HLL Riemann Solver Research Articles

A high-resolution shallow water model using unstructured quadrilateral grids

A two-dimensional cell-centred finite volume model for quadrilateral grids is presented. The solution methodology of the depth-averaged shallow water equations is based upon a Godunov-type upwind finite volume formulation, whereby the inviscid fluxes of the system of equations are obtained using the HLL Riemann solver. A simple yet precise analytical expression is presented to compute hydrostatic flux through an interface of a quadrilateral cell in order to achieve exact balance between flux gradient and bed slope source terms under still water condition. A multidimensional gradient reconstruction procedure and a continuously differentiable multidimensional slope limiter based on a wide computational stencil are proposed to maintain second-order spatial accuracy. The proposed second-order scheme is shown to be more accurate even when distorted grids are used and is therefore more suitable for practical applications. The presented model is verified and validated by solving a wide variety of test cases having analytical solutions and laboratory measurements.

A one-dimensional shock-capturing model for long wave run-up on sloping beaches

This paper presents a one-dimensional finite-volume model to investigate long wave run-up under non-breaking and breaking conditions. A conservative form of the non-linear shallow water equations is solved using a high-resolution Godunov-type scheme coupled with the HLL Riemann solver. The surface gradient method leads to a well-balanced formulation between the flux and the source terms. The explicit time discretisation is first-order accurate, but a piecewise linear reconstruction of numerical data at cell interfaces helps achieve second-order accuracy in space. The computed surface elevation, flow velocity and run-up show very good agreement with available analytical solutions and experimental observations. The model accurately describes propagation of non-breaking and breaking waves on a sloping beach.

A two-dimensional HLLC Riemann solver for conservation laws: Application to Euler and magnetohydrodynamic flows

In this paper we present a genuinely two-dimensional HLLC Riemann solver. On logically rectangular meshes, it accepts four input states that come together at an edge and outputs the multi-dimensionally upwinded fluxes in both directions. This work builds on, and improves, our prior work on two-dimensional HLL Riemann solvers. The HLL Riemann solver presented here achieves its stabilization by introducing a constant state in the region of strong interaction, where four one-dimensional Riemann problems interact vigorously with one another. A robust version of the HLL Riemann solver is presented here along with a strategy for introducing sub-structure in the strongly-interacting state. Introducing sub-structure turns the two-dimensional HLL Riemann solver into a two-dimensional HLLC Riemann solver. The sub-structure that we introduce represents a contact discontinuity which can be oriented in any direction relative to the mesh. [Display omitted] The Riemann solver presented here is general and can work with any system of conservation laws. We also present a second order accurate Godunov scheme that works in three dimensions and is entirely based on the present multidimensional HLLC Riemann solver technology. The methods presented are cost-competitive with traditional higher order Godunov schemes.The two-dimensional HLLC Riemann solver is shown to work robustly for Euler and Magnetohydrodynamic (MHD) flows. Several stringent test problems are presented to show that the inclusion of genuinely multidimensional effects into higher order Godunov schemes indeed produces some very compelling advantages. For two dimensional problems, we were routinely able to run simulations with CFL numbers of ∼0.7, with some two-dimensional simulations capable of reaching higher CFL numbers. For three dimensional problems, CFL numbers as high as ∼0.6 were found to be stable. We show that on resolution-starved meshes, the scheme presented here outperforms unsplit second order Godunov schemes that are based on conventional one-dimensional Riemann solver technology. Strong discontinuities are shown to propagate very isotropically using the methods presented here. The present Riemann solver provides an elegant resolution to the problem of obtaining multi-dimensionally upwinded electric fields in MHD without resorting to a doubling of the dissipation in each dimension.

Discontinuous Galerkin Method for 1D Shallow Water Flows in Natural Rivers

A numerical model is proposed for the solution of one-dimensional shallow water flow equations for natural rivers. This model is based on the total variation diminishing Runge-Kutta discontinuous Galerkin finite element method. In natural rivers, the cross-section shape and bed slope can be quite irregular, which requires a compatible discretization scheme for the bed slope term and net pressure force term. Therefore, in this model, the hydrostatic pressure force term and the wall pressure force term are combined and a new discretization for the resulting term is introduced. This formulation is shown to prevent unphysical flow due to improper treatment of bottom slope term. The mass and momentum flux term are calculated by HLL Riemann solver. A scheme is presented to model flow over dry bed. To evaluate the numerical scheme, tests are conducted for idealized dambreak problems in parabolic (with wet and dry beds), and rectangular channels, hydraulic jump in a rectangular channel, dambreak in the Teton River (Idaho, USA) and the Toce River (Northern Alps, Italy), and flooding event in the East Fork River (Wyoming, USA). The comparison of the computational results with analytical and laboratory results of dam break flows shows that the model is capable of handling flow over dry areas. The simulation results for hydraulic jump show the discharge conservation property and shock prediction capability of the model. The dambreak and flood simulations in natural channels show that the model is capable of handling flows in highly varying bed topography and channel geometry.

The early influence of peter lax on computational hydrodynamics and an application of lax-friedrichs and lax-wendroff on triangular grids in lagrangian coordinates

We give a brief discussion of some of the contributions of Peter Lax to Computational Fluid Dynamics. These include the Lax-Friedrichs and Lax-Wendroff numerical schemes. We also mention his collaboration in the 1983 HLL Riemann solver. We develop two-dimensional Lax-Friedrichs and Lax-Wendroff schemes for the Lagrangian form of the Euler equations on triangular grids. We apply a composite scheme that uses a Lax-Friedrichs time step as a dissipative filter after several Lax-Wendroff time steps. Numerical results for Noh's infinite strength shock problem, the Sedov blast wave problem, and the Saltzman piston problem are presented.

A multi-dimensional HLL-Riemann solver for Euler equations of gas dynamics

This article presents a numerical model that enables to solve on unstructured triangular meshes and with a high-order of accuracy, a multi-dimensional Riemann problem that appears when solving hyperbolic problems. For this purpose, we use a MUSCL-like procedure in a “cell-vertex” finite-volume framework. In the first part of this procedure, we devise a four-state bi-dimensional HLL solver (HLL-2D). This solver is based upon the Riemann problem generated at the centre of gravity of a triangular cell, from surrounding cell-averages. A new three-wave model makes it possible to solve this problem, approximately. A first-order version of the bi-dimensional Riemann solver is then generated for discretizing the full compressible Euler equations. In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third-order accuracy. The resulting over determined system is solved by using a least-square methodology. To enforce monotonicity conditions into the polynomial interpolation, we develop a simplified central WENO (CWENO) procedure. Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model.

Quasi-steady Wave Propagation 알고리듬을 이용한 2차원 수치모형의 하상경사항 처리

Two dimensional numerical model of high-order accuracy is developed to analyze complex flow including transition flow, discontinuous flow, and wave propagation to dry bed emerging at natural river flow. The bed slope term of two dimensional shallow water equation consisting of integral conservation law is treated efficiently by applying quasi-steady wave propagation scheme. In order to apply Finite Volume Method using Fractional Step Method, MUSCL scheme is applied based on HLL Riemann solver, which is second-order accurate in time and space. The TVD method is applied to prevent numerical oscillations in the second-order accurate scheme. The developed model is verified by comparing observed data of two dimenstional levee breach experiment and dam breach experiment containing structure at lower section of channel. Also effect of the source term is verified by applying to dam breach experiment considering the adverse slope channel.

A high-order multi-dimensional HLL-Riemann solver for non-linear Euler equations

This article presents a numerical model that enables to solve on unstructured triangular meshes and with a high-order of accuracy, a multi-dimensional Riemann problem that appears when solving hyperbolic problems. For this purpose, we use a MUSCL-like procedure in a “cell-vertex” finite-volume framework. In the first part of this procedure, we devise a four-state bi-dimensional HLL solver (HLL-2D). This solver is based upon the Riemann problem generated at the centre of gravity of a triangular cell, from surrounding cell-averages. A new three-wave model makes it possible to solve this problem, approximately. A first-order version of the bi-dimensional Riemann solver is then generated for discretizing the full compressible Euler equations. In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third-order accuracy. The resulting over determined system is solved by using a least-square methodology. To enforce monotonicity conditions into the polynomial interpolation, we develop a simplified central WENO (CWENO) procedure. Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model.

A multidimensional HLL-Riemann solver for non-linear hyperbolic systems

This article presents a numerical model that enables to solve on unstructured triangular meshes and with a high order of accuracy, Riemann problems that appear when solving hyperbolic systems. For this purpose, we use a MUSCL-like procedure in a ‘cell-vertex’ finite-volume framework. In the first part of this procedure, we devise a four-state bi-dimensional HLL solver (HLL-2D). This solver is based upon the Riemann problem generated at the barycenter of a triangular cell, from the surrounding cell-averages. A new three-wave model makes it possible to solve this problem, approximately. A first-order version of the bi-dimensional Riemann solver is then generated for discretizing the full compressible Euler equations. In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third-order accuracy. The resulting over determined system is solved by using a least-square methodology. To enforce monotonicity conditions into the polynomial interpolation, we use and adapt the monotonicity-preserving limiter, initially devised by Barth (AIAA Paper 90-0013, 1990). Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model. Copyright © 2010 John Wiley & Sons, Ltd.

Towards a compact high-order method for non-linear hyperbolic systems, II. The Hermite-HLLC scheme

In a finite-volume framework, we develop an approximate HLL Riemann solver specific to weakly hyperbolic systems. Those systems are obtained by considering not only the variable but also its first spatial derivative, as unknowns. To this aim, we rely upon the theory of “ δ-shock waves”, newly developed in the scalar case. First, we demonstrate that the extended version of the HLLE scheme to weakly hyperbolic systems is compatible with the existence of Dirac measures in the solution. Then, we develop a specific Hermite Least-Square (HLSM) interpolation that enables to generate a high-order and compact scheme, without creating spurious oscillations in the reconstruction of the variable or its first derivative. Extensive numerical experiments make it possible to validate the method and to check convergence to entropy solutions. Relying upon those results, we construct a new HLL Riemann solver, suited for the extended one-dimensional Euler equations. For this purpose, we introduce the contribution of a contact discontinuity inside the definition of the solver. By using a formal analogy with the scalar study, we demonstrate that this solver tolerates the existence of “ δ-shock waves” in the solution. Numerical experiments that follow help to validate some of the assumptions made to generate this scheme.

Improved implementation of the HLL approximate Riemann solver for one-dimensional open channel flows

Several new techniques are proposed to overcome the deficiencies in the conventional formulation of the approximate Riemann solvers for onedimensional open channel flows, which include numerical imbalance and inaccuracy in the solution of discharge. The former arises in the case of irregular geometry and the latter in the presence of a hydraulic jump. These new techniques include: (1) adopting the form of the saint venant equations that include both gravity and pressure in one source term; (2) using water surface level as one of the primitive variables, in stead of cross-sectional area; (3) defining discharge at interface and evaluating it according to the flux obtained by the HLL Riemann solver; and (4) estimating water surface gradient based on piecewise linearly reconstructed variables in the second-order scheme. The performance of the resulting schemes is evaluated by means of theoretical analysis and various test examples, including ideal dam-break flows with dry and wet bed, hydraulic jump, steady flow over bump with hydraulic jump, wave interactions, tidal flow in an open channel, and wave propagation. it is demonstrated that the schemes have excellent numerical balance and mass conservation property and are capable of satisfactorily reproducing various open channel flows.

A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics

A new multi-state Harten–Lax–van Leer (HLL) approximate Riemann solver for the ideal magnetohydrodynamic (MHD) equations is developed based on the assumption that the normal velocity is constant over the Riemann fan. This assumption is same as that used in the HLLC (“C” denotes Contact) approximate Riemann solver for the Euler equations. From the assumption, it is naturally derived that the Riemann fan should consist of four intermediate states for B x ≠ 0, whereas the number of the intermediate states is reduced to two when B x = 0. Since the intermediate states satisfied with all jump conditions in this approximate Riemann system are analytically obtained, the multi-state HLL Riemann solver can be constructed straightforwardly. It is shown that this solver can exactly resolve isolated discontinuities formed in the MHD system, and hence named as HLLD Riemann solver. (Here, “D” stands for Discontinuities.) It is also analytically proved that the HLLD Riemann solver is positively conservative like the HLLC Riemann solver. Indeed, the HLLD Riemann solver corresponds to the HLLC Riemann solver when the magnetic field vanishes. Numerical tests demonstrate that the HLLD Riemann solver is more robust and efficient than the linearized Riemann solver, and its resolution is equally good. It indicates that the HLLD solver must be useful in practical applications for the ideal MHD equations.

A fully conservative model for compressible two‐fluid flow

AbstractA five‐equation model for compressible two‐fluid flow is proposed, that is based on physical flow equations only. The model is conservative and pressure‐oscillation free. Equations for continuous flow and jump conditions for discontinuities are given, as well as a discretization of the equations and an adaptation of the HLL Riemann solver to two‐fluid flow. Numerical tests in 1D and 2D show the accuracy of the method. Copyright © 2005 John Wiley & Sons, Ltd.

Flux and source term discretization in two-dimensional shallow water models with porosity on unstructured grids

Two-dimensional shallow water models with porosity appear as an interesting path for the large-scale modelling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two-dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. This paper presents a discretization of the modified fluxes using a modified HLL Riemann solver on unstructured grids. The source term arising from the gradients in the topography and in the porosity is treated in an upwind fashion so as to enhance the stability of the solution. The Riemann solver is tested against new analytical solutions with variable porosity. A new formulation is proposed for the macroscopic head loss in urban areas. An application example is presented, where the large scale model with porosity is compared to a refined flow model containing obstacles that represent a schematic urban area. The quality of the results illustrates the potential usefulness of porosity-based shallow water models for large scale floodplain simulations. Copyright © 2005 John Wiley & Sons, Ltd.

Finite-volume WENO schemes for three-dimensional conservation laws

In this paper we firstly carry out an extension of the finite-volume WENO schemes to three space dimensions and higher orders of accuracy. Secondly, we propose to use more accurate fluxes as the building block. These are the HLLC and MUSTA fluxes [Multi-stage predictor–corrector fluxes for hyperbolic equations, 2003; Restoration of the contact surface in the HLL Riemann solver, Report CoA 9204, June 1992; J. Shock Waves 4 (1994) 25]. The numerical results suggest that the new WENO-HLLC and WENO-MUSTA schemes compare satisfactorily with the state-of-the-art finite-volume scheme of Shi et al. [J. Comput. Phys. 175 (2002) 108].

Finite volume method for simulating extreme flood events in natural channels

The need for mitigating damages produced by extreme hydrologie events has stimulated the European Community to fund several projects. The Concerted Action on Dam-break Modelling workgroup (CADAM) performed a considerable work for the development of new codes and for the adequate verification of their performance. In the context of the CADAM project, a new 2D computer code is developed, tested and applied, as described in the present paper. The algorithm is obtained through the spatial discretisation of the shallow water equations by a finite volume method, based on the Godunov approach. The HLL Riemann solver is used. A second order accuracy in space and time is achieved, respectively by MUSCL and predictor-corrector techniques. The high resolution requirement is ensured by satisfaction of TVD property. Particular attention is posed to the numerical treatment of source terms. Accuracy, stability and the reliability of the code are tested on a selected set of study cases. A grid refinement analysis is performed. Numerical results are compared with experimental data, obtained by the physical modelling of a submersion wave on a portion of the Toce river valley, Italy, performed by ENEL-HYDRO and considered as representative of a real life flood occurrence.

A practical, general‐purpose, two‐state HLL Riemann solver for hyperbolic conservation laws

AbstractIn this paper, we construct a general‐purpose Riemann solver for hyperbolic conservation laws that does not involve extensive characteristic analysis of governing equations but can nevertheless sharply resolve discontinuities. To achieve this goal, we revisit the class of two‐state HLL schemes and show that inexpensive, accurate solvers within this class can be designed using only geometric interpretations of the Rankine–Hugoniot conditions. We argue that the small cost and nearly uniform algorithmic complexity of resulting methods make them attractive for quick computations of practical, especially very complex, problems for which more accurate solvers are either not available or their development cost is not justified. Copyright © 2002 John Wiley & Sons, Ltd.

On Riemann solvers for compressible liquids

A number of Riemann solvers are proposed for the solution of the Riemann problem in a compressible liquid. Both the Tait and Tammann equations of state are used to describe the liquid. Along with exact Riemann solvers, a detailed description of a primitive variable Riemann solver, a two-shock Riemann solver, a two-rarefaction Riemann solver and an extension to the HLL Riemann solver, namely the HLLC Riemann solver, are presented. It is shown how these Riemann solvers may be implemented into Godunov-type numerical methods. The appropriateness of each of the Riemann solvers for a number of flow situations is demonstrated by applying Godunov's method to some revealing shock tube test problems. © 1998 John Wiley & Sons, Ltd.

An approximate Riemann solver for a two-phase flow model with numerically given slip relation

In this paper an approximate Riemann solver is constructed for solving one class of two-phase models. These models describe the gas–liquid flow in a long tube where the flow behaviour perpendicular to the tube axis is averaged, so that the model is essentially one-dimensional in the direction of the axis. The model consists of equations for the conservation of mass for each of the phases and the conservation of momentum of the mixture. In addition, an equation is supplied which relates the velocities of the two phases at any point, the slip relation, which may have a different shape for each of the flow regimes of interest and change in time. Generally, a slip relation will not be known entirely in algebraic form, but given partly in numerical form. Under certain restrictions the resulting system of conservation laws is hyperbolic and allows discontinuous solutions. The general idea is that the numerical algorithm for solving this model must be able to handle any valid slip relation. As the slip relation affects the Jacobian of the flux function to a large extent, this means that flux vector or flux difference splittings cannot be based on algebraic manipulation of the Jacobian. We propose to use a first-order upwind scheme of Roe-type, where the construction of the approximate Riemann solver is fully numerical. This basic scheme, however, has the same limitation as the original Roe method, namely, that for systems the positivity of the solution is not guaranteed. The modification of the basic scheme to ensure positivity of the solution, is based on the HLL Riemann solver.

Restoration of the contact surface in the HLL-Riemann solver

The missing contact surface in the approximate Riemann solver of Harten, Lax, and van Leer is restored. This is achieved following the same principles as in the original solver. We also present new ways of obtaining wave-speed estimates. The resulting solver is as accurate and robust as the exact Riemann solver, but it is simpler and computationally more efficient than the latter, particulaly for non-ideal gases. The improved Riemann solver is implemented in the second-order WAF method and tested for one-dimensional problems with exact solutions and for a two-dimensional problem with experimental results.