Well-balanced Scheme Research Articles

An approximate well‐balanced upgrade of Godunov‐type schemes for the isothermal Euler equations and the drift flux model with laminar friction and gravitation

AbstractIn this article, approximate well‐balanced (WB) finite‐volume schemes are developed for the isothermal Euler equations and the drift flux model (DFM), widely used for the simulation of single‐ and two‐phase flows. The proposed schemes, which are extensions of classical schemes, effectively enforce the WB property to obtain a higher accuracy compared with classical schemes for both the isothermal Euler equations and the DFM in case of nonzero flow in the presences of both laminar friction and gravitation. The approximate WB property also holds for the case of zero flow for the isothermal Euler equations. This is achieved by defining a relevant average of the source terms which exploits the steady‐state solution of the system of equations. The new extended schemes reduce to the original classical scheme in the absence of source terms in the system of equations. The superiority of the proposed WB schemes to classical schemes, in terms of accuracy and computational effort, is illustrated by means of numerical test cases with smooth steady‐state solutions. Furthermore, the new schemes are shown numerically to be approximately first‐order accurate.

Simulation of THz Oscillations in Semiconductor Devices Based on Balance Equations

Instabilities of electron plasma waves in high-mobility semiconductor devices have recently attracted a lot of attention as a possible candidate for closing the THz gap. Conventional moments-based transport models usually neglect time derivatives in the constitutive equations for vectorial quantities, resulting in parabolic systems of partial differential equations (PDE). To describe plasma waves however, such time derivatives need to be included, resulting in hyperbolic rather than parabolic systems of PDEs; thus the fundamental nature of these equation systems is changed completely. Additional nonlinear terms render the existing numerical stabilization methods for semiconductor simulation practically useless. On the other hand there are plenty of numerical methods for hyperbolic systems of PDEs in the form of conservation laws. Standard numerical schemes for conservation laws, however, are often either incapable of correctly handling the large source terms present in semiconductor devices due to built-in electric fields, or rely heavily on variable transformations which are specific to the equation system at hand (e.g. the shallow water equations), and can not be generalized easily to different equations. In this paper we develop a novel well-balanced numerical scheme for hyperbolic systems of PDEs with source terms and apply it to a simple yet non-linear electron transport model.

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 efficient finite difference method for the shallow water equations

A high-order explicit finite difference scheme is derived solving the shallow water equations. The boundary closures are based on the diagonal-norm summation-by-parts (SBP) framework and the boundary conditions are imposed using a penalty (SAT) technique. Flux-splitting combined with upwind SBP operators is used to naturally introduce artificial dissipation. The scheme is tested against various benchmark problems where high-order convergence is verified for smooth solutions. A particular discretization of the source term is used leading to a well-balanced scheme. We also present an application: A simplified incident wave simulation with wave-channel interaction using a multi-block setup. Experiments suggest that a bathymetry consisting of many spikes could provide a dispersing effect on an incoming wave.

A fully well-balanced scheme for the 1D blood flow equations with friction source term

We consider the 1D blood flow equations with friction source term. The main purpose of this work is the derivation of a positivity preserving and fully well-balanced scheme, which correctly approximates the solutions of this system, exactly preserves all the associated steady states, including the moving ones, and ensures the positivity of the cross-sectional area. To address such issues, a study of the moving steady states related to the friction source term is first performed. Afterwards, a Godunov-type scheme is constructed, by deriving a two-state approximate Riemann solver and by introducing a relevant discretization of the source term, to enforce the well-balanced property. The scheme is then adapted to the generalized model including a space-variable viscous resistance; a second-order well-balanced MUSCL extension of the scheme is also proposed. Numerical experiments are finally carried out in order to validate the scheme and highlight its properties.

An asymptotic preserving well-balanced scheme for the isothermal fluid equations in low-temperature plasmas at low-pressure

We present a novel numerical scheme for the efficient and accurate solution of the isothermal two-fluid (electron + ion) equations coupled to Poisson's equation for low-temperature plasmas at low-pressure. The model considers electrons and ions as separate fluids, comprising the electron inertia and space charge regions. The discretization of this system with standard explicit schemes is constrained by very restrictive time steps and cell sizes related to the resolution of the Debye length, electron plasma frequency, and electron sound waves. Both sheath and electron inertia are fundamental to fully explain the physics in low-pressure and low-temperature plasmas. However, most of the phenomena of interest for fluid models occur at speeds much slower than the electron thermal speed and are quasi-neutral, except in small charged regions. A numerical method that is able to simulate efficiently and accurately all these regimes is a challenge due to the multiscale character of the problem. In this work, we present a scheme based on the Lagrange-projection operator splitting that preserves the asymptotic regime where the plasma is quasi-neutral with massless electrons. As a result, the quasi-neutral regime is treated without the need of an implicit solver nor the resolution of the Debye length and electron plasma frequency. Additionally, the scheme proves to accurately represent the dynamics of the electrons both at low speeds and when the electron speed is comparable to the thermal speed. In addition, a well-balanced treatment of the ion source terms is proposed in order to tackle problems where the ion temperature is very low compared to the electron temperature. The scheme significantly improves the accuracy both in the quasi-neutral limit and in the presence of plasma sheaths when the Debye length is resolved. In order to assess the performance of the scheme in low-temperature plasmas conditions, we propose two specifically designed test-cases: a quasi-neutral two-stream periodic perturbation with analytical solution and a low-temperature discharge that includes sheaths. The numerical strategy, its accuracy, and computational efficiency are assessed on these two discriminating configurations.

Well-balanced central schemes for the one and two-dimensional Euler systems with gravity

In this paper we develop a family of central schemes for the one and two-dimensional systems of Euler equations with gravitational source term. The proposed schemes are unstaggered, second-order, central finite volume schemes that avoid solving Riemann problems at the cell interfaces and avoid switching between an original and a staggered grid. The main feature of the schemes developed here is that they are capable of preserving any steady state of the Euler with gravity system up to machine accuracy by updating the numerical solution in terms of a relevant reference solution. The methodology proposed results in a well-balanced scheme capable of capturing any steady state. Our scheme is then implemented and used to solve classical problems from the recent literature.

A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem

A novel notion for constructing a well-balanced scheme – a gradient-robust scheme – is introduced and a showcase application for the steady compressible, isothermal Stokes equations in a nearly-hydrostatic situation is presented. Gradient-robustness means that gradient fields in the momentum balance are well-balanced by the discrete pressure gradient — which is possible on arbitrary, unstructured grids. The scheme is asymptotic-preserving in the sense that it reduces for low Mach numbers to a recent inf–sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM–FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible.

Discontinuous Galerkin well-balanced schemes using augmented Riemann solvers with application to the shallow water equations

Abstract High order methods are becoming increasingly popular in shallow water flow modeling motivated by their high computational efficiency (i.e. the ratio between accuracy and computational cost). In particular, Discontinuous Galerkin (DG) schemes are very well suited for the resolution of the shallow water equations (SWE) and related models, being a competitive alternative to the traditional finite volume schemes. In this work, a novel framework for the construction of DG schemes using augmented Riemann solvers is proposed. Such solvers incorporate the source term at cell interfaces in the definition of the Riemann problem, allowing definition of two different inner states in the so-called star region. The benefits of this family of solvers lie in the exact preservation of the Rankine–Hugoniot condition at cell interfaces at the discrete level, ensuring the preservation of equilibrium solutions (i.e. the well-balanced property) without requiring extra corrections of the numerical fluxes. The semi-discrete DG operator becomes nil automatically under equilibrium conditions, provided the use of suitable quadrature rules. The proposed scheme is applied to the Burgers' equation with geometric source term and to the SWE. The numerical results evidence that the proposed scheme achieves the prescribed convergence rates and preserves the equilibrium states of relevance with machine precision.

Fokker--Planck Approach of Ostwald Ripening: Simulation of a Modified Lifshitz--Slyozov--Wagner System with a Diffusive Correction

We propose a well-balanced scheme for the modified Lifshitz--Slyozov equation, which incorporates a size-diffusion term. The method uses the Fokker--Planck structure of the equation. In turn, large time simulations can be performed with a reduced computational cost, since the time-step constraints are relaxed. The simulations bring out the critical mass threshold and the relaxation to equilibrium, which can be expected from the formal analogies with the Becker--Döring system.

Hydrostatic equilibrium preservation in MHD numerical simulation with stratified atmospheres

Context. Many astrophysical processes involving plasma flows are produced in the context of a gravitationally stratified atmosphere in hydrostatic equilibrium, in which strong gradients can exist with gas properties that vary in small regions by several orders of magnitude. The standard Godunov-type schemes with polynomial reconstruction used to numerically solve these problems fail to preserve the hydrostatic equilibrium owing to the appearance of spurious fluxes generated by the numerical unbalance between gravitational forces and pressure gradients. Aims. The aim of this work is to present local hydrostatic reconstruction techniques that can be implemented in existing codes with Godunov-type methods to obtain well-balanced schemes that numerically satisfy the hydrostatic equilibrium for various conditions. Methods. The proposed numerical scheme is based on the Godunov method with second order MUSCL-type reconstruction, as is extensively used in astrophysical applications. The difference between the scheme and the standard formulations is only given by calculating the pressure and density Riemann states on each intercell face and by computing the gravitational source term on each cell. Results. The local hydrostatic reconstruction scheme is implemented in the FLASH code to verify the well-balanced property for hydrostatic equilibrium with constant or linearly variable temperature and constant or variable gravity. In addition, the behavior of the scheme for hydrostatic equilibrium with arbitrary temperature distributions is also analyzed together with the ability to propagate low-amplitude waves and to capture shock waves. Conclusions. The scheme is demonstrated to be robust and relatively simple to implement in existing codes. This approach produces good results in hydrostatic equilibrium preservation, satisfying the well-balanced property for the preset conditions and strongly reducing the spurious fluxes for extreme configurations.

A novel well-balanced scheme for spatial and temporal bed evolution in rapidly varying flow

This paper presents a novel method of preserving well-balanced conditions for dam-break flows on irregular beds. The research on preservation of well-balanced conditions commonly deals only with spatial variation of beds. It is challenging to design a well-balanced scheme that can model both spatial and temporal variation of beds particularly on domains discretized with unstructured meshes. This challenge arises due to the non-trivial nature of the governing equations modeling hydro-morpho dynamics. This study proposes a new and simple well-balanced method applicable for both spatial and temporal bed variations without requiring additional treatment and complicated discretization of slope source terms. The work relies on a cell-centered Godunov-type finite volume method for solving coupled Saint Venant and sediment continuity equations. The two most important issues, i.e., the ‘lake at rest condition’ and ‘water depth positivity’ of shallow-water flows over irregular beds, are tested and verified in the proposed numerical scheme. The authors employed Riemann solvers to compute fluxes through the control volume surfaces. The proposed scheme is evaluated for various benchmark cases of dam-break flows on rigid and erodible beds. Comparisons between the results of the proposed numerical scheme, analytical solution, experimental data, and previous numerical observations show significant improvement for some cases and a good agreement in the rest.

A well-balanced scheme for the simulation tool-kit A-MaZe: implementation, tests, and first applications to stellar structure

Characterizing stellar convection in multiple dimensions is a topic at the forefront of stellar astrophysics. Numerical simulations are an essential tool for this task. We present an extension of the existing numerical tool-kit A-MaZe that enables such simulations of stratified flows in a gravitational field. The finite-volume based, cell-centered, and time-explicit hydrodynamics solver of A-MaZe was extended such that the scheme is now well-balanced in both momentum and energy. The algorithm maintains an initially static balance between gravity and pressure to machine precision. Quasi-stationary convection in slab-geometry preserves gas energy (internal plus kinetic) on average, despite strong local up- and down-drafts. By contrast, a more standard numerical scheme is demonstrated to result in substantial gains of energy within a short time on purely numerical grounds. The test is further used to point out the role of dimensionality, viscosity, and Rayleigh number for compressible convection. Applications to a young sun in 2D and 3D, covering a part of the inner radiative zone, as well as the outer convective zone, demonstrate that the scheme meets its initial design goal. Comparison with results obtained for a physically identical setup with a time-implicit code show qualitative agreement.

Discontinuous Galerkin method for solving incompressible two-phase shallow granular flow model

This article deals with the numerical study of two-phase shallow flow model describing the mixture of fluid and solid granular particles. The model under investigation consists of coupled mass and momentum equations for solid granular material and fluid particles through non-conservative momentum exchange terms. The non-conservativity of model equations poses major challenges for any numerical scheme, such as well balancing, positivity preservation, accurate approximation of non-conservative terms, and achievement of steady-state conditions. Thus, in order to approximate the present model an accurate, well-balanced, robust, and efficient numerical scheme is required. For this purpose, in this article, Runge–Kutta discontinuous Galerkin method is applied successfully for the first time to solve the model equations. Several test problems are also carried out to check the performance and accuracy of our proposed numerical method. To compare the results, the same model is solved by staggered central Nessyahu–Tadmor scheme. A good comparison is found between two schemes, but the results obtained by Runge–Kutta discontinuous Galerkin scheme are found superior over the central Nessyahu–Tadmor scheme.

High order well-balanced finite difference WENO schemes for shallow water flows along channels with irregular geometry

In this paper, we present high order finite difference weighted essentially non-oscillatory (WENO) schemes for the shallow water flows along open channels with irregular geometry and over a non-flat bottom topography. The proposed schemes maintain the well-balanced property for the still water steady state solutions, namely preserve steady state at the discrete level, when there is a exact balance between the flux gradient and the source term. Compared with the traditional shallow water equations with constant cross section, the construction of the well-balanced schemes is not a trivial work due to the effect induced by the irregular geometry of the channels. To preserve the well-balanced property, we first reformulate the source term, then propose to construct the numerical fluxes by means of a flux modification technique, and finally discrete the source term with the help of a novel source term approximation. Benchmark numerical examples are applied to validate the good performances of the resulting schemes: well-balanced property, high order accuracy, and high resolution for the discontinuous solutions.

Multi-exponential wave solutions to two extended Jimbo–Miwa equations and the resonance behavior

Resonance phenomena occur widely in fluid, physics and other fields, e.g., they are related with the optical elements, the well-balanced scheme for shallow water with discontinuous topography, and some phenomena in chaotic dynamics and fluid dynamics. Application of the principle of linear superposition to the Hirota bilinear equation gives rise to a sufficient and necessary condition for the existence of multi-exponential wave solutions. We study the resonance behavior based on the construction of multi-exponential wave solution to two extended (3 + 1)- dimensional Jimbo–Miwa equations. The resonance characteristics are analyzed and simulated for some resonant two-wave and three-wave solutions.

A High-performance and Portable All-Mach Regime Flow Solver Code with Well-balanced Gravity. Application to Compressible Convection

Abstract Convection is an important physical process in astrophysics well-studied using numerical simulations under the Boussinesq and/or anelastic approximations. However, these approaches reach their limits when compressible effects are important in the high-Mach flow regime, e.g., in stellar atmospheres or in the presence of accretion shocks. In order to tackle these issues, we propose a new high-performance and portable code called “ARK” with a numerical solver well suited for the stratified compressible Navier–Stokes equations. We take a finite-volume approach with machine precision conservation of mass, transverse momentum, and total energy. Based on previous works in applied mathematics, we propose the use of a low-Mach correction to achieve a good precision in both low and high-Mach regimes. The gravity source term is discretized using a well-balanced scheme in order to reach machine precision hydrostatic balance. This new solver is implemented using the Kokkos library in order to achieve high-performance computing and portability across different architectures (e.g., multi-core, many-core, and GP-GPU). We show that the low-Mach correction allows to reach the low-Mach regime with a much better accuracy than a standard Godunov-type approach. The combined well-balanced property and the low-Mach correction allowed us to trigger Rayleigh–Bénard convective modes close to the critical Rayleigh number. Furthermore, we present 3D turbulent Rayleigh–Bénard convection with low diffusion using the low-Mach correction leading to a higher kinetic energy power spectrum. These results are very promising for future studies of high Mach and highly stratified convective problems in astrophysics.

Pressure-induced locking in mixed methods for time-dependent (Navier–)Stokes equations

We consider inf-sup stable mixed methods for the time-dependent incompressible Stokes and Navier--Stokes equations, extending earlier work on the steady (Navier-)Stokes Problem. A locking phenomenon is identified for classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart elements by a novel, elegant and simple numerical analysis and corresponding numerical experiments, whenever the momentum balance is dominated by forces of a gradient type. More precisely, a reduction of the $L^2$ convergence order for high order methods, and even a complete stall of the $L^2$ convergence order for lowest-order methods on preasymptotic meshes is predicted by the analysis and practically observed. On the other hand, it is also shown that (structure-preserving) pressure-robust mixed methods do not suffer from this locking phenomenon, even if they are of lowest-order. A connection to well-balanced schemes for (vectorial) hyperbolic conservation laws like the shallow water or the compressible Euler equations is made.

Some examples of kinetic schemes whose diffusion limit is Il’in’s exponential-fitting

This paper is concerned with diffusive approximations of peculiar numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov-Fokker-Planck plasma modeling. The well-balanced method applied to such kinetic equations leads to time-marching schemes involving a " scattering S-matrix " , itself derived from a normal modes decomposition of the stationary solution. One common feature these models share is the type of diffusive approximation: their macroscopic densities solve drift-diffusion systems, for which a distinguished numerical scheme is Il'in/Scharfetter-Gummel's " exponential fitting " discretization. We prove that the well-balanced schemes relax, within a parabolic rescaling, towards the Il'in exponential-fitting discretization by means of an appropriate decomposition of the S-matrix. This is the so-called asymptotic preserving (or uniformly accurate) property.

Capturing Near-Equilibrium Solutions: A Comparison Between High-Order Discontinuous Galerkin Methods and Well-Balanced Schemes

Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.