experimental cases and a full-scale glacial outburst flood. A non-uniform, layer-based morphodynamic model is presented which is composed of a combination of three modules: a hydrodynamic model governed by the two-dimensional shallow water equations involving sediment effects; a sediment transport model controlling the mass conservation of sediment; and a bed deformation model for updating the bed elevation. The model is solved by a second-order Godunov-type numerical scheme. Through the modelling of the selected sediment-laden flow events, the interactions of flow and sediment transport and geomorphic processes within flood events are elucidated. It is found that the inclusion of sediment transport increases peak flow discharge, water level and water depth in dam-break flows over a flat bed. For a partial dam breach, sediment material has a blockage effect on the flood dynamics. In comparison with the ‘sudden collapse’ of a dam, a gradual dam breach significantly delays the arrival time of peak flow, and the flow hydrograph is changed similarly. Considerable bed erosion and deposition occur within the rapid outburst flood, which scours the river channel severely. It is noted that the flood propagation is accelerated after the incorporation of sediment transport, and the water level in most areas of the channel is reduced.

A GPU-accelerated shallow flow model for tsunami simulations

This work presents a numerical model for tsunami propagation and inundation, which solves two-dimensional shallow water equations using a finite-volume Godunov-type numerical scheme incorporating a Harten–Lax–van Leer–contact (HLLC) approximate Riemann solver. To improve the computational efficiency for high-resolution tsunami simulations that commonly cover large spatial domains, the model is implemented for graphics processing units using compute unified device architecture (CUDA). The accelerated tsunami model is used to simulate an idealised dam-break case involving 1 million uniform computational cells and a laboratory tsunami propagation and run-up test with complex domain topographic features. Compared with a Fortran code for the same numerical scheme running on a standard personal computer, the model is able to reduce computational time by more than 40 times.

Riemann solver for a kinematic wave traffic model with discontinuous flux

We investigate a model for traffic flow based on the Lighthill–Whitham–Richards model that consists of a hyperbolic conservation law with a discontinuous, piecewise-linear flux. A mollifier is used to smooth out the discontinuity in the flux function over a small distance ϵ≪1 and then the analytical solution to the corresponding Riemann problem is derived in the limit as ϵ→0. For certain initial data, the Riemann problem can give rise to zero waves that propagate with infinite speed but have zero strength. We propose a Godunov-type numerical scheme that avoids the otherwise severely restrictive CFL constraint that would arise from waves with infinite speed by exchanging information between local Riemann problems and thereby incorporating the effects of zero waves directly into the Riemann solver. Numerical simulations are provided to illustrate the behavior of zero waves and their impact on the solution. The effectiveness of our approach is demonstrated through a careful convergence study and comparisons to computations using a third-order WENO scheme.

Numerical simulations of acoustic waves with the graphic acceleration GAMER code

Abstract We present results of numerical simulations of acoustic waves with the use of the Graphics Processing Unit (GPU) acceleration GAMER code which implements a second-order Godunov-type numerical scheme and adaptive mesh refinement (AMR). The AMR implementation is based on constructing a hierarchy of grid patches with an octree data structure. In this code a hybrid model is adopted, in which the time-consuming solvers are dealt with GPUs and the complex AMR data structure is manipulated by Central Processing Units (CPUs). The code is highly parallelized with the Hilbert space-filling curve method. These implementations allow us to resolve well desperate spatial scales that are associated with acoustic waves. We show that a localized velocity (gas pressure) pulse that is initially launched within a uniform and still medium triggers acoustic waves simultaneously with a vortex (an entropy mode). In a flowing medium, acoustic waves experience amplitude growth or decay, a scenario which depends on a location of the flow and relative direction of wave propagation. The amplitude growth results from instabilities which are associated with negative energy waves.

Hyperconcentrated 1D Shallow Flows on Fixed Bed with Geometrical Source Term Due to a Bottom Step

Free-surface, two-phase flows constituted by water and a significant concentration of sediments (i.e. hyperconcentrated flows), are often studied under the classical 1D shallow-water assumptions. This paper deals with the problem of the numerical simulation of this type of flows in presence of a geometrical source term caused by a discontinuity in the bed. Since we work in the framework of the finite-volume Godunov-type numerical schemes, we focus on the exact and approximated solution to Riemann Problems at cell interfaces with a step-like bed discontinuity. The paper has a twofold aim. First, it provides insights into the properties of the Step Riemann Problems (SRPs), highlighting in particular that a contact wave with special features may develop. Second, it presents a two-step predictor-corrector algorithm for the solution of the SRPs. In the predictor step, a Generalized Roe solver is used in order to provide an estimation of the wave structure arising from the cell-interface, while in the corrector step the exact SRP is solved by means of an efficient iterative procedure. The capabilities of the proposed solver are tested by comparing numerical and exact solutions of some SRPs.

Flood Simulation Using a Well-Balanced Shallow Flow Model

This work extends and improves a one-dimensional shallow flow model to two-dimensional (2D) for real-world flood simulations. The model solves a prebalanced formulation of the fully 2D shallow water equations, including friction source terms using a finite volume Godunov-type numerical scheme. A reconstruction method ensuring nonnegative depth is used along with a Harten, Lax, and van Leer approximate Riemann solver with the contact wave restored for calculation of interface fluxes. A local bed modification method is proposed to maintain the well-balanced property of the algorithm for simulations involving wetting and drying. Second-order accurate scheme is achieved by using the slope limited linear reconstruction together with a Runge-Kutta time integration method. The model is applicable to calculate different types of flood wave ranging from slow-varying inundations to extreme and violent floods, propagating over complex domains including natural terrains and dense urban areas. After validating against...

Generation of warm dense matter and strongly coupled plasmas using the High Radiation on Materials facility at the CERN Super Proton Synchrotron

A dedicated facility named High Radiation on Materials (HiRadMat) is being constructed at CERN to study the interaction of the 450 GeV protons generated by the Super Proton Synchrotron (SPS) with fixed solid targets of different materials. The main purpose of these future experiments is to study the generation and propagation of thermal shock waves in the target in order to assess the damage caused to the equipment, including collimators and absorbers, in case of an accident involving an uncontrolled release of the entire beam at a given point. Detailed numerical simulations of the beam-target interaction of several cases of interest have been carried out. In this paper we present simulations of the thermodynamic and the hydrodynamic response of a solid tungsten cylindrical target that is facially irradiated with the SPS beam with nominal parameters. These calculations have been carried out in two steps. First, the energy loss of the protons is calculated in the solid target using the FLUKA code [Fasso et al., “FLUKA: A multi-particle transport code,” Report Nos. CERN-2005-10, INFN/TC-05/11, and SLAC-R-773, 2005; Fasso et al., Conference on Computing in High Energy and Nuclear Physics, La Jolla, CA, 24–28 March 2003] and this energy loss data is used as input to a sophisticated two-dimensional hydrodynamic code, BIG2 [Fortov et al., Nucl. Sci. Eng. 123, 169 (1996)], which is based on a Godunov-type numerical scheme. The transverse intensity distribution in the beam focal spot is Gaussian. We consider three different sizes of the focal spot that are characterized by standard deviations, σ=0.088, 0.28, and 0.88 mm, respectively. This study has shown that the target is severely damaged in all the three cases and the material in the beam-heated region is transformed into warm dense matter including a strongly coupled plasma state. This new experimental facility can therefore also be used for dedicated experiments to study high energy density matter.