HLL Schemes Research Articles

A Well-Balanced HLL Scheme for Hyperbolic Conservation Systems With Source Terms

This paper presents new finite volume schemes for hyperbolic conservation system with source terms. The classical finite volume schemes could not accurately simulate the dynamic problems caused by the balance between flux term and source terms. In order to deal with this problem, an approximate Riemann solver with source terms was designed in accordance with the classical HLL approximation Riemann solver. The well-balanced HLL schemes (WB-HLL) was obtained by modifying the flux calculation schemes for one-dimensional Euler equations and ideal MHD with gravity source terms, and a proof for well-balanced property of the new schemes has been presented.Two numerical examples of one-dimensional Euler equation and ideal MHD demonstrated that the WB-HLL scheme has higher accuracy and faster convergence than the classical HLL.

Implementation of exactly well‐balanced numerical schemes in the event of shockwaves: A 1D approach for the shallow water equations

AbstractThis article presents a numerical technique that ensures the exact solution of stationary shockwaves, known as the hydraulic jump, for the one‐dimensional shallow water equations with the geometric source term. The mathematical description of the hydraulic jump is given by the Rankine–Hugoniot condition, at the interface of a control volume, where one variable in the system, the discharge, is supposed to be constant while the remaining variables are discontinuous. However, at the discrete level, the solution could be, for instance, a 3‐state jump, consisting of extra (intermediate) states resulting in the so‐called numerical anomaly. The anomaly is due to the nonlinearity of the Hugoniot locus and the fact that, at these points, a naive formulation is unlikely to avoid the inaccurate representation. The concept used here is simple, limit all discontinuous variables, that is, the surface flow data inside the cell that contains a shock with the data of the nearby cells where the shock is traveling to, allowing the solution to be linear. When used with a number of well‐known numerical solvers, namely Lax–Friedrichs, HLL, and Roe schemes, this approach is confirmed to accurately capture the moving and stationary shockwaves in the trans‐critical flow over nonflat beds.

A comparison of approximate non-linear Riemann solvers for Relativistic MHD

ABSTRACT We compare a particular selection of approximate solutions of the Riemann problem in the context of ideal relativistic magnetohydrodynamics. In particular, we focus on Riemann solvers not requiring a full eigenvector structure. Such solvers recover the solution of the Riemann problem by solving a simplified or reduced set of jump conditions, whose level of complexity depends on the intermediate modes that are included. Five different approaches – namely the HLL, HLLC, HLLD, HLLEM, and GFORCE schemes – are compared in terms of accuracy and robustness against one – and multidimensional standard numerical benchmarks. Our results demonstrate that – for weak or moderate magnetizations – the HLLD Riemann solver yields the most accurate results, followed by HLLC solver(s). The GFORCE approach provides a valid alternative to the HLL solver being less dissipative and equally robust for strongly magnetized environments. Finally, our tests show that the HLLEM Riemann solver is not cost-effective in improving the accuracy of the solution and reducing the numerical dissipation.

An Efficient Hybrid Method for Solving Euler Equations

In this paper, a hybrid method suitable for solving the Euler equations using high order methods has been proposed. The method was implemented and validated with a seventh order WENO scheme in OpenFOAM®. The hybrid method combines a simple MUSCL-type flux approach and a characteristic flux approach. In the MUSCL-type flux approach, the inviscid fluxes are computed using approximate Riemann solvers HLL and HLLC schemes based on the WENO-reconstructed state variables. Hence, this is dubbed as the VF (variable-based flux) approach. In critical regions where VF may produce spurious oscillations, a novel, low-dissipation HLL-based CF (characteristic flux) approach is applied. Critical regions were identified using a modified Bhagatwala–Lele shock sensor. The VF/CF hybrid method has been shown to produce high-resolution, essentially non-oscillatory results for a number of 1D and 2D problems at a fraction of the cost of a pure CF approach. Moreover, a 2D advection problem was designed to investigate the choice of state variables and flux schemes. The results have shed more light on the relation between Kelvin–Helmholtz roll-ups and numerical instabilities along slip lines.

Formulation of exactly balanced solvers for blood flow in elastic vessels and their application to collapsed states

In this work, numerical solvers based on extensions of the Roe and HLL schemes are adapted to deal with test cases involving extreme collapsing conditions in elastic vessels. To achieve this goal, the system is transformed to provide a conservation–law form, allowing to define Rankine–Hugoniot conditions. The approximate solvers allow to describe the inner states of the solution. Therefore, source term fixes can be used to prevent unphysical values of vessel area and, at the same time, the eigenvalues of the system control stability. Numerical solvers of different order are tested using a wide variety of Riemann problems, including extreme vessel collapse and blockage. In all cases, the robustness of the approximate solvers presented here is checked using first and third order methods in time and space, using the WENO reconstruction scheme in combination with the TVDRK3 method.

Effect of Reconstruction Scheme on the Plasma Fluid Modeling With Original and Reformulated Ion-Related Modeling Equations Using HLL Flux Scheme

In this paper, we would like to investigate the effect of spatial reconstruction schemes on the plasma fluid model with original and reformulated ion-related modeling equations by using the Harten–Lax–van Leer (HLL) flux scheme. The results based on the 1-D fluid model equations for electropositive plasma show that the appearance of unphysical oscillations near-sheath edges strongly depends on the reconstruction schemes (e.g., $\kappa =1/2,1 /3$ , and first-order scheme) when the original ion equations utilize the incorrect ion sound speed. The unphysical oscillations disappear no matter what the spatial reconstruction scheme is applied if the reformulated ion equations with correct ion sound speed are used instead. For electronegative plasma, the kappa reconstruction scheme causes numerical instability when using either the incorrect or correct numerical ion sound speed, but the min-mod reconstruction scheme does not. The numerical results show that for either electropositive or electronegative discharges, both the second-order HLL scheme with min-mod reconstruction and the first-order HLL schemes can produce stable solution, while the accuracy of the former is slightly better than the later. Finally, some examples with an extension to the 2-D-axisymmetric electropositive and electronegative plasmas are demonstrated.

A New Flux Splitting Scheme Based on Toro-Vazquez and HLL Schemes for the Euler Equations

This paper presents a new flux splitting scheme for the Euler equations. The proposed scheme termed TV-HLL is obtained by following the Toro-Vazquez splitting (Toro and Vázquez-Cendón, 2012) and using the HLL algorithm with modified wave speeds for the pressure system. Here, the intercell velocity for the advection system is taken as the arithmetic mean. The resulting scheme is more accurate when compared to the Toro-Vazquez schemes and also enjoys the property of recognition of contact discontinuities and shear waves. Accuracy, efficiency, and other essential features of the proposed scheme are evaluated by analyzing shock propagation behaviours for both the steady and unsteady compressible flows. The accuracy of the scheme is shown in 1D test cases designed by Toro.

An efficient shock capturing algorithm to the extended Boussinesq wave equations

A hybrid finite-volume and finite-difference method is proposed for numerically solving the two-dimensional (2D) extended Boussinesq equations. The governing equations are written in such a way that the convective flux is approximated using finite volume (FV) method while the remaining terms are discretized using finite difference (FD) method. Multi-stage (MUSTA) scheme, instead of commonly used HLL or Roe schemes, is adopted to evaluate the convective flux as it has the simplicity of centred scheme and accuracy of upwind scheme. The third order Runge–Kutta method is used for time marching. Wave breaking and wet–dry interface are also treated in the model. In addition to model validation, the emphasis is given to compare the merits and limitations of using MUSTA scheme and HLL scheme in the model. The analytical and experimental data available in the literature have been used for the assessment. Numerical tests demonstrate that the developed model has the advantages of stability preserving, shock-capturing and numerical efficiency when applied in the complex nearshore region. Compared with that using HLL scheme, the proposed model has comparable numerical accuracy, but requires slightly less computation time and is much simpler to code.

1D and 2D Numerical Modeling for Solving Dam‐Break Flow Problems Using Finite Volume Method

The purpose of this study is to model the flow movement in an idealized dam‐break configuration. One‐dimensional and two‐dimensional motion of a shallow flow over a rigid inclined bed is considered. The resulting shallow water equations are solved by finite volumes using the Roe and HLL schemes. At first, the one‐dimensional model is considered in the development process. With conservative finite volume method, splitting is applied to manage the combination of hyperbolic term and source term of the shallow water equation and then to promote 1D to 2D. The simulations are validated by the comparison with flume experiments. Unsteady dam‐break flow movement is found to be reasonably well captured by the model. The proposed concept could be further developed to the numerical calculation of non‐Newtonian fluid or multilayers fluid flow.

Development of an improved spatial reconstruction technique for the HLL method and its applications

The integral form of the conventional HLL fluxes are presented by taking integrals around the control volume centred on each cell interface. These integrals are demonstrated to reduce to the conventional HLL flux through simplification by assuming spatially constant conserved properties. The integral flux expressions are then modified by permitting the analytical inclusion of spatially linearly varying conserved quantities. The newly obtained fluxes (which are named HLLG fluxes for clarification, where G stands for gradient inclusion) demonstrate that conventional reconstructions at cell interfaces are invalid and can produce unstable results when applied to conventional HLL schemes. The HLLG method is then applied to the solution of the Euler Equations and Shallow Water Equations for various common benchmark problems and finally applied to a 1D fluid modeling for an argon RF discharge at low pressure. Results show that the correct inclusion of flow gradients is shown to demonstrate superior transient behavior when compared to the existing HLL solver and conventional spatial reconstruction without significantly increasing computational expense.

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

Robust HLLC Riemann solver with weighted average flux scheme for strong shock

Many researchers have reported failures of the approximate Riemann solvers in the presence of strong shock. This is believed to be due to perturbation transfer in the transverse direction of shock waves. We propose a simple and clear method to prevent such problems for the Harten–Lax–van Leer contact (HLLC) scheme. By defining a sensing function in the transverse direction of strong shock, the HLLC flux is switched to the Harten–Lax–van Leer (HLL) flux in that direction locally, and the magnitude of the additional dissipation is automatically determined using the HLL scheme. We combine the HLLC and HLL schemes in a single framework using a switching function. High-order accuracy is achieved using a weighted average flux (WAF) scheme, and a method for v-shear treatment is presented. The modified HLLC scheme is named HLLC–HLL. It is tested against a steady normal shock instability problem and Quirk’s test problems, and spurious solutions in the strong shock regions are successfully controlled.

Finite‐volume multi‐stage schemes for shallow‐water flow simulations

AbstractA finite‐volume multi‐stage (FMUSTA) scheme is proposed for simulating the free‐surface shallow‐water flows with the hydraulic shocks. On the basis of the multi‐stage (MUSTA) method, the original Riemann problem is transformed to an independent MUSTA mesh. The local Lax–Friedrichs scheme is then adopted for solving the solution of the Riemann problem at the cell interface on the MUSTA mesh. The resulting first‐order monotonic FMUSTA scheme, which does not require the use of the eigenstructure and the special treatment of entropy fixes, has the generality as well as simplicity. In order to achieve the high‐resolution property, the monotonic upstream schemes for conservation laws (MUSCL) method are used. For modeling shallow‐water flows with source terms, the surface gradient method (SGM) is adopted. The proposed schemes are verified using the simulations of six shallow‐water problems, including the 1D idealized dam breaking, the steady transcritical flow over a hump, the 2D oblique hydraulic jump, the circular dam breaking and two dam‐break experiments. The simulated results by the proposed schemes are in satisfactory agreement with the exact solutions and experimental data. It is demonstrated that the proposed FMUSTA schemes have superior overall numerical accuracy among the schemes tested such as the commonly adopted Roe and HLL schemes. Copyright © 2007 John Wiley & Sons, Ltd.

An upstream flux-splitting finite-volume scheme for 2D shallow water equations

An upstream flux-splitting finite-volume (UFF) scheme is proposed for the solutions of the 2D shallow water equations. In the framework of the finite-volume method, the artificially upstream flux vector splitting method is employed to establish the numerical flux function for the local Riemann problem. Based on this algorithm, an UFF scheme without Jacobian matrix operation is developed. The proposed scheme satisfying entropy condition is extended to be second-order-accurate using the MUSCL approach. The proposed UFF scheme and its second-order extension are verified through the simulations of four shallow water problems, including the 1D idealized dam breaking, the oblique hydraulic jump, the circular dam breaking, and the dam-break experiment with 45° bend channel. Meanwhile, the numerical performance of the UFF scheme is compared with those of three well-known upwind schemes, namely the Osher, Roe, and HLL schemes. It is demonstrated that the proposed scheme performs remarkably well for shallow water flows. The simulated results also show that the UFF scheme has superior overall numerical performances among the schemes tested. Copyright © 2005 John Wiley & Sons, Ltd.

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.