Godunov-type Scheme Research Articles

Cosmological Radiation Simulation Analysis: Taking Arepo-RT as an Example

With the fast-growing progress in computational physics, astrophysical simulations are made possible and continually upgraded with novel physical models. Contemporarily, many modern simulations feature the capability of cosmological radiation. In order to collate the recent improvements in this field, the numerical solver Arepo-RT is discussed as an important example. This study reorganizes and interprets the main physical models and numerical techniques used in Arepo-RT and similar solvers, with some details simplified to suit common readers. This paper describes simply and briefly the finite volume method, the radiative transfer model, the radiation source model, the Godunov-type schemes and the time integration, all of which are frequently used in modern radiation simulations. Apart from the realizations of the solvers, the results and limitations are also outlined in this paper, including the justification of reduced speed of light approximation, the results compared with JWST, and the issue of missing photons. These results shed light on guiding further improvements for Arepo-RT.

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

A Godunov-type scheme for the Saint-Venant-Exner equations with a moving steady states

The Saint-Venant-Exner system is widely used in industrial codes to model the transport of bed sediments. In practice, most of the methods used for its numerical resolution suffer from significant stability problems due to the fact that they are mainly guaranteed by separation techniques which allow weak coupling between hydraulic and morphodynamic software. The search for an adequate alternative method to this problem is a focus towards which several approaches converge. Recently, many authors have proposed acceptable methods but they are not so trivial to implement in an industrial context and some do not take into account all aspects such as mobile steady states. In this work, we derive a well-balanced scheme that takes into account all stable states, to approach weak solutions of the sediment transport model in shallow waters. We demonstrate that it captures exactly all regular steady solutions with non-evanescent velocities, retains water level positivity and is able to degenerate to a classical shallow water model when the sedimentary transport flow is null. A number of numerical test cases are also presented to illustrate these properties.

Computation of Unsteady Swirling Flows in Nozzles and Pipes by Applying a New Locally Implicit Godunov-Type Scheme

Computation of Unsteady Swirling Flows in Nozzles and Pipes by Applying a New Locally Implicit Godunov-Type Scheme

Godunov-type scheme for air–water transient pipe flow considering variable heat transfer and laboratorial validation

A robust comprehensive energy dissipation model is developed to investigate the rapid filling process of a T-shaped bifurcated pipeline with entrapped air pocket, while an experimental system is designed to validate the numerical model. In this work, a self-adapting heat transfer model is proposed to describe the energy exchange during transient event, fully considering the heat transfer of entrapped air pocket and hydraulic losses caused by steady friction and unsteady friction in the section of the filling water column. Importantly, the related heat transfer coefficient is variable and determined by mechanism formula of media characteristics, which is obviously different from the constant heat transfer coefficient in conventional model relying on the trial-and-error method and experimental data. Moreover, the second-order Godunov-type scheme is introduced to solve the governing equations of filling water column, while a virtual plug approach is proposed to track the air–water interface. The resulting predictions are compared to those obtained via a conventional empirical polytropic model and constant coefficient heat transfer model, and to experimental data. The proposed model accurately reproduces the experimental pressure oscillations. The results display that when an air pocket content exceeds a certain threshold (>1.7%), the comprehensive model using forced convection can reproduce the pressure fluctuations. When the air content is lower (<0.9%), the comprehensive model using natural convection can simulate pressure changes more accurately. The conventional empirical polytropic model underestimates the pressure damping. The constant coefficient heat transfer model previously cannot be employed if there is no experimental data to calibrate the constant coefficient, but now the constant coefficient can be obtained from the results of the proposed model. Significantly, the comprehensive model can directly and accurately simulate the energy dissipation during the rapid filling process.

Numerical stability analysis of Godunov-type schemes for high Mach number flow simulations

Modern shock-capturing schemes often suffer from numerical shock instabilities when simulating strong shocks, limiting their application in supersonic or hypersonic flow simulations. In the current study, we devote our efforts to investigating the shock instability problem for second-order schemes, which has not gotten enough attention in previous research but is crucial to address. To this end, we develop the matrix stability analysis method for the finite-volume Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) approach, taking into account the influence of reconstruction. With the help of this newly developed method, the shock instability of second-order schemes is investigated quantitatively and efficiently. The results demonstrate that when second-order schemes are employed, whether shock instabilities will occur is closely related to the property of Riemann solvers, just like the first-order case. However, enhancing spatial accuracy still impacts the shock instability problem, and the impact can be categorized into two types depending on the dissipation of Riemann solvers. Furthermore, the research emphasizes the impact of the numerical shock structure, highlighting both its role as the source of instability and the influence of its state on the occurrence of instability. One of the most significant contributions of this study is the confirmation of the multidimensional coupled nature of shock instability. Both one-dimensional and multidimensional instabilities are proven to influence the instability problem, and they have different properties. Moreover, this paper reveals that increasing the aspect ratio and distortion angle of the computational grid can help mitigate shock instabilities. The current work provides an effective tool for quantitatively investigating the shock instabilities for second-order schemes, revealing the inherent mechanism and thereby contributing to the elimination of shock instability.

A well-defined Godunov-type scheme in a Navier-Stokes model with a friction term

A well-defined Godunov-type scheme in a Navier-Stokes model with a friction term

Graphics processing unit (GPU)-enhanced nonhydrostatic model with grid nesting for global tsunami propagation and coastal inundation

Nonhydrostatic models have proven their superiority in describing tsunami propagation over trans-oceanic distances and nearshore transformation because of their good dispersion and nonlinearity properties. The novel one-layer nonhydrostatic formulations proposed by Wang et al. [Phys. Fluids 35, 076610 (2023)] have been rederived in the spherical coordinate system incorporating Coriolis effects to enable the application of basin-wide tsunami propagation. The model was implemented using the fractional step method, where the hydrostatic step was solved by a Godunov-type finite-volume scheme, and the nonhydrostatic step was obtained with the finite-difference method. Additionally, a two-way grid-nesting scheme was employed to adapt the topographic features for efficient computation of tsunami propagation in deep ocean and coastal inundation. Furthermore, graphics processing unit (GPU)-parallelism technique was incorporated to further optimize the model performance. An idealized benchmark test as well as three experiments of regular and irregular waves, solitary, and N-waves transformations have been simulated to demonstrate the superior performance of the current GPU-accelerated grid-nesting nonhydrostatic model. Finally, the model has been applied to reproduce the 1964 Prince William Sound Tsunami, its propagation across the North Pacific and induced inundation in the Seaside.

A Godunov-type stabilization scheme for the Boltzmann transport equation of III-V devices with a 3D k-space

This paper presents a deterministic approach for solving the Boltzmann transport equation (BTE) together with the Poisson equation (PE) for III-V semiconductor devices with a three-dimensional k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {k}}$$\\end{document}-space. The BTE is stabilized using Godunov’s scheme, whose linearity in the distribution function simplifies the application of the Newton–Raphson method to the coupled discrete BTE and PE. The formulation of the discrete equations ensures the nonnegativity of the distribution function regardless of the scattering rate, which can include the Pauli exclusion principle, and exhibits excellent numerical stability under steady state as well as transient conditions. In the latter case, both implicit and explicit time integration methods can be used and even slow processes (e.g., recombination) can be handled using this approach. In addition, the direct solution of the BTE can be easily extended to the small-signal case for arbitrary frequencies. Exemplary BTE results are shown for a GaAs N+NN+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{N}}^{+}{\ extrm{NN}}^{+}$$\\end{document}-structure, revealing, inter alia, that the approximations of the drift-diffusion model can fail for large built-in fields in III-V devices.

Hyperbolic modeling of gradient damage and one-dimensional finite volume simulations

This study provides a new formulation of gradient damage model which allows an efficient explicit numerical solution of dynamics problems. The proposed methodology is based on an ”extended Lagrangian approach” developed by one of the authors for the nondissipative and dispersive shallow water equation. By using this strategy, the global minimization problem commonly derived for gradient damage models is recast as a purely local hyperbolic one with source terms that can be easily solved using finite volumes. The numerical solution of the governing system is then based on a fractional-step method consisting of a classical Godunov-type scheme and an implicit Ordinary Differential Equation solver for the local source terms. Numerical results are presented on one-dimensional multi-fragmentation and spalling tests for illustration purpose.

A cell-centred Eulerian volume-of-fluid method for compressible multi-material flows

We present a practical cell-centred volume-of-fluid method developed within a pure Eulerian setting for the simulation of compressible solid-fluid problems. The method builds on a previously published diffuse-interface Godunov-type scheme through the addition of a specialised mixed-cell update that is capable of maintaining sharp interfaces indefinitely. The mixed-cell update is local and may be viewed as an interface-sharpening extension to the underlying diffuse-interface scheme serving the same purpose as other techniques such as Tangent of Hyperbola INterface Capturing (THINC), and hence the method can be straightforwardly extended to include other coupled physics. We validate the method on a range of challenging test problems including a collapsing metal shell, cylinder impacts and the three-dimensional simulation of a buried explosive charge. Finally we demonstrate the robustness of the method, and its use in a multi-physics context, by modelling the BRL 105 mm unconfined shaped charge with reactive high-explosive burn and rate-sensitive plasticity.

Sediment transport and bed erosion during storm surge using a coupled hydrodynamic and morphodynamic model considering wave and current interaction

Storm surge, a major disaster in coastal areas, is generated by tropical cyclones. Strong currents and waves during storm surges often cause sediment transport and beach erosion, significantly damaging the coastal ecosystem. This paper presents a flow-wave-sediment-morphodynamic coupled model to predict morphodynamic changes caused by overwash during storm surges. In the model, the flow field is calculated using the generalized shallow water equations coupled with the Simulating Waves Nearshore (SWAN) model, while sediment transport and bed changes are computed using a non-equilibrium total-load sediment transport model. The model solves the governing equations by employing the explicit finite-volume approach based on a rectangular mesh. The Godunov-type central upwind scheme is adopted to calculate the interface fluxes, thereby solving the complex Riemann problem. A series of numerical experiments considering wave-induced longshore current and short-term bed erosion are conducted to validate the established model and demonstrate its ability to simulate waves, flow fields, and bed changes during storm surge. Specifically, we used the model to simulate the morphological evolution of the Santa Rosa barrier island caused by Hurricane Ivan's storm surge, and the results indicate that the model accurately predicts the real-case erosion process caused by storm surge and overwash during the hurricane. The simulated results clearly explain overwash development, including washover fans, foredune erosion, and back-barrier deposition. It demonstrates that the front dune significantly eroded with the Santa Rosa barrier island coastline retreating by a maximum of about 80 m during the storm surge. More than three breaches with a width of up to 135 m were formed on the sandy barrier due to wave overwash, while the topography at the breach changed significantly.

A Random Choice Scheme for Transient Mixed Flows

A novel numerical model for describing highly transient mixed flows is presented based on the Preissmann slot approach. To overcome the spurious oscillation problems caused by this approach, a numerical scheme named “random choice method” (RCM), in which flow variables of the next time level are obtained by picking a local Riemann solution state at random, is applied herein. Three numerical tests are performed to verify the ability of the proposed model in simulating from single flows to mixed flows. The results show that the RCM gives sharper shock resolutions as compared with the Godunov-type scheme, which causes the smearing of discontinuities. The proposed model can eliminate the numerical oscillations under flow conditions of switching from free-surface flow to pressurized flow, because of its unconditional stability. On further analysis of the experimental verifications, a hybrid method is presented to improve the performance of RCM in the smooth (nonuniform) parts of the flow.

An entropy consistent and symmetric seven-equation model for compressible two-phase flows

We propose new seven-equation model and related solution methods for compressible two-phase flows. The model is built on new closure laws for the interaction between the coexisting two phases. It takes a symmetric form for the two phases and shares some favorable properties with the well-known Baer-Nunziato model [4]: the equations system is unconditionally hyperbolic and is conservative for the total momentum and energy; the interfacial interaction between phases is an isentropic process for each phase. Moreover, it permits to obtain a linear degenerate field associated with the wave which separates the mixture, and the full set of the associated Riemann invariants can be derived explicitly. The pressure and velocity relaxation terms are included for the additional interaction between phases. Numerical approximation of the entire model is performed in an operator splitting manner – two Godunov-type schemes utilizing respectively a HLL solver and a composite approximate Riemann solver are used to discretize the homogeneous system; stable implicit algorithms are explored for the non-instantaneous pressure and velocity relaxations. Several examples are provided to illustrate the property of the new model and the capability of the solution methods.

Direct numerical simulation of three-dimensional isotropic turbulence with smoothed particle hydrodynamics

This study presents an investigation of the capability of smoothed particle hydrodynamics (SPH) to simulate three-dimensional isotropic turbulence. The effect of the error introduced by the particle disorder is assessed by comparing the standard Lagrangian SPH with an Eulerian adaptation. For the free decay of isotropic turbulence in a triple periodic box, the Eulerian SPH shows very good agreement with the reference solution, while the particle disorder in Lagrangian simulations yields an incorrect prediction of turbulent energy spectra. For the first time, a SPH investigation on linearly forced isotropic turbulence is also conducted with a focus on how the numerical dissipation affects the obtained solution. It is found that by using a Godunov-type SPH scheme for the continuity equation and by employing Roe's approximate solver for the Riemann problem at the interface of each neighboring particle, a stable solution is obtained, which is also in agreement with the results predicted by the theory of homogeneous isotropic turbulence. The efficacy of the particle shifting technique applied to turbulent SPH flows is studied in the end. Numerical findings indicate that corrective terms derived from the arbitrary Lagrangian–Eulerian theory are essential for a proper estimation of turbulence characteristics.

Numerical studies of two-phase water hammer flows using Godunov methods

In this paper, an explicit finite volume Godunov-type scheme is applied for a two-phase homogeneous water hammer flow model. The developed finite volume method is based on an exact Riemann solver for a gas–liquid two-phase flow model. This paper investigates the case of a water hammer problem caused by a rapid closing valve located in upstream of the pipeline. The performance of the proposed numerical method is validated against experimental data from literature. The good agreement found between the numerical and experimental results confirms the accuracy and the robustness of both the numerical methods and the proposed exact Riemann solver.

A hyperbolic generalized Zener model for nonlinear viscoelastic waves

A macroscopic model describing nonlinear viscoelastic waves is derived in Eulerian formulation, through the introduction of relaxation tensors. It accounts for both constitutive and geometrical nonlinearities. In the case of small deformations, the governing equations recover those of the linear generalized Zener model (GZM) with memory variables, which is widely used in acoustics and seismology. The structure of the relaxation terms implies that the model is dissipative. The chosen family of specific internal energies ensures also that the model is unconditionally hyperbolic. A Godunov-type scheme with relaxation is implemented. A procedure for maintaining isochoric transformations at the discrete level is introduced. Numerical examples are proposed to illustrate the properties of viscoelastic waves and nonlinear wave phenomena.

Riemann problem and Godunov-type scheme for a two-layer blood flow model

We establish and investigate a two-layer blood flow model with two velocities in this paper. The mass exchange of the flows and the acceleration offset term between each layer have been incorporated into the model as source terms. We derive the elementary waves and solve the Riemann problem constructively. Besides, we develop a Godunov scheme for the new model. Numerical simulations are given to verify the effective and versatility of the model. The results may contribute to the application in more layers blood flow and the design of high order numerical schemes.

The Riemann problem and a Godunov-type scheme for a traffic flow model on two lanes with two velocities

We establish and investigate a traffic flow model on two lanes with two velocities in this paper. The model takes account of the conservation of mass and acceleration equation on each lane. We derive the elementary waves and solve the Riemann problem of the new model in 1-D case constructively. The interaction of elementary waves are then discussed subsequently. Besides, we develop a Godunov-type scheme with an exact Riemann solver established on the Riemann solutions. Numerical simulations generated from the real traffic flow situations are given to verify the effectiveness and versatility of the model.

A coupled model for macroscopic behavior of crowd in flood induced evacuation

Floods are regarded as critical disasters that often lead to catastrophic situations and provoke potential injuries and/or fatalities. The consequences of the flood on the exposed population can be realized by exploring the complex behavioral phenomena of the crowd in interaction with the flood hazard. A profound understanding of the time–space distribution of evacuation patterns and vulnerability in floodwater would help in developing a robust risk reduction strategy and mitigating the flood impact. This paper presents a model that combines a macroscopic description of a crowd evacuation model with a hydrodynamic model of flood inundation described by a two-dimensional depth-averaged Shallow Water Equation (SWE), considering two-way dynamic coupling to study the collective behavior of the crowd in a flooded evacuation scenario. Behavioral rules, such as reduced motion in floodwater, route choice through safe areas, and stress due to perception of danger in threatening flood, are imposed to account for the evacuees’ reaction to floodwater. Moreover, the disruption in free flood propagation due to the pedestrians’ presence in the floodplain has also been modeled. The pedestrian model is solved using a third-order Weighted Essentially Non-Oscillatory (WENO3) scheme, and the hydrodynamic model is numerically handled using a high-resolution Godunov type scheme with HLLC Riemann solver. The potential of the developed model is illustrated through two designed hypothetical test cases, which include the crowd evacuation from a supermarket area exposed to a sudden flood.