Well-balanced Finite Volume Scheme Research Articles

A Well-Balanced Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at All Mach Numbers

We propose a second-order accurate semi-implicit and well-balanced finite volume scheme for the equations of ideal magnetohydrodynamics including gravitational source terms. The scheme treats all terms associated with the acoustic pressure implicitly while keeping the remaining terms part of the explicit sub-system. This semi-implicit approach makes the method particularly well suited for problems in the low Mach regime. We combine the semi-implicit scheme with the deviation well-balancing technique and prove that it maintains equilibrium solutions for the magnetohydrostatic case up to rounding errors. In order to preserve the divergence-free property of the magnetic field enforced by the solenoidal constraint, we incorporate a constrained transport method in the semi-implicit framework. Second order of accuracy is achieved by means of a standard spatial reconstruction technique with total variation diminishing property, and by an asymptotic preserving time stepping algorithm built upon the implicit-explicit Runge–Kutta time integrators. Numerical tests in the low Mach regime and near magnetohydrostatic equilibria support the low Mach and well-balanced properties of the numerical method.

A robust unstructured finite volume scheme for urban flash flood simulation: Duhok city as a case study

ABSTRACT A robust well-balanced finite volume scheme is proposed for simulating the flash floods due to rainfall-runoff process. The solver’s implementation on unstructured triangular meshes is based on a Godunov-type second-order upwind finite volume formulation, and Roe’s average scheme is used to determine the fluxes. The bed slope, friction and rainfall source terms are discretized using an upwind approach. The suggested algorithm uses the slope limiter and Hancock’s predictor-corrector method in order to secure numerical stability. The proposed model overwhelms the drawbacks of many commercial codes that use the finite difference, finite elements and finite volume in their scheme, especially in dealing with rapid change in topography. The prepared scheme solves the two-dimensional shallow water equations over a real topography of Hishkaro basin located in Duhok city in Kurdistan of Iraq where the variation of bed elevation reaches 717 m. The scheme is validated against real observed data in Duhok city provided by the Sewage directory. The data of five observation points over the study area were used to validate the code. The results showed that the max water level reaches 3.67, 2.12, 1.20, 0.67 and 0.3 m at observed points 3, 2, 4, 1 and 5, respectively.

A well-balanced finite volume scheme based on planar Riemann solutions for 2D shallow water equations with bathymetry

We consider in this paper a finite volume scheme based on local planar Riemann solutions for the two-dimensional shallow water equations with bathymetry. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The scheme to be presented is a development of the preliminary works that will be cited below. Our foremost purpose is to extend those results to two-dimensional formalism while preserving the physical and mathematical properties, including the well-balancedness. The proposed scheme is applied to some specific families of solutions, especially lake at rest and partially well-balanced solution. The numerical results show that this approach can give a good accuracy, except for resonant cases. Furthermore, it is proved that our finite volume scheme can preserve the C-property in the sense that it can capture exactly the lake at rest solution.

A new well-balanced finite-volume scheme on unstructured triangular grids for two-dimensional two-layer shallow water flows with wet-dry fronts

In this study, a novel two-dimensional finite-volume method on unstructured triangular meshes for two-layer shallow water flows is developed. First, the one-dimensional relaxation approach developed in Abgrall and Karni (2009) [1] is extended to the currently studied two-dimensional two-layer shallow water equations to build an unconditionally hyperbolic system. Next, a two-dimensional finite-volume HLL scheme on triangular grids for such a non-conservative hyperbolic system is constructed. Special attention is paid to guaranteeing the well-balanced property even in the presence of the wet-dry fronts. To this end, techniques such as spatial reconstruction for the secondary variables, special cell classifications for the two-layer fluid system, and special local reconstruction of auxiliary surface variables are proposed. These special techniques are further combined with new discrete formulas for the nonconservative products and the bed-slope terms, and it thus leads to an exactly well-balanced numerical scheme for the studied two-layer system even in the presence of wet-dry fronts. The developed numerical model is second order accurate and preserves the nonnegative layer depths. The performances of the proposed numerical model are investigated by several numerical experiments.

High-Order Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations With Nonlocal Free Energy

We propose high-order well-balanced finite-volume schemes for a broad class of hydrodynamic systems with attractive-repulsive interaction forces and linear and nonlinear damping. Our schemes are suitable for free energies containing convolutions of an interaction potential with the density, which are essential for applications such as the Keller--Segel model, more general Euler--Poisson systems, or dynamic-density functional theory. Our schemes are also equipped with a nonnegative-density reconstruction which allows for vacuum regions during the simulation. We provide several prototypical examples from relevant applications highlighting the benefit of our algorithms and also elucidate some of our analytical results.

Well-balanced finite volume schemes for nearly steady adiabatic flows

We present well-balanced finite volume schemes designed to approximate the Euler equations with gravitation. They are based on a novel local steady state reconstruction. The schemes preserve a discrete equivalent of steady adiabatic flow, which includes non-hydrostatic equilibria. The proposed method works in Cartesian, cylindrical and spherical coordinates. The scheme is not tied to any specific numerical flux and can be combined with any consistent numerical flux for the Euler equations, which provides great flexibility and simplifies the integration into any standard finite volume algorithm. Furthermore, the schemes can cope with general convex equations of state, which is particularly important in astrophysical applications. Both first- and second-order accurate versions of the schemes and their extension to several space dimensions are presented. The superior performance of the well-balanced schemes compared to standard schemes is demonstrated in a variety of numerical experiments. The chosen numerical experiments include simple one-dimensional problems in both Cartesian and spherical geometry, as well as two-dimensional simulations of stellar accretion in cylindrical geometry with a complex multi-physics equation of state.

An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity

We present an implicit-explicit well-balanced finite volume scheme for the Euler equations with a gravitational source term which is able to deal also with low Mach flows. To visualize the different scales we use the non-dimensionalized equations on which we apply a pressure splitting and a Suliciu relaxation. On the resulting model, we apply a splitting of the flux into a linear implicit and an non-linear explicit part that leads to a scale independent time-step. The explicit step consists of a Godunov type method based on an approximative Riemann solver where the source term is included in the flux formulation. We develop the method for a first order scheme and give an extension to second order. Both schemes are designed to be well-balanced, preserve the positivity of density and internal energy and have a scale independent diffusion. We give the low Mach limit equations for well-prepared data and show that the scheme is asymptotic preserving. These properties are numerically validated by various test cases.

A Second-Order Well-Balanced Finite Volume Scheme for the Multilayer Shallow Water Model with Variable Density

In this work, we consider a multilayer shallow water model with variable density. It consists of a system of hyperbolic equations with non-conservative products that takes into account the pressure variations due to density fluctuations in a stratified fluid. A second-order finite volume method that combines a hydrostatic reconstruction technique with a MUSCL second order reconstruction operator is developed. The scheme is well-balanced for the lake-at-rest steady state solutions. Additionally, hints on how to preserve a general class of stationary solutions corresponding to a stratified density profile are also provided. Some numerical results are presented, including validation with laboratory data that show the efficiency and accuracy of the approach introduced here. Finally, a comparison between two different parallelization strategies on GPU is presented.

Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations with General Free Energy

Well-balanced and free energy dissipative first- and second-order accurate finite-volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The variation of the natural Lyapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analogous property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. Performing a careful validation in a battery of relevant test cases, we show that these schemes can accurately analyze the stability properties of stationary states in challenging problems such as phase transitions in collective behavior, generalized Euler--Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories.

Second Order Finite Volume Scheme for Euler Equations with Gravity which is Well-Balanced for General Equations of State and Grid Systems

We develop a second order well-balanced finite volume scheme for compressible Euler equations with a gravitational source term. The well-balanced property holds for arbitrary hydrostatic solutions of the corresponding Euler equations without any restriction on the equation of state. The hydrostatic solution must be known a priori either as an analytical formula or as a discrete solution at the grid points. The scheme can be applied on curvilinear meshes and in combination with any consistent numerical flux function and time stepping routines. These properties are demonstrated on a range of numerical tests.

A second-order, discretely well-balanced finite volume scheme for euler equations with gravity

We present a well-balanced, second order, Godunov-type finite volume scheme for compressible Euler equations with gravity. By construction, the scheme admits a discrete stationary solution which is a second order accurate approximation to the exact stationary solution. Such a scheme is useful for problems involving complex equations of state and/or hydrostatic solutions which are not known in closed form expression. No á priori knowledge of the hydrostatic solution is required to achieve the well-balanced property. The performance of the scheme is demonstrated on several test cases in terms of preservation of hydrostatic solution and computation of small perturbations around a hydrostatic solution.

High-order well-balanced finite volume schemes for the Euler equations with gravitation

A high-order well-balanced scheme for the Euler equations with gravitation is presented. The scheme is able to preserve a spatially high-order accurate discrete representation of isentropic hydrostatic equilibria. It is based on a novel local hydrostatic reconstruction, which, in combination with any standard high-order accurate reconstruction procedure, achieves genuine high-order accuracy for smooth solutions close or away from equilibrium. The resulting scheme is very simple and can be implemented into any existing finite volume code with minimal effort. Moreover, the scheme is not tied to any particular form of the equation of state, which is crucial for example in astrophysical applications. Several numerical experiments were performed with a third-order accurate reconstruction. They demonstrate the robustness and high-order accuracy of the scheme nearby and out of hydrostatic equilibrium.

Numerical simulations of hydraulic jumps with the Shear Shallow Water model

An extension and numerical approximation of the shear shallow water equations model recently proposed in [21] is considered in this work. The model equations are able to describe the oscillatory nature of turbulent hydraulic jumps and as such correct the deficiency of the classical shallow water equations in describing such phenomena. The model equations, orig- inally developed for horizontal flow or flows occurring over small constant slopes, are straight- forwardly extended here for modeling flows over non-constant slopes and numerically solved by a second-order well-balanced finite volume scheme. Further, a new set of exact solutions to the extended model equations are derived and several numerical tests are performed to validate the numerical scheme and its ability to predict the oscillatory nature of hydraulic jumps under different conditions.

Reprint of: Finite volume methods for multi-component Euler equations with source terms

A first-order well-balanced finite volume scheme for the solution of a multi-component gas flow model in a pipe on non-flat topography is introduced. The mathematical model consists of Euler equations with source terms which arise from heat exchange, and gravity and viscosity forces, coupled with the mass conservation equations of species. We propose a segregated scheme in which the Euler and species equations are solved separately. This methodology leads to a flux vector in the Euler equations which depends not only on the conservative variables but also on time and space variables through the gas composition. This fact makes necessary to add some artificial viscosity to the usual numerical flux which is done by introducing an additional source term. Besides, in order to preserve the positivity of the species concentrations, we discretize the flux in the mass conservation equations for species, in accordance with the upwind discretization of the total mass conservation equation in the Euler system. Moreover, as proposed in a previous reference by the authors, [5], the discretizations of the flux and source terms are made so as to ensure that the full scheme is well-balanced. Numerical tests including both academic and real gas network problems are solved, showing the performance of the proposed methodology.

A Well-balanced Finite Volume Scheme for Shallow Water Equations with Porosity: Application to Modelling Flow through Rigid Vegetation

Strong interactions exist between flow dynamics and vegetation in open-channel. Depth-averaged shallow water equations can be used for such a study. However, explicit representation of vegetation can lead to very high resolution of the mesh since the vegetation is often modelled as vertical cylinders. Our work aims to study the ability of a single porosity-based shallow water model for these applications. More attention on flux and source terms discretizations are required in order to archive the well-balancing and shock capturing properties. We present a new Godunov-type finite volume scheme based on a simple-wave approximation and compare it with some other methods in the literature. A first application with experimental data was performed.

Finite volume methods for multi-component Euler equations with source terms

A first-order well-balanced finite volume scheme for the solution of a multi-component gas flow model in a pipe on non-flat topography is introduced. The mathematical model consists of Euler equations with source terms which arise from heat exchange, and gravity and viscosity forces, coupled with the mass conservation equations of species. We propose a segregated scheme in which the Euler and species equations are solved separately. This methodology leads to a flux vector in the Euler equations which depends not only on the conservative variables but also on time and space variables through the gas composition. This fact makes necessary to add some artificial viscosity to the usual numerical flux which is done by introducing an additional source term. Besides, in order to preserve the positivity of the species concentrations, we discretize the flux in the mass conservation equations for species, in accordance with the upwind discretization of the total mass conservation equation in the Euler system. Moreover, as proposed in a previous reference by the authors, [5], the discretizations of the flux and source terms are made so as to ensure that the full scheme is well-balanced. Numerical tests including both academic and real gas network problems are solved, showing the performance of the proposed methodology.

Runout analysis of a potential debris flow in the Dongwopu gully based on a well-balanced numerical model over complex topography

Debris flows could cause large amounts of damage in the alluvial fan where many houses and transport infrastructures exist. Thus, runout analysis of potential debris flows in these areas is necessary for protecting human lives, houses and infrastructure. Currently, the numerical simulation method is an effective way to intuitively display the runout of a potential debris flow. In this paper, a well-balanced finite volume scheme over complex topography was employed for studying the runout of a potential debris flow in the Dongwopu gully. Through model testing and validation, the numerical model had a good potential to adapt to the complex topography and could reasonably simulate the runout of debris flow events. Then, runout distance and inundation path, as well as flow depths and flow velocities, for the potential debris flow were obtained using the model. The simulation results appropriately reflected the runout characteristics.

A well-balanced finite volume scheme for the Euler equations with gravitation

Context. Many problems in astrophysics feature flows which are close to hydrostatic equilibrium. However, standard numerical schemes for compressible hydrodynamics may be deficient in approximating this stationary state, where the pressure gradient is nearly balanced by gravitational forces.Aims. We aim to develop a second-order well-balanced scheme for the Euler equations. The scheme is designed to mimic a discrete version of the hydrostatic balance. It therefore can resolve a discrete hydrostatic equilibrium exactly (up to machine precision) and propagate perturbations, on top of this equilibrium, very accurately.Methods. A local second-order hydrostatic equilibrium preserving pressure reconstruction is developed. Combined with a standard central gravitational source term discretization and numerical fluxes that resolve stationary contact discontinuities exactly, the well-balanced property is achieved.Results. The resulting well-balanced scheme is robust and simple enough to be very easily implemented within any existing computer code that solves time explicitly or implicitly the compressible hydrodynamics equations. We demonstrate the performance of the well-balanced scheme for several astrophysically relevant applications: wave propagation in stellar atmospheres, a toy model for core-collapse supernovae, convection in carbon shell burning, and a realistic proto-neutron star.

A Well-Balanced Finite Volume Scheme for a Mixed Hyperbolic/Parabolic System to Model Chemotaxis

This work is concerned by the numerical approximation of the weak solutions of a system of partial differential equations arising when modeling the movements of cells according to a chemoattractant concentration. The adopted PDE system turns out to couple a hyperbolic system with a diffusive equation. The solutions of such a model satisfy several properties to be preserved at the numerical level. Indeed, the solutions may contain vacuum, satisfy steady regimes and asymptotic regimes. By deriving a judicious approximate Riemann solver, a finite volume method is designed in order to exactly preserve the steady regimes of particular physical interest. Moreover, the scheme is able to deal with vacuum regions and it preserves the asymptotic regimes. Numerous numerical experiments illustrate the relevance of the proposed numerical procedure.

A Second Order Well-Balanced Finite Volume Scheme for Euler Equations with Gravity

We present a well-balanced, second order, Godunov-type finite volume scheme for compressible Euler equations with gravity. By construction, the scheme admits a discrete stationary solution which is a second order accurate approximation to the exact stationary solution. Such a scheme is useful for problems involving complex equations of state and/or hydrostatic solutions which are not known in closed form expression. No \'a priori knowledge of the hydrostatic solution is required to achieve the well-balanced property. The performance of the scheme is demonstrated on several test cases in terms of preservation of hydrostatic solution and computation of small perturbations around a hydrostatic solution.