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.     

High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

In this paper, we present a class of high‐order accurate cell‐centered arbitrary Lagrangian–Eulerian (ALE) one‐step ADER weighted essentially non‐oscillatory (WENO) finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two‐dimensional unstructured triangular meshes. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space–time Galerkin predictor allows the schemes to be high order accurate also in time by using an element‐local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained either (i) as an arithmetic average among the states surrounding the node, as suggested by Cheng and Shu, or (ii) as a solution of multiple one‐dimensional half‐Riemann problems around a vertex, as suggested by Maire, or (iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional Harten–Lax–van Leer Riemann solver recently proposed by Balsara et al. Once the vertex velocity and thus the new node location have been determined by the node solver, the local mesh motion is then constructed by straight edges connecting the vertex positions at the old time level tⁿ with the new ones at the next time level t^{n + 1}. If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unnecessary, as the ALE fluxes already properly take into account the rezoned geometry. In this sense, our scheme falls into the category of direct ALE methods. Furthermore, the geometric conservation law is satisfied by the scheme by construction. We apply the high‐order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic magnetohydrodynamic equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration. Copyright © 2014 John Wiley & Sons, Ltd.     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.     

A sharp‐interface immersed boundary framework for simulations of high‐speed inviscid compressible flows

A new finite‐volume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of high‐speed inviscid compressible flows. The IB method adopts a sharp‐interface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solid‐fluid interface. We show that proposed reconstruction leads to second‐order spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder lift‐off and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shape‐changing geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes.     

An efficient discontinuous Galerkin method for aeroacoustic propagation

An efficient discontinuous Galerkin formulation is applied to the solution of the linearized Euler equations and the acoustic perturbation equations for the simulation of aeroacoustic propagation in two‐dimensional and axisymmetric problems, with triangular and quadrilateral elements. To improve computational efficiency, a new strategy of variable interpolation order is proposed in addition to a quadrature‐free approach and parallel implementation. Moreover, an accurate wall boundary condition is formulated on the basis of the solution of the Riemann problem for a reflective wall. Time discretization is based on a low dissipation formulation of a fourth‐order, low storage Runge–Kutta scheme. Along the far‐field boundaries a perfectly matched layer boundary condition is used. For the far‐field computations, the integral formulation of Ffowcs Williams and Hawkings is coupled with the near‐field solver. The efficiency and accuracy of the proposed variable order formulation is assessed for realistic geometries, namely sound propagation around a high‐lift airfoil and the Munt problem. Copyright © 2011 John Wiley & Sons, Ltd.     

A nodal Godunov method for Lagrangian shock hydrodynamics on unstructured tetrahedral grids

We present a nodal Godunov method for Lagrangian shock hydrodynamics. The method is designed to operate on three‐dimensional unstructured grids composed of tetrahedral cells. A node‐centered finite element formulation avoids mesh stiffness, and an approximate Riemann solver in the fluid reference frame ensures a stable, upwind formulation. This choice leads to a non‐zero mass flux between control volumes, even though the mesh moves at the fluid velocity, but eliminates volume errors that arise due to the difference between the fluid velocity and the contact wave speed. A monotone piecewise linear reconstruction of primitive variables is used to compute interface unknowns and recover second‐order accuracy. The scheme has been tested on a variety of standard test problems and exhibits first‐order accuracy on shock problems and second‐order accuracy on smooth flows using meshes of up to O(10⁶) tetrahedra. Copyright © 2014 John Wiley & Sons, Ltd.     

Solution of the hyperbolic mild‐slope equation using the finite volume method

A finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.     

Adaptive Q‐tree Godunov‐type scheme for shallow water equations

This paper presents details of a second‐order accurate, Godunov‐type numerical model of the two‐dimensional shallow water equations (SWEs) written in matrix form and discretized using finite volumes. Roe's flux function is used for the convection terms and a non‐linear limiter is applied to prevent unwanted spurious oscillations. A new mathematical formulation is presented, which inherently balances flux gradient and source terms. It is, therefore, suitable for cases where the bathymetry is non‐uniform, unlike other formulations given in the literature based on Roe's approximate Riemann solver. The model is based on hierarchical quadtree (Q‐tree) grids, which adapt to inherent flow parameters, such as magnitude of the free surface gradient and depth‐averaged vorticity. Validation tests include wind‐induced circulation in a dish‐shaped basin, two‐dimensional frictionless rectangular and circular dam‐breaks, an oblique hydraulic jump, and jet‐forced flow in a circular reservoir. Copyright © 2001 John Wiley & Sons, Ltd.     

A compact high‐order HLL solver for nonlinear hyperbolic systems

We present a new finite‐volume method for calculating complex flows on non‐uniform meshes. This method is designed to be highly compact and to accurately capture all discontinuities that may arise within the solution of a nonlinear hyperbolic system. In the first step, we devise a fourth‐degree Hermite polynomial to interpolate the solution. The coefficients defining this polynomial are calculated by using a least‐square method. To introduce monotonicity conditions within the procedure, two constraints are added into the least‐square system. Those constraints are derived by locally matching the high‐order Hermite polynomial with a low‐order TVD polynomial. To emulate these constraints only in regions of discontinuities, data‐depending weights are defined; these weights are based upon normalized indicators of smoothness of the solution and are parameterized by an O(1) quantity. The reconstruction so generated is highly compact and is fifth‐order accurate when the solution is smooth; this reconstruction becomes first order in regions of discontinuities. In the second step, this reconstruction is inserted in an HLL approximate Riemann solver. This solver is designed to correctly capture all discontinuities that may arise into the solution. To this aim, we introduce the contribution of a possible contact discontinuity into the HLL Riemann solver. Thus, a spatially fifth‐order non‐oscillatory method is generated. This method evolves in time the solution and its first derivative. In a one‐dimensional context, a linear spectral analysis and extensive numerical experiments make it possible to assess the robustness and the advantages of the method in computing multi‐scale problems with embedded discontinuities. Copyright © 2008 John Wiley & Sons, Ltd.     

A well‐balanced stable generalized Riemann problem scheme for shallow water equations using adaptive moving unstructured triangular meshes

We propose a well‐balanced stable generalized Riemann problem (GRP) scheme for the shallow water equations with irregular bottom topography based on moving, adaptive, unstructured, triangular meshes. In order to stabilize the computations near equilibria, we use the Rankine–Hugoniot condition to remove a singularity from the GRP solver. Moreover, we develop a remapping onto the new mesh (after grid movement) based on equilibrium variables. This, together with the already established techniques, guarantees the well‐balancing. Numerical tests show the accuracy, efficiency, and robustness of the GRP moving mesh method: lake at rest solutions are preserved even when the underlying mesh is moving (e.g., mesh points are moved to regions of steep gradients), and various comparisons with fixed coarse and fine meshes demonstrate high resolution at relatively low cost. Copyright © 2013 John Wiley & Sons, Ltd.     

An unstructured finite volume model for dam‐break floods with wet/dry fronts over complex topography

A robust, well‐balanced, unstructured, Godunov‐type finite volume model has been developed in order to simulate two‐dimensional dam‐break floods over complex topography with wetting and drying. The model is based on the nonlinear shallow water equations in hyperbolic conservation form. The inviscid fluxes are calculated using the HLLC approximate Riemann solver and a second‐order spatial accuracy is achieved by implementing the MUSCL reconstruction technique. To prevent numerical oscillations near shocks, slope‐limiting techniques are used for controlling the total variation of the reconstructed field. The model utilizes an explicit two‐stage Runge–Kutta method for time stepping, whereas implicit treatments for friction source terms. The novelties of the model include the flux correction terms and the water depth reconstruction method both for partially and fully submerged cells, and the wet/dry front treatments. The proposed flux correction terms combined with the water depth reconstruction method are necessary to balance the bed slope terms and flux gradient in the hydrostatical steady flow condition. Especially, this well‐balanced property is also preserved in partially submerged cells. It is found that the developed wet/dry front treatments and implicit scheme for friction source terms are stable. The model is tested against benchmark problems, laboratory experimental data, and realistic application related to dam‐break flood wave propagation over arbitrary topography. Numerical results show that the model performs satisfactorily with respect to its effectiveness and robustness and thus has bright application prospects. Copyright © 2010 John Wiley & Sons, Ltd.     

Edge‐based reconstruction schemes for unstructured tetrahedral meshes

In this paper, we consider edge‐based reconstruction (EBR) schemes for solving the Euler equations on unstructured tetrahedral meshes. These schemes are based on a high‐accuracy quasi‐1D reconstruction of variables on an extended stencil along the edge‐based direction. For an arbitrary tetrahedral mesh, the EBR schemes provide higher accuracy in comparison with most second‐order schemes at rather low computational costs. The EBR schemes are built in the framework of vertex‐centered formulation for the point‐wise values of variables. Here, we prove the high accuracy of EBR schemes for uniform grid‐like meshes, introduce an economical implementation of quasi‐one‐dimensional reconstruction and the resulting new scheme of EBR family, estimate the computational costs, and give new verification results. Copyright © 2015 John Wiley & Sons, Ltd.     

Stabilized finite‐element computation of compressible flow with linear and quadratic interpolation functions

The unsteady compressible flow equations are solved using a stabilized finite‐element formulation with C⁰ elements. In 2D, the performance of three‐noded linear and six‐noded quadratic triangular elements is compared. In 3D, the relative performance is evaluated for 6‐noded linear and 18‐noded quadratic wedge elements. Results are compared for the solutions to Euler, laminar, and turbulent flows at different Mach numbers for several flow problems. The finite‐element meshes considered for comparison have same location of nodes for the linear and quadratic interpolations. For the turbulent flow, the Spalart–Allmaras model is used for closure. It is found that the quadratic elements yield better performance than the linear elements. This is attributed to accurate representation of the stabilization terms that involve second‐order derivatives in the formulation. When these terms are dropped from the formulation with quadratic interpolation, the numerical results are similar to those obtained with linear interpolation. The absence of these terms result in added numerical diffusion that seems to give the effect of a relatively reduced Reynolds number. For the same location of nodes, the computations with the linear triangular and wedge elements are approximately 20% and 100% faster than those with quadratic triangular and wedge elements, respectively. However, if the same quadrature rule for numerical integration is used for both interpolations, the computations with quadratic elements are approximately 20% and 45% faster in 2D and 3D, respectively. Copyright © 2014 John Wiley & Sons, Ltd.     

Roe‐type Riemann solvers for general hyperbolic systems

We present a Roe‐type weak formulation Riemann solver where the average coefficient matrix is computed numerically. The novelty of this approach is that it is general enough that can be applied to any hyperbolic system while retaining the accuracy of the original Roe solver. We show applications to the compressible Euler equations with general equation of state. An alternative version of the method uses directly the eigenvectors in the averaging process, simplifying the algorithm. These new solvers are applied in conservative and path‐conservative schemes with high‐order accuracy and on unstructured meshes. Copyright © 2014 John Wiley & Sons, Ltd.     

Unsteady CFD computations using vertex‐centered finite volumes for unstructured grids on Graphics Processing Units

This paper presents a Navier–Stokes solver for steady and unsteady turbulent flows on unstructured/hybrid grids, with triangular and quadrilateral elements, which was implemented to run on Graphics Processing Units (GPUs). The paper focuses on programming issues for efficiently porting the CPU code to the GPU, using the CUDA language. Compared with cell‐centered schemes, the use of a vertex‐centered finite volume scheme on unstructured grids increases the programming complexity since the number of nodes connected by edge to any other node might vary a lot. Thus, delicate GPU memory handling is absolutely necessary in order to maximize the speed‐up of the GPU implementation with respect to the Fortran code running on a single CPU core. The developed GPU‐enabled code is used to numerically study steady and unsteady flows around the supercritical airfoil OAT15A, by laying emphasis on the transonic buffet phenomenon. The computations were carried out on NVIDIA's Ge‐Force GTX 285 graphics cards and speed‐ups up to ～46 × (on a single GPU, with double precision arithmetic) are reported. Copyright © 2010 John Wiley & Sons, Ltd.     

High‐order ENO and WENO schemes for unstructured grids

This work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd.     

A cell‐based smoothed finite element method with semi‐implicit CBS procedures for incompressible laminar viscous flows

In this paper, the cell‐based smoothed finite element method (CS‐FEM) with the semi‐implicit characteristic‐based split (CBS) scheme (CBS/CS‐FEM) is proposed for computational fluid dynamics. The 3‐node triangular (T3) element and 4‐node quadrilateral (Q4) element are used for present CBS/CS‐FEM for two‐dimensional flows. The 8‐node hexahedral element (H8) is used for three‐dimensional flows. Two types of CS‐FEM are implemented in this paper. One is standard CS‐FEM with quadrilateral gradient smoothing cells for Q4 element and hexahedron cells for H8 element. Another is called as n‐sided CS‐FEM (nCS‐FEM) whose gradient smoothing cells are triangles for Q4 element and pyramids for H8 element. To verify the proposed methods, benchmarking problems are tested for two‐dimensional and three‐dimensional flows. The benchmarks show that CBS/CS‐FEM and CBS/nCS‐FEM are capable to solve incompressible laminar flow and can produce reliable results for both steady and unsteady flows. The proposed CBS/CS‐FEM method has merits on better robustness against distorted mesh with only slight more computation time and without losing accuracy, which is important for problems with heavy mesh distortion. The blood flow in carotid bifurcation is also simulated to show capabilities of proposed methods for realistic and complicated flow problems.     

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.     

Multigrid third‐order least‐squares solution of Cauchy–Riemann equations on unstructured triangular grids

In this paper, a multigrid algorithm is developed for the third‐order accurate solution of Cauchy–Riemann equations discretized in the cell‐vertex finite‐volume fashion: the solution values stored at vertices and the residuals defined on triangular elements. On triangular grids, this results in a highly overdetermined problem, and therefore we consider its solution that minimizes the residuals in the least‐squares norm. The standard second‐order least‐squares scheme is extended to third‐order by adding a high‐order correction term in the residual. The resulting high‐order method is shown to give sufficiently accurate solutions on relatively coarse grids. Combined with a multigrid technique, the method then becomes a highly accurate and efficient solver. We present some results to demonstrate its accuracy and efficiency, including both structured and unstructured triangular grids. Copyright © 2006 John Wiley & Sons, Ltd.     

Riemann solvers and boundary conditions for two‐dimensional shallow water simulations

Most existing algorithms for two‐dimensional shallow water simulations treat multi‐dimensional waves using wave splitting or time splitting. This often results in anisotropy of the computed flow. Both wave splitting and time splitting are based on a local decomposition of the multi‐dimensional problem into one‐dimensional, orthogonal problems. Therefore, these algorithms handle boundary conditions in a very similar way to classical one‐dimensional algorithms. This should be expected to trigger a dependence of the number of boundary conditions on the direction of the flow at the boundaries. However, most computational codes based on alternate directions do not exhibit such sensitivity, which seems to contradict the theory of existence and uniqueness of the solution. The present paper addresses these issues. A Riemann solver is presented that aims to convert two‐dimensional Riemann problems into a one‐dimensional equivalent Riemann problem (ERP) at the interfaces between the computational cells. The ERP is derived by applying the theory of bicharacteristics at each end of the interface and by performing a linear averaging along the interface. The proposed approach is tested against the traditional one‐dimensional approach on the classical circular dambreak problem. The results show that the proposed solver allows the isotropy of the solution to be better preserved. Use of the two‐dimensional solver with a first‐order scheme may give better results than use of a second‐order scheme with a one‐dimensional solver. The theory of bicharacteristics is also used to discuss the issue of boundary conditions. It is shown that, when the flow is subcritical, the number of boundary conditions affects the accuracy of the solution, but not its existence and uniqueness. When only one boundary condition is to be prescribed, it should not be the velocity in the direction parallel to the boundary. When two boundary conditions are to be prescribed, at least one of them should involve the component of the velocity in the direction parallel to the boundary. Copyright © 2003 John Wiley & Sons, Ltd.