Harten-Lax-van Leer-Contact Research Articles

Hydrodynamic modelling in support of flash flood warning

A new approach is proposed to determine the distributed threshold rainfall for flash flooding to support practical flood warning. It is built upon a two-dimensional full hydrodynamic model incorporating rainfall, infiltration loss and boundary resistance. Numerical solution of the governing equations of the model is achieved under an operator-splitting framework, the second-order total-variation-diminishing version of the weighted-average-flux method along with the Harten-Lax–van Leer contact wave (HLLC) approximate Riemann solver for the homogeneous equations, and a Runge–Kutta scheme for the ordinary differential equations of the source terms. The applicability of the new approach is demonstrated as applied to real flash flooding-prone areas in Hunan Province, China. The threshold rainfall for flash flooding, resulting from rainfall of durations from 1 h to 6 h, is found to be locally less than 100 mm or even 50 mm. This means a high risk of flash flooding in the local areas, and therefore effective measures are necessary to avoid loss of property and life. The present work should find general application in support of practical flash flood warning in catchments with the necessary topography and soil and geological data available.

Gridless Method for Unsteady Flows Involving Moving Discrete Points and its Applications

A gridless method for dealing with unsteady flow involving moving discrete points is investigated. The moving principle of discrete points for gridless method is proposed. The spring analogy is modified to control the movements of points caused by the motion and torsion of boundaries, so the adaptation and precision of the analogy are enhanced. The algorithm for solving Arbitrary Lagrangian-Eulerian (ALE) equations based on gridless method is accomplished. On the base of clouds in the computational region, the spatial derivatives are approximated by local least-square curve fits, HLLC (Harten, Lax, van Leer, Contact) scheme is extended into the gridless method to calculate the numerical flux and a method of flux limiter is employed in order to improve the accuracy, a multistage Runge-Kutta algorithm is used to advance the equation in time. The oscillatory pitching movement of representative airfoils is simulated, the computational results agree well with the experimental data, and the plunging movement is simulated too. It indicates that the method is successful and good at catching shock waves.

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.

Realization of contact resolving approximate Riemann solvers for strong shock and expansion flows

AbstractThe Harten–Lax–van Leer contact (HLLC) and Roe schemes are good approximate Riemann solvers that have the ability to resolve shock, contact, and rarefaction waves. However, they can produce spurious solutions, called shock instabilities, in the vicinity of strong shock. In strong expansion flows, the Roe scheme can admit nonphysical solutions such as expansion shock, and it sometimes fails. We carefully examined both schemes and propose simple methods to prevent such problems. High‐order accuracy is achieved using the weighted average flux (WAF) and MUSCL‐Hancock schemes. Using the WAF scheme, the HLLC and Roe schemes can be expressed in similar form. The HLLC and Roe schemes are tested against Quirk's test problems, and shock instability appears in both schemes. To remedy shock instability, we propose a control method of flux difference across the contact and shear waves. To catch shock waves, an appropriate pressure sensing function is defined. Using the proposed method, shock instabilities are successfully controlled. For the Roe scheme, a modified Harten–Hyman entropy fix method using Harten–Lax–van Leer‐type switching is suggested. A suitable criterion for switching is established, and the modified Roe scheme works successfully with the suggested method. Copyright © 2009 John Wiley & Sons, Ltd.

A Godunov-Type Scheme for Atmospheric Flows on Unstructured Grids: Euler and Navier-Stokes Equations

In recent years there has been a growing interest in using Godunov-type methods for atmospheric flow problems. Godunov's unique approach to numerical modeling of fluid flow is characterized by introducing physical reasoning in the development of the numerical scheme (van Leer, 1999). The construction of the scheme itself is based upon the physical phenomenon described by the equation sets. These finite volume discretizations are conservative and have the ability to resolve regions of steep gradients accurately, thus avoiding dispersion errors in the solution. Positivity of scalars (an important factor when considering the transport of microphysical quantities) is also guaranteed by applying the total variation diminishing condition appropriately. This paper describes the implementation of a Godunov-type finite volume scheme based on unstructured adaptive grids for simulating flows on the meso-, micro- and urban-scales. The Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver used to calculate the Godunov fluxes is described in detail. The higher-order spatial accuracy is achieved via gradient reconstruction techniques after van Leer and the total variation diminishing condition is enforced with the aid of slope-limiters. A multi-stage explicit Runge-Kutta time marching scheme is used for maintaining higher-order accuracy in time. The scheme is conservative and exhibits minimal numerical dispersion and diffusion. The subgrid scale diffusion in the model is parameterized via the Smagorinsky-Lilly turbulence closure. The scheme uses a non-staggered mesh arrangement of variables (all quantities are cell-centered) and requires no explicit filtering for stability. A comparison with exact solutions shows that the scheme can resolve the different types of wave structures admitted by the atmospheric flow equation set. A qualitative evaluation for an idealized test case of convection in a neutral atmosphere is also presented. The scheme was able to simulate the onset of Kelvin-Helmholtz type instability and shows promise in simulating atmospheric flows characterized by sharp gradients without using explicit filtering for numerical stability.

Propagation and run-up of nearshore tsunamis with HLLC approximate Riemann solver

A numerical model describing the propagation and run-up process of nearshore tsunamis in the vicinity of shorelines is developed based on an approximate Riemann solver. The governing equations of the model are the nonlinear shallow-water equations. The governing equations are discretized explicitly by using a finite volume method. The nonlinear terms in the momentum equations are solved with the Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver. The developed model is first applied to prediction of water motions in a parabolic basin, and propagation and subsequent run-up process of nearshore tsunamis around a circular island. Computed results are then compared with available analytical solutions and laboratory measurements. Very reasonable agreements are observed.

An HLLC Riemann solver for relativistic flows – II. Magnetohydrodynamics

An approximate Riemann solver for the equations of relativistic magnetohydrodynamics (RMHD) is derived. The Harten–Lax–van Leer contact wave (HLLC) solver, originally developed by Toro, Spruce and Spears, generalizes the algorithm described in a previous paper to the case where magnetic fields are present. The solution to the Riemann problem is approximated by two constant states bounded by two fast shocks and separated by a tangential wave. The scheme is Jacobian-free, in the sense that it avoids the expensive characteristic decomposition of the RMHD equations and it improves over the HLL scheme by restoring the missing contact wave. Multidimensional integration proceeds via the single step, corner transport upwind (CTU) method of Colella, combined with the constrained transport (CT) algorithm to preserve divergence-free magnetic fields. The resulting numerical scheme is simple to implement, efficient and suitable for a general equation of state. The robustness of the new algorithm is validated against one- and two-dimensional numerical test problems.

A discontinuous Galerkin method for two-temperature plasmas

We develop a formulation for the single-fluid/two-temperature equations for simulating two-species, compressible, non-equilibrium plasma flows. The divergence-free condition of the magnetic field is enforced via the characteristic decomposition of an extended nine-wave system. The source terms are modified appropriately to improve energy and momentum conservation accuracy. A spectral/ hp element algorithm is employed in the discretization combined with a discontinuous Galerkin formulation for the advective and diffusive contributions. The formulation is conservative, and monotonicity is enforced by appropriately lowering the spectral order around discontinuities. A new MHD flux introduced here is the MHD-HLLC (Harten–Lax–van Leer Contact wave) flux that preserves monotonicity and resolves contact discontinuities better. Exponential convergence is demonstrated for a magneto-hydrostatic problem. Two tests are presented using the new MHD-HLLC flux. Also, the differences between the single-temperature and the two-temperature models are presented for two-dimensional plasma flows around bluff bodies.

An HLLC-Type Approximate Riemann Solver for Ideal Magnetohydrodynamics

This paper presents a solver based on the HLLC (Harten--Lax--van Leer contact wave) approximate nonlinear Riemann solver for gas dynamics for the ideal magnetohydrodynamics (MHD) equations written in conservation form. It is shown how this solver also can be considered a modification of Linde's "adequate" solver. This approximation method is intended to resolve slow, Alfvén, and contact waves better than the original HLL (Harten--Lax--van Leer) solver. Compared to exact nonlinear solvers and Roe's solver, this new solver is computationally inexpensive. In addition, the method will exactly resolve isolated contacts and fast shocks. The method also preserves positive density and pressure with two caveats: first, the numerical signal velocities (the eigenvalues of the Roe average matrix) do not underestimate the physical signal velocities, and second, in a very few cases it may be required to change the wavespeeds of the Riemann fan for the underlying HLL method to guarantee positive pressures. These conditions are less restrictive on the definitions of the wavespeeds than the conditions needed to make the HLLC method positively conservative for gas dynamics. While the method is intended for a three-dimensional MHD problem, the simulation results concentrate on one-dimensional test cases.