Well-balanced Numerical Scheme Research Articles

A Steady-State-Preserving Numerical Scheme for One-Dimensional Blood Flow Model

In this work, an entropy-stable and well-balanced numerical scheme for a one-dimensional blood flow model is presented. Such a scheme was obtained from an explicit entropy-conservative flux along with a second-order discretisation of the source term by using centred finite differences. We prove that the scheme is entropy-stable and preserves steady-state solutions. In addition, some numerical examples are included to test the performance of the proposed scheme.

High-order well-balanced numerical schemes for one-dimensional shallow-water systems with Coriolis terms

The goal of this work is to develop high-order well-balanced schemes for the one-dimensional shallow-water equations with Coriolis terms. The main contribution is the development of general numerical methods that allow the achievement of arbitrary high-order for the shallow-water equations with Coriolis terms while preserving all the stationary solutions.

A scalable well-balanced numerical scheme for the simulation of fast landslides with efficient time stepping

We consider a single-phase depth-averaged model for the numerical simulation of fast-moving landslides with the goal of constructing a well-balanced, yet scalable and efficient, second-order time-stepping algorithm. We apply a Strang splitting approach to distinguish between parabolic and hyperbolic problems. For the parabolic contribution, we adopt a second-order Implicit-Explicit Runge-Kutta-Chebyshev scheme, while we use a two-stage Taylor discretization combined with a path-conservative strategy, to deal with the purely hyperbolic contribution. The proposed strategy allows to decouple hyperbolic from parabolic-reaction stiff contributions resulting in an overall well-balanced scheme subject just to stability restrictions of the hyperbolic term. The spatial discretization we adopt is based on a standard finite element method, associated with a hierarchically refined Cartesian grid. After providing numerical evidence of the well-balancing property, we demonstrate the capability of the proposed approach to select time steps larger than the ones adopted by a classical Taylor-Galerkin scheme. Finally, we provide some meaningful scaling results on ideal and realistic scenarios.

Numerical schemes for mixture theory models with filling constraint: application to biofilm ecosystems

The mixture theory framework is a powerful way to describe multi-phasic systems at an intermediary scale between microscopic and macroscopic scales. In particular, mixture theory reveals a powerful approach to represent microbial biofilms where a consortium of cells is embedded in a polymeric structure. To simulate a model of microalgal biofilm, we propose an upgraded numerical scheme, consolidating the one proposed by Berthelin et al. (2016) to enforce the volume-filling constraint in mixture models including mass exchanges. The strategy consists in deducing the discrete version of the incompressibility constraint from the discretized mass balance equations. Numerical simulations show that this method constrains the total volume filling constraint, even at the discrete level. Moreover, we add viscous terms in the biofilm model to properly represent biofilms interactions with its fluidic environment. It turns out that a well-balanced numerical scheme becomes of outmost importance to capture the biofilm dynamic when including the viscosity. This modelling upgrade also involves recalibrating model parameters. In particular, the elastic tensors to recover realistic front features. With the new parameters, the numerical set-up becomes more demanding to reach convergence.

Numerical Simulation of Bed Load and Suspended Load Sediment Transport Using Well-Balanced Numerical Schemes

Numerical Simulation of Bed Load and Suspended Load Sediment Transport Using Well-Balanced Numerical Schemes

Well-balanced numerical resolution of the two-layer shallow water equations under rigid-lid with wet–dry fronts

In this paper, we present a well-balanced numerical scheme for the frictional two-layer shallow water equations (2LSWE) over variable bottom topography and under a rigid-lid (RL) to simulate internal waves propagating over wet and dry areas. Following the idea of Liang and Borthwick (2009) for the derivation of a pre-balanced formulation for one-layer flows, we derive a new formulation of the 2LSWE-RL using the interface elevation above datum and horizontal momentum as conservative variables. This new formulation mathematically balances the flux gradient and source terms so that the lake-at-rest steady state is automatically preserved in wet-bed applications. A proper discretization of the slope source term is adopted to produce well-balanced solutions in dry-bed areas where the lower layer depth vanishes. Using the Harten, Lax, and van Leer (HLL) approximate Riemann solver, we adopt a finite volume MUSCL-RK2 method to construct a second-order well-balanced numerical scheme ensuring the non-negativity of the layers depths with a special treatment of source terms. Finally, various numerical experiments are tested in both one and two-layer cases validating the properties of the numerical scheme.

Well-balanced treatment of gravity in astrophysical fluid dynamics simulations at low Mach numbers

Context.Accurate simulations of flows in stellar interiors are crucial to improving our understanding of stellar structure and evolution. Because the typically slow flows are merely tiny perturbations on top of a close balance between gravity and the pressure gradient, such simulations place heavy demands on numerical hydrodynamics schemes.Aims.We demonstrate how discretization errors on grids of reasonable size can lead to spurious flows orders of magnitude faster than the physical flow. Well-balanced numerical schemes can deal with this problem.Methods.Three such schemes were applied in the implicit, finite-volume SEVEN-LEAGUEHYDROcode in combination with a low-Mach-number numerical flux function. We compare how the schemes perform in four numerical experiments addressing some of the challenges imposed by typical problems in stellar hydrodynamics.Results.We find that theα-βand deviation well-balancing methods can accurately maintain hydrostatic solutions provided that gravitational potential energy is included in the total energy balance. They accurately conserve minuscule entropy fluctuations advected in an isentropic stratification, which enables the methods to reproduce the expected scaling of convective flow speed with the heating rate. The deviation method also substantially increases accuracy of maintaining stationary orbital motions in a Keplerian disk on long timescales. The Cargo–LeRoux method fares substantially worse in our tests, although its simplicity may still offer some merits in certain situations.Conclusions.Overall, we find the well-balanced treatment of gravity in combination with low Mach number flux functions essential to reproducing correct physical solutions to challenging stellar slow-flow problems on affordable collocated grids.

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.

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.

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.

A well-balanced numerical scheme for a model of two-phase flows with treatment of nonconservative terms

We construct a numerical scheme for a model of two-phase flows. The keystone is the treatment of nonconservative terms using stationary contact discontinuities. The nonconservative terms are absorbed into left-hand and right-hand states of the local stationary contact discontinuity at each grid node. The numerical scheme is built by taking into accounts the states on both sides of local contact discontinuities as supplemented facts of a standard numerical approximation for the conservative part of the system. The scheme is shown to possess interesting properties: it preserves the positivity of the volume fractions in both phases and the positivity of the gas density, it is well-balanced and satisfying the numerical minimum entropy principle in the gas phase.

Well-Balanced Numerical Schemes for Shallow Water Equations with Horizontal Temperature Gradient

A class of well-balanced numerical schemes for the one-dimensional shallow water equations with temperature gradient is constructed. The construction of the schemes is based on two steps: the first step to absorb the nonconservative term and the second step to deal with the evolution of the system. Algorithms for computing contact waves which absorb the nonconservative term are developed. Furthermore, to improve the accuracy, the underlying numerical fluxes can be formed as convex combinations of a pair of numerical fluxes of a low and stable scheme and a higher and fast scheme. The schemes are well balanced and can retain the positivity of the water height and the water temperature. Numerical tests show that the schemes are stable and have a good accuracy.

Estimation of the area of sediment deposition by debris flow using a physical-based modeling approach

Debris flow triggered by shallow landslides in hillslope catchments is a main geological phenomenon driving landscape changes, and represents an important natural hazard. Numerous studies have assessed sediment transport and deposition by debris flows in hillslope catchments. Thus, the objective of this study is a development of two-dimensional debris flow model to estimate sediment transport and deposition in hillslope catchments. To simulate debris flow, we implemented a vertically integrated shallow-water governing equation based on the Voellmy rheological model and a simple entrainment model. In addition, we applied a quadtree grid structure to support adaptive mesh refinement, where the mesh for the simulation was automatically generated as the debris flow proceeded. Finally, a well-balanced numerical scheme for wet–dry transition treatment was implemented and implicit discretization of the general source terms, including the rheological term, was included for numerical stability. The developed model was verified based on a debris flow triggered by the 2011 Mt. Umyeon landslides in the Republic of Korea. Sediment transport was successfully generated and the sediment deposition area generally matched the field survey data well. Overall, the simulated sediment volume was in good agreement with the survey results, with an error below 1%.

Well-balanced schemes for the shallow water equations with Coriolis forces

In the present paper we study shallow water equations with bottom topography and Coriolis forces. The latter yield non-local potential operators that need to be taken into account in order to derive a well-balanced numerical scheme. In order to construct a higher order approximation a crucial step is a well-balanced reconstruction which has to be combined with a well-balanced update in time. We implement our newly developed second-order reconstruction in the context of well-balanced central-upwind and finite-volume evolution Galerkin schemes. Theoretical proofs and numerical experiments clearly demonstrate that the resulting finite-volume methods preserve exactly the so-called jets in the rotational frame. For general two-dimensional geostrophic equilibria the well-balanced methods, while not preserving the equilibria exactly, yield better resolution than their non-well-balanced counterparts.

A well-balanced numerical scheme for debris flow run-out prediction in Xiaojia Gully considering different hydrological designs

To simulate debris flow run-out, the governing equations for free-surface shallow flow are corrected by setting the basal flow resistance coefficients with the quadratic rheological friction model. A well-balanced numerical scheme is developed for its run-out simulation over irregular topography. A linear reconstruction is adopted for improving the spatial accuracy of the numerical scheme. Considering the complex friction terms of governing equations of debris flow, they are estimated with a full implicit scheme for ensuring stability of the numerical scheme. The validity check of run-out simulation is implemented based on general knowledge of fluid, and a well-studied case occurred in the Xiezi Gully in Yingxiu Town, Sichuan Province of China. For practical purpose, the present numerical scheme is used for run-out prediction of debris flow in Xiaojia Gully of Panzhihua City, Sichuan Province of China. Our work aims to present a well-balanced numerical scheme for debris flow run-out simulation prediction, which can be applied quite conveniently to solve other kinds of debris flow models and helpful to promote the development in debris flow numerical calculation.

A multi well-balanced scheme for the shallow water MHD system with topography

The shallow water magnetohydrodynamic system involves several families of physically relevant steady states. In this paper we design a well-balanced numerical scheme for the one-dimensional shallow water magnetohydrodynamic system with topography, that resolves exactly a large range of steady states. Two variants are proposed with slightly different families of preserved steady states. They are obtained by a generalized hydrostatic reconstruction algorithm involving the magnetic field and with a cutoff parameter to remove singularities. The solver is positive in height and semi-discrete entropy satisfying, which ensures the robustness of the method.

Low-Shapiro hydrostatic reconstruction technique for blood flow simulation in large arteries with varying geometrical and mechanical properties

The purpose of this work is to construct a simple, efficient and accurate well-balanced numerical scheme for one-dimensional (1D) blood flow in large arteries with varying geometrical and mechanical properties. As the steady states at rest are not relevant for blood flow, we construct two well-balanced hydrostatic reconstruction techniques designed to preserve low-Shapiro number steady states that may occur in large network simulations. The Shapiro number Sh=u/c is the equivalent of the Froude number for shallow water equations and the Mach number for compressible Euler equations. The first is the low-Shapiro hydrostatic reconstruction (HR-LS), which is a simple and efficient method, inspired from the hydrostatic reconstruction technique (HR). The second is the subsonic hydrostatic reconstruction (HR-S), adapted here to blood flow and designed to exactly preserve all subcritical steady states. We systematically compare HR, HR-LS and HR-S in a series of single artery and arterial network numerical tests designed to evaluate their well-balanced and wave-capturing properties. The results indicate that HR is not adapted to compute blood flow in large arteries as it is unable to capture wave reflections and transmissions when large variations of the arteries' geometrical and mechanical properties are considered. On the contrary, HR-S is exactly well-balanced and is the most accurate hydrostatic reconstruction technique. However, HR-LS is able to compute low-Shapiro number steady states as well as wave reflections and transmissions with satisfying accuracy and is simpler and computationally less expensive than HR-S. We therefore recommend using HR-LS for 1D blood flow simulations in large arterial network simulations.

Water ow on vegetated hill. 1D shallow water equation type model

Abstract A mathematical model for the water ow on a hill covered by variable distributed vegetation is proposed in this article. The model takes into account the variation of the geometrical properties of the terrain surface, but it assumes that the surface exhibits large curvature radius. After describing some theoretical properties for this model, we introduce a simplified model and a well-balanced numerical approximation scheme for it. Some mathematical properties with physical relevance are discussed and finally, some numerical results are presented.

A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

This work is devoted to the derivation of a fully well-balanced numerical scheme for the well-known shallow-water model. During the last two decades, several well-balanced strategies have been introduced with special attention to the exact capture of the stationary states associated with the so-called lake at rest. By fully well-balanced, we mean here that the proposed Godunov-type method is also able to preserve stationary states with non zero velocity. The numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality.

Overland Flow Modeling with the Shallow Water Equations Using a Well-Balanced Numerical Scheme: Better Predictions or Just More Complexity

In the last decades, several physically based hydrological modeling approaches of various complexities have been developed that solve shallow water equations or their approximations using various numerical methods. Users of the model may not necessarily know the different hypotheses underlying these development and simplifications, and it might therefore be difficult to judge if a code is well adapted to their objectives and test case configurations. This paper aims to compare the predictive abilities of different models and evaluate potential gain by using an advanced numerical scheme for modeling runoff. Four different codes are presented, each based on either shallow water or kinematic wave equations, and using either the finite volume or finite difference method. These four numerical codes are compared with different test cases, allowing to emphasize their main strengths and weaknesses. Results show that, for relatively simple configurations, kinematic wave equations solved with the finite volume method represent an interesting option. Nevertheless, as it appears to be limited in case of discontinuous topography or strong spatial heterogeneities, for these cases they advise the use of shallow water equations solved with the finite volume method.