Discretization Of Source Terms Research Articles

A numerical model for three-dimensional shallow water flows with sharp gradients over mobile topography

Abstract This study aims to develop a three-dimensional (3-D) numerical model for shallow water flows over mobile topography, which is capable of simulating morphological evolution under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations (SWEs) using a finite volume method on prismatic cells in σ-coordinates. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. Using a relaxation approach, a hyperbolic system is built for hydrodynamic system, which allows for using a Godunov-type central-upwind method to capture the shocks and approximate the numerical fluxes. Consequently, the 3D-SWEs-Exner model proposed in the present study can stably and accurately solve the dam-break flows over mobile beds. A spatially and temporally second-order “prismatic” central-upwind method is used to approximate the numerical fluxes through cell interfaces. The Exner equation is solved using an upwind method. Using spatially linear reconstruction and explicit two-stage Runge–Kutta time discretization, second order accuracy is achieved in space and time. The proposed model can preserve the well-balanced property due to the special discretization of bed-slope source terms. The proposed model is validated by several tests with experimental measurements, and is compared with the simulated results using reported two-dimensional (2-D) models.

An efficient and stable hydrodynamic model with novel source term discretization schemes for overland flow and flood simulations

AbstractNumerical models solving the full 2‐D shallow water equations (SWEs) have been increasingly used to simulate overland flows and better understand the transient flow dynamics of flash floods in a catchment. However, there still exist key challenges that have not yet been resolved for the development of fully dynamic overland flow models, related to (1) the difficulty of maintaining numerical stability and accuracy in the limit of disappearing water depth and (2) inaccurate estimation of velocities and discharges on slopes as a result of strong nonlinearity of friction terms. This paper aims to tackle these key research challenges and present a new numerical scheme for accurately and efficiently modeling large‐scale transient overland flows over complex terrains. The proposed scheme features a novel surface reconstruction method (SRM) to correctly compute slope source terms and maintain numerical stability at small water depth, and a new implicit discretization method to handle the highly nonlinear friction terms. The resulting shallow water overland flow model is first validated against analytical and experimental test cases and then applied to simulate a hypothetic rainfall event in the 42 km2 Haltwhistle Burn, UK.

Overcoming numerical shockwave anomalies using energy balanced numerical schemes. Application to the Shallow Water Equations with discontinuous topography

When designing a numerical scheme for the resolution of conservation laws, the selection of a particular source term discretization (STD) may seem irrelevant whenever it ensures convergence with mesh refinement, but it has a decisive impact on the solution. In the framework of the Shallow Water Equations (SWE), well-balanced STD based on quiescent equilibrium are unable to converge to physically based solutions, which can be constructed considering energy arguments. Energy based discretizations can be designed assuming dissipation or conservation, but in any case, the STD procedure required should not be merely based on ad hoc approximations. The STD proposed in this work is derived from the Generalized Hugoniot Locus obtained from the Generalized Rankine Hugoniot conditions and the Integral Curve across the contact wave associated to the bed step. In any case, the STD must allow energy-dissipative solutions: steady and unsteady hydraulic jumps, for which some numerical anomalies have been documented in the literature. These anomalies are the incorrect positioning of steady jumps and the presence of a spurious spike of discharge inside the cell containing the jump. The former issue can be addressed by proposing a modification of the energy-conservative STD that ensures a correct dissipation rate across the hydraulic jump, whereas the latter is of greater complexity and cannot be fixed by simply choosing a suitable STD, as there are more variables involved. The problem concerning the spike of discharge is a well-known problem in the scientific community, also known as slowly-moving shock anomaly, it is produced by a nonlinearity of the Hugoniot locus connecting the states at both sides of the jump. However, it seems that this issue is more a feature than a problem when considering steady solutions of the SWE containing hydraulic jumps. The presence of the spurious spike in the discharge has been taken for granted and has become a feature of the solution. Even though it does not disturb the rest of the solution in steady cases, when considering transient cases it produces a very undesirable shedding of spurious oscillations downstream that should be circumvented. Based on spike-reducing techniques (originally designed for homogeneous Euler equations) that propose the construction of interpolated fluxes in the untrustworthy regions, we design a novel Roe-type scheme for the SWE with discontinuous topography that reduces the presence of the aforementioned spurious spike. The resulting spike-reducing method in combination with the proposed STD ensures an accurate positioning of steady jumps, provides convergence with mesh refinement, which was not possible for previous methods that cannot avoid the spike.

Well-Balanced Nodal Discontinuous Galerkin Method for Euler Equations with Gravity

We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss---Lobatto---Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh composed of quadrilateral cells and for any gravitational potential. It is applied on several examples to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.

Source Term Discretization Effects on the Steady-State Accuracy of Finite Volume Schemes

Source terms often arise in Computational Fluid Dynamics to describe a variety of physical phenomena, including turbulence, chemical reactions, and certain methods used for code verification, such as the method of manufactured solutions. While much has already been published on the treatment of source terms, here we follow an uncommon approach, designing compatible source term discretizations in terms of spatial truncation error for finite volume schemes in multiple dimensions. In this work we examine the effect of source term discretization on three finite volume flux schemes applied to steady flows: constant reconstruction, linear reconstruction, and a recently published third-order flux correction method. Three source term discretization schemes are considered, referred to as point, Galerkin, and corrected. The corrected source discretization is a new method that extends our previous work on flux correction to equations with source terms. In all cases, computational grid refinement studies confirm the compatibility (or lack thereof) of various flux-source combinations predicted through detailed truncation error analysis.

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.

Moving-water equilibria preserving central-upwind schemes for the shallow water equations

We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions and a draining time-step technique ensures positivity of the water depth. Several numerical experiments are performed to verify the well-balanced and positivity preserving properties as well as the ability of the proposed scheme to accurately capture small perturbations of moving-water steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart.

Very efficient high-order hyperbolic schemes for time-dependent advection–diffusion problems: Third-, fourth-, and sixth-order

In this paper, we construct very efficient high-order schemes for general time-dependent advection–diffusion problems, based on the first-order hyperbolic system method. Extending the previous work on the second-order time-dependent hyperbolic advection–diffusion scheme (Mazaheri and Nishikawa, NASA/TM-2014-218175, 2014), we construct third-, fourth-, and sixth-order accurate schemes by modifying the source term discretization. In this paper, two techniques for the source term discretization are proposed; (1) reformulation of the source terms with their divergence forms and (2) correction to the trapezoidal rule for the source term discretization. We construct spatially third- and fourth-order schemes from the former technique. These schemes require computations of the gradients and second-derivatives of the source terms. From the latter technique, we construct spatially third-, fourth-, and sixth-order schemes by using the gradients and second-derivatives for the source terms, except the fourth-order scheme, which does not require the second derivatives of the source term and thus is even less computationally expensive than the third-order schemes. We then construct high-order time-accurate schemes by incorporating a high-order backward difference formula as a source term. These schemes are very efficient in that high-order accuracy is achieved for both the solution and the gradient only by the improved source term discretization. A very rapid Newton-type convergence is achieved by a compact second-order Jacobian formulation. The numerical results are presented for both steady and time-dependent linear and nonlinear advection–diffusion problems, demonstrating these powerful features.

A Well-Balanced 2-D Model for Dam-Break Flow with Wetting and Drying

This paper presents a two-dimension (2-D) numerical model for simulating dam-break flow involving wet-dry fronts over irregular topography. The Central- upwind scheme is chosen to calculate interface fluxes for each cell edge. A second order spatial linear reconstruction with multidimensional limiter and second order TVD Runge-Kutta scheme are chosen to acquire high order accuracy in space and time. Non-negative water reconstruction of variables at cell interfaces and compatible discretization of slope source term lead to stable and well-balanced scheme for hydraulics over irregular topography. The friction term is discretized with a semi-implicit scheme for numerical stability when very small water depth exists. An accurate and effective technique is presented for tracking wetting-drying interfaces during the process of wave front propagation on dry bed. The capacity and accuracy of current model are verified by several benchmark tests as well as a real dam-break case, and good performances are achieved in tests.

Unsteady Non-Newtonian Solver on Unstructured Grid for the Simulation of Blood Flow

Blood is in fact a suspension of different cells with yield stress, shear thinning, and viscoelastic properties, which can be represented by different non-Newtonian models. Taking Casson fluid as an example, an unsteady solver on unstructured grid for non-Newtonian fluid is developed to simulate transient blood flow in complex flow region. In this paper, a steady solver for Newtonian fluid is firstly developed with the discretization of convective flux, diffusion flux, and source term on unstructured grid. For the non-Newtonian characteristics of blood, the Casson fluid is approximated by the Papanastasiou's model and treated as Newtonian fluid with variable viscosity. Then considering the transient property of blood flow, an unsteady non-Newtonian solver based on unstructured grid is developed by introducing the temporal term by first-order upwind difference scheme. Using the proposed solver, the blood flows in carotid bifurcation of hypertensive patients and healthy people are simulated. The result shows that the possibility of the genesis and development of atherosclerosis is increased, because of the increase in incoming flow shock and backflow areas of the hypertensive patients, whose WSS was 20~87.1% lower in outer vascular wall near the bifurcation than that of the normal persons and 3.7~5.5% lower in inner vascular wall downstream the bifurcation.

On the well-balanced numerical discretization of shallow water equations on unstructured meshes

We consider in this work a finite volume numerical approximation of weak solutions of the shallow water equations with varying topography, on unstructured meshes. Relying on an alternative formulation of the shallow water equations that involves the free surface as a conservative variable, instead of the water height, we introduce a simple discretization of the bed slope source term, together with some suitable conservative variables reconstructions. The resulting scheme is automatically consistent and well-balanced, for any given consistent numerical flux for the homogeneous system. We obtain a very simple formulation, which do not need to be modified when second order accuracy MUSCL reconstructions are adopted. Additionally, the positivity of the water height is preserved under a relevant stability condition, as soon as the numerical flux for the associated homogeneous system does. Numerical assessments, involving dry areas and complex geometry are performed.

Wave Riemann description of friction terms in unsteady shallow flows: Application to water and mud/debris floods

In this work, the source term discretization in hyperbolic conservation laws with source terms is considered using an approximate augmented Riemann solver. The technique is applied to the shallow water equations with bed slope and friction terms with the focus on the friction discretization. The augmented Roe approximate Riemann solver provides a family of weak solutions for the shallow water equations, that are the basis of the upwind treatment of the source term. This has proved successful to explain and to avoid the appearance of instabilities and negative values of the thickness of the water layer in cases of variable bottom topography. Here, this strategy is extended to capture the peculiarities that may arise when defining more ambitious scenarios, that may include relevant stresses in cases of mud/debris flow. The conclusions of this analysis lead to the definition of an accurate and robust first order finite volume scheme, able to handle correctly transient problems considering frictional stresses in both clean water and debris flow, including in this last case a correct modelling of stopping conditions.

A 2D shallow flow model for practical dam-break simulations

Dam-break flows usually occur in domains with complex geometric and topographic features and involve abrupt flow patterns. A dam-break model must therefore be able to effectively handle different flow types including transcritical flows or hydraulic jumps, deal with complex domain topography, capture repeating wet–dry interface and represent high roughness values in the floodplain. Herein, all of these objectives are achieved by extending a recent one-dimensional finite volume Godunov-type model into two dimensions for solving the shallow-water equations. While doing so, a much simplified condition to maintain well-balanced solutions around a wet–dry front is proposed and a two-dimensional friction source term discretization is derived under a suitable stability condition in relation to practical simulations. The two-dimensional model is successfully validated against three analytical benchmark tests and then assessed for predicting realistic dam-break flood events.

An efficient scheme on wet/dry transitions for shallow water equations with friction

The present work concerns the derivation of a suitable discretization to approximate the friction source terms in the shallow-water model. Such additional source terms are known to be very stiff as soon as the water height is vanishing. The proposed numerical procedure comes from a relevant correction of a Godunov-type scheme that approximates the solutions of hyperbolic systems of conservation laws. The adopted correction gives a discretization of the source term which preserves the robustness and does not change the CFL condition. The scheme is shown to be particularly efficient for wet/dry transition simulations. In addition, this numerical procedure can be used together with any robust and well-balanced discretization of the topography source term. Second order extension is also investigated. Extensive numerical validations illustrate the interest of this new approach.

A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain

An unstructured Godunov-type finite volume model is developed for the numerical simulation of geometrically challenging two-dimensional shallow water flows with wetting and drying over convoluted topography. In the framework of sloping bottom model, a modified formulation of shallow water equations is used to preserve mass conservation during flooding and recession. The key ingredient of the model is the use of this combination of the sloping bottom model and the modified shallow water equations to provide a robust technique for wet/dry fronts tracking and, together with centered discretization of the bed slope source term, to exactly preserve the static flow on irregular topographies. The variable reconstruction technique ensures nonnegative reconstructed water depth and reasonable reconstructed velocity, and the friction terms are solved by semi-implicit scheme that does not invert the direction of velocity components. The robustness and accuracy of the proposed model are assessed by comparing numerical and reference results of extensive test cases. Moreover, the results of a dam-break flooding over real topography are presented to show the capability of the model on field-scale application.

Locally Limited and Fully Conserved RKDG2 Shallow Water Solutions with Wetting and Drying

This work extends a well-balanced second-order Runge-Kutta discontinuous Galerkin (RKDG2) scheme to provide conservative simulations for shallow flows involving wetting and drying over irregular topographies with friction effects. For this purpose, a wetting and drying technique designed originally for a finite volume (FV) scheme is improved and implemented, which includes the discretization of friction source terms via a splitting implicit integration approach. Another focus of this work is to design a fully conserved RKDG2 scheme to provide conservative solutions for both mass and momentum through a local slope limiting process. Several steady and transient benchmark tests with/without friction effects are simulated to validate the new solver and demonstrate the effects of different slope limiting processes, i.e. globally and locally slope limiting processes.

Asymptotic preserving HLL schemes

This work concerns the derivation of HLL schemes to approximate the solutions of systems of conservation laws supplemented by source terms. Such a system contains many models such as the Euler equations with high friction or the M1 model for radiative transfer. The main difficulty arising from these models comes from a particular asymptotic behavior. Indeed, in the limit of some suitable parameter, the system tends to a diffusion equation. This article is devoted to derive HLL methods able to approximate the associated transport regime but also to restore the suitable asymptotic diffusive regime. To access such an issue, a free parameter is introduced into the source term. This free parameter will be a useful correction to satisfy the expected diffusion equation at the discrete level. The derivation of the HLL scheme for hyperbolic systems with source terms comes from a modification of the HLL scheme for the associated homogeneous hyperbolic system. The resulting numerical procedure is robust as the source term discretization preserves the physical admissible states. The scheme is applied to several models of physical interest. The numerical asymptotic behavior is analyzed and an asymptotic preserving property is systematically exhibited. The scheme is illustrated with numerical experiments. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1396–1422, 2011

Network Implementation of the Two-Component Pressure Approach for Transient Flow in Storm Sewers

The two-component pressure approach (TPA) is an alternative to the Preissman slot method (PSM) for modeling highly transient sewer flow, including transitions between free-surface and pressurized conditions. TPA and PSM resolve intralink wave action by discretizing sewers with numerous elements and solving one-dimensional flow equations in contrast to link-node models, such as the popular storm water management model, which resolve only interlink wave action. Here, improvements of TPA are reported to support storm sewer network modeling. These include a source term discretization to preserve stationarity, a wetting and drying scheme, and a local time-stepping scheme to coordinate solution updates across many links and enable coupling to a two-dimensional overland flow model. A unique variant of the Harten, Lax and van Leer (HLL) Riemann solver is also introduced, and a boundary solver is developed to accommodate the wide range of possible flow regimes and transitions. The boundary solver is explicit to facilitate the extension of TPA to large networks and coupling with an overland flow model. Promising results are obtained in a varied set of test problems involving simple sewer networks.

Finite-volume solution of one-dimensional shallow-water sensitivity equations

Solving the Shallow-Water-Sensitivity Equations for discontinuous flows involves the discretization of a Dirac source term accounting for discontinuities. Failing to account for this source term usually results in solution instability, with empirical sensitivity solutions exhibiting artificial peaks in the neighbourhood of shocks. An extension of the Harten-Lax-van Leer approximate Riemann solver is presented that allows the one-dimensional shallow-water-sensitivity equations to be discretized more accurately than in previously published versions. A discretization of boundary conditions and source terms is also provided. The proposed discretization allows for discontinuities in both the hydraulic and sensitivity boundary conditions. Numerical experiments indicate the superiority of the proposed approach for sensitivity analysis over the classical, empirical approach if the flow solution is discontinuous. A numerical convergence analysis demonstrates that the numerical and analytical solutions converge.

A primitive-variable Riemann method for solution of the shallow water equations with wetting and drying

A Riemann flux that uses primitive variables rather than conserved variables is developed for the shallow water equations with nonuniform bathymetry. This primitive-variable flux is both conservative and well behaved at zero depth. The unstructured finite-volume discretization used is suitable for highly nonuniform grids that provide resolution of complex geometries and localized flow structures. A source-term discretization is derived for nonuniform bottom that balances the discrete flux integral both for still water and in dry regions. This primitive-variable formulation is uniformly valid in wet and dry regions with embedded wetting and drying fronts. A fully nonlinear implicit scheme and both nonlinear and time-linearized explicit schemes are developed for the time integration. The implicit scheme is solved by a parallel Newton-iterative algorithm with numerically computed flux Jacobians. A concise treatment of characteristic-variable boundary conditions with source terms is also given. Computed results obtained for the one-dimensional dam break on wet and dry beds and for normal-mode oscillations in a circular parabolic basin are in very close agreement with the analytical solutions. Other results for a forced breaking wave with friction interacting with a sloped bottom demonstrate a complex wave motion with wetting, drying and multiple interacting wave fronts. Finally, a highly nonuniform, coastline-conforming unstructured grid is used to demonstrate an unsteady simulation that models an artificial coastal flooding due to a forced wave entering the Gulf of Mexico.