The composite finite volume method on unstructured meshes for the two‐dimensional shallow water equations

A composite finite volume method (FVM) is developed on unstructured triangular meshes and tested for the two‐dimensional free‐surface flow equations. The methodology is based on the theory of the remainder effect of finite difference schemes and the property that the numerical dissipation and dispersion of the schemes are compensated by each other in a composite scheme. The composite FVM is formed by global composition of several Lax–Wendroff‐type steps followed by a diffusive Lax–Friedrich‐type step, which filters out the oscillations around shocks typical for the Lax–Wendroff scheme. To test the efficiency and reliability of the present method, five typical problems of discontinuous solutions of two‐dimensional shallow water are solved. The numerical results show that the proposed method, which needs no use of a limiter function, is easy to implement, is accurate, robust and is highly stable. Copyright © 2001 John Wiley & Sons, Ltd.     

Improving simple explicit methods for unsteady open channel and river flow

A rigorous study of the explicit Lax–Friedrichs scheme for its application to one‐dimensional shallow water flows is presented. The deficiencies of this method are identified and the way to overcome them are presented. It is compared to the explicit first order upwind scheme and to the explicit second order Lax–Wendroff scheme by means of the simulation of several test cases with exact solution. All three schemes in their best balanced version are applied to the simulation of a real river flood wave leading to very satisfactory results. Copyright © 2004 John Wiley & Sons, Ltd.     

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

A 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.     

Comparisons of numerical methods with respect to convectively dominated problems

A series of numerical schemes: first‐order upstream, Lax–Friedrichs; second‐order upstream, central difference, Lax–Wendroff, Beam–Warming, Fromm; third‐order QUICK, QUICKEST and high resolution flux‐corrected transport and total variation diminishing (TVD) methods are compared for one‐dimensional convection–diffusion problems. Numerical results show that the modified TVD Lax–Friedrichs method is the most competent method for convectively dominated problems with a steep spatial gradient of the variables. Copyright © 2001 John Wiley & Sons, Ltd.     

Multidimensional upwind schemes for the shallow water equations

A multidimensional discretisation of the shallow water equations governing unsteady free-surface flow is proposed. The method, based on a residual distribution discretisation, relies on a characteristic eigenvector decomposition of each cell residual, and the use of appropriate distribution schemes. For uncoupled equations, multidimensional convection schemes on compact stencils are used, while for coupled equations, either system distribution schemes such as the Lax–Wendroff scheme or scalar schemes may be used. For steady subcritical flows, the equations can be partially diagonalised into a purely convective equation of hyperbolic nature, and a set of coupled equations of elliptic nature. The multidimensional discretisation, which is second-order-accurate at steady state, is shown to be superior to the standard Lax–Wendroff discretisation. For steady supercritical flows, the equations can be fully diagonalised into a set of convective equations corresponding to the steady state characteristics. Discontinuities such as hydraulic jumps, are captured in a sharp and non-oscillatory way. For unsteady flows, the characteristic equations remain coupled. An appropriate treatment of the coupling terms allows the discretisation of these equations at the scalar level. Although presently only first-order-accurate in space and time, the classical dam-break problem demonstrates the validity of the approach. © 1998 John Wiley & Sons, Ltd.     

A preconditioning technique for steady Euler solutions

This paper presents a preconditioning technique for solving a two‐dimensional system of hyperbolic equations. The main attractive feature of this approach is that, unlike a technique based on simply extending the solver for a one‐dimensional hyperbolic system, convergence and stability analysis can be investigated. This method represents a genuine numerical algorithm for multi‐dimensional hyperbolic systems. In order to demonstrate the effectiveness of this approach, applications to solving a two‐dimensional system of Euler equations in supersonic flows are reported. It is shown that the Lax–Friedrichs scheme diverges when applied to the original Euler equations. However, convergence is achieved when the same numerical scheme is employed using the same CFL number to solve the equivalent preconditioned Euler system. Copyright © 1999 John Wiley & Sons, Ltd.     

ANALYSIS AND IMPLEMENTATION OF THE GAS-KINETIC BGK SCHEME FOR COMPUTATIONAL GAS DYNAMICS

Gas-kinetic schemes based on the BGK model are proposed as an alternative evolution model which can cure some of the limitations of current Riemann solvers. To analyse the schemes, simple advection equations are reconstructed and solved using the gas-kinetic BGK model. Results for gas-dynamic application are also presented. The final flux function derived in this model is a combination of a gas-kinetic Lax– Wendroff flux of viscous advection equations and kinetic flux vector splitting. These two basic schemes are coupled through a non-linear gas evolution process and it is found that this process always satisfies the entropy condition. Within the framework of the LED (local extremum diminishing) principle that local maxima should not increase and local minima should not decrease in interpolating physical quantities, several standard limiters are adopted to obtain initial interpolations so as to get higher-order BGK schemes. Comparisons for well-known test cases indicate that the gas-kinetic BGK scheme is a promising approach in the design of numerical schemes for hyperbolic conservation laws. © 1997 by John Wiley & Sons, Ltd.     

Optimised composite numerical schemes in 2‐D for hyperbolic conservation laws

One of the techniques available for optimising parameters that regulate dispersion and dissipation effects in finite difference schemes is the concept of minimised integrated exponential error for low dispersion and low dissipation. In this paper, we work essentially with the two‐dimensional (2D) Corrected Lax–Friedrichs and Lax–Friedrichs schemes applied to the 2D scalar advection equation. We examine the shock‐capturing properties of these two numerical schemes, and observe that these methods are quite effective from the point of being able to control computational noise and having a large range of stability. To improve the shock‐capturing efficiency of these two methods, we derive composite methods using the idea of predictor/corrector or a linear combination of the two schemes. The optimal cfl number for some of these composite schemes are computed. Some numerical experiments are carried out in two dimensions such as cylindrical explosion, shock‐focusing, dam‐break and Riemann gas dynamics tests. The modified equations of some of the composite schemes when applied to the 2D scalar advection equation are obtained. We also perform some convergence tests to obtain the order of accuracy and show that better results in terms of shock‐capturing property are obtained when the optimal cfl obtained using minimised integrated exponential error for low dispersion and low dissipation is used. Copyright © 2011 John Wiley & Sons, Ltd.     

Toward a reduction of mesh imprinting

This paper is the initial investigation into a new Lagrangian cell‐centered hydrodynamic scheme that is motivated by the desire for an algorithm that resists mesh imprinting and has reduced complexity. Key attributes of the new approach include multidimensional construction, the use of flux‐corrected transport (FCT) to achieve second order accuracy, automatic determination of the mesh motion through vertex fluxes, and vorticity control. Toward this end, vorticity preserving Lax–Wendroff (VPLW) type schemes for the two‐dimensional acoustic equations were analyzed and then implemented in a FCT algorithm. Here, mesh imprinting takes the form of anisotropic dispersion relationships. If the stencil for the LW methods is limited to nine points, four free parameters exist. Two parameters were fixed by insisting that no spurious vorticity be created. Dispersion analysis was used to understand how the remaining two parameters could be chosen to increase isotropy. This led to new VPLW schemes that suffer less mesh imprinting than the rotated Richtmyer method. A multidimensional, vorticity preserving FCT implementation was then sought using the most promising VPLW scheme to address the problem of spurious extrema. A well‐behaved first order scheme and a new flux limiter were devised in the process. The flux limiter is unique in that it acts on temporal changes and does not place a priori bounds on the solution. Numerical results have demonstrated that the vorticity preserving FCT scheme has comparable performance to an unsplit MUSCL‐H algorithm at high Courant numbers but with reduced mesh imprinting and superior symmetry preservation. Copyright © 2014 John Wiley & Sons, Ltd.     

Some applications of the concept of minimized integrated exponential error for low dispersion and low dissipation

Several techniques to optimize parameters that regulate dispersion and dissipation effects in finite difference schemes have been devised in our previous works. They all use the notion that dissipation neutralizes dispersion. These techniques are the minimized integrated square difference error (MISDE) and the minimized integrated exponential error for low dispersion and low dissipation (MIEELDLD). It is shown in this work based on several numerical schemes tested that the technique of MIEELDLD is more accurate than MISDE to optimize the parameters that regulate dispersion and dissipation effects with the aim of improving the shock‐capturing properties of numerical methods. First, we consider the family of third‐order schemes proposed by Takacs. We use the techniques MISDE and MIEELDLD to optimize two parameters, namely, the cfl number and another variable which also controls dispersion and dissipation. Second, these two techniques are used to optimize a numerical scheme proposed by Gadd. Moreover, we compute the optimal cfl for some multi‐level schemes in 1D. Numerical tests for some of these numerical schemes mentioned above are performed at different cfl numbers and it is shown that the results obtained are dependent on the cfl number chosen. The errors from the numerical results have been quantified into dispersion and dissipation using a technique devised by Takacs. Finally, we make use of a composite scheme made of corrected Lax–Friedrichs and the two‐step Lax–Friedrichs schemes like the CFLF4 scheme at its optimal cfl number, to solve some problems in 2D, namely: solid body rotation test, acoustics and the circular Riemann problem. Copyright © 2011 John Wiley & Sons, Ltd.     

VALIDATION OF A NAVIER–STOKES SOLUTION ALGORITHM WITH EXPERIMENTAL VALUES IN A SUPERSONIC WAKE

This paper describes the validation of a finite element solver for an axisymmetric compressible flow with experimental values, especially velocities measured with a laser Doppler anemometer in the near wake of a circular cylinder. The equations under consideration are the Navier-Stokes equations with turbulent terms. A time-stepping scheme for the solution of these equations can be produced by applying a forward-time Taylor series expansion including time derivatives of second order. These time derivatives are evaluated in terms of space derivatives in the Lax–Wendroff fashion. The method is based on unstructured triangular grids with a high resolution in the radial direction. In order to predict the measured turbulent intensites more exactly, a modification of the Baldwin–Lomax model is necessary.     

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

The 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.     

Fluid–structural interaction with application to rocket engines

The objective of this work is to develop a finite element model for studying fluid–structure interaction. The geometrically non‐linear structural behaviour is considered and based on large rotations and large displacements. An arbitrary Lagrangian–Eulerian (ALE) formulation is used to represent the compressible inviscid flow with moving boundaries. The structural response is obtained using Newmark‐type time integration and fluid response employs the Lax–Wendroff scheme. A number of numerical examples are presented to validate the structural model, moving mesh implantation of the ALE model and complete fluid–structure interaction. Copyright © 1999 John Wiley & Sons, Ltd.     

Preventing numerical oscillations in the flux‐split based finite difference method for compressible flows with discontinuities,II

Problems in the characteristic‐wise flux‐split based finite difference method when compressible flows with contact discontinuities or material interfaces are computed were presented and analyzed. The current analysis showed the following: (i) Even with the local characteristic decomposition technique, numerical errors could be caused by point‐wise flux vector splitting (FVS) methods, such as the Steger–Warming FVS or the van Leer FVS. Therefore, the Lax–Friedrichs type FVS method is required. (ii) If the isobars of a material are vertical lines, the combination of using the local characteristic decomposition and the global Lax–Friedrichs FVS can avoid velocity and pressure oscillations of contact discontinuities in this material for weighted essentially non‐oscillatory (WENO) schemes. (iii) For problems with material interfaces, the quasi‐conservative approach can be realized using characteristic‐wise flux‐split based finite difference WENO schemes if nonlinear WENO schemes in genuinely nonlinear characteristic fields can be guaranteed to be the same and the decomposition equation representing material interfaces is discretized properly. Copyright © 2015 John Wiley & Sons, Ltd.     

Compact high‐resolution algorithms for time‐dependent advection on unstructured grids

A technique for constructing monotone, high resolution, multi‐dimensional upwind fluctuation distribution schemes for the scalar advection equation is presented. The method combines the second‐order Lax–Wendroff scheme with the upwind positive streamwise invariant (PSI) scheme via a fluctuation redistribution step, which ensures monotonicity (and which is a generalization of the flux‐corrected transport approach for fluctuation distribution schemes). Furthermore, the concept of a distribution point is introduced, which, when related to the equivalent equation for the scheme, leads to a ‘preferred direction’ for the limiting procedure, and hence to a new distribution of the fluctuation, which retains second‐order accuracy from the Lax–Wendroff scheme, even when the solution contains turning points. Experimental comparisons show that the new method compares favourably in terms of speed, accuracy and robustness with other, similar, techniques. Copyright © 2000 John Wiley & Sons, Ltd.     

Convergence issues in using high‐resolution schemes and lower–upper symmetric Gauss–Seidel method for steady shock‐induced combustion problems

This paper reports numerical convergence study for simulations of steady shock‐induced combustion problems with high‐resolution shock‐capturing schemes. Five typical schemes are used: the Roe flux‐based monotone upstream‐centered scheme for conservation laws (MUSCL) and weighted essentially non‐oscillatory (WENO) schemes, the Lax–Friedrichs splitting‐based non‐oscillatory no‐free parameter dissipative (NND) and WENO schemes, and the Harten–Yee upwind total variation diminishing (TVD) scheme. These schemes are implemented with the finite volume discretization on structured quadrilateral meshes in dimension‐by‐dimension way and the lower–upper symmetric Gauss–Seidel (LU–SGS) relaxation method for solving the axisymmetric multispecies reactive Navier–Stokes equations. Comparison of iterative convergence between different schemes has been made using supersonic combustion flows around a spherical projectile with Mach numbers M = 3.55 and 6.46 and a ram accelerator with M = 6.7. These test cases were regarded as steady combustion problems in literature. Calculations on gradually refined meshes show that the second‐order NND, MUSCL, and TVD schemes can converge well to steady states from coarse through fine meshes for M = 3.55 case in which shock and combustion fronts are separate, whereas the (nominally) fifth‐order WENO schemes can only converge to some residual level. More interestingly, the numerical results show that all the schemes do not converge to steady‐state solutions for M = 6.46 in the spherical projectile and M = 6.7 in the ram accelerator cases on fine meshes although they all converge on coarser meshes or on fine meshes without chemical reactions. The result is based on the particular preconditioner of LU–SGS scheme. Possible reasons for the nonconvergence in reactive flow simulation are discussed.Copyright © 2012 John Wiley & Sons, Ltd.     

Solving the Savage–Hutter equations for granular avalanche flows with a second‐order Godunov type method on GPU

In this article, we apply Davis's second‐order predictor‐corrector Godunov type method to numerical solution of the Savage–Hutter equations for modeling granular avalanche flows. The method uses monotone upstream‐centered schemes for conservation laws (MUSCL) reconstruction for conservative variables and Harten–Lax–van Leer contact (HLLC) scheme for numerical fluxes. Static resistance conditions and stopping criteria are incorporated into the algorithm. The computation is implemented on graphics processing unit (GPU) by using compute unified device architecture programming model. A practice of allocating memory for two‐dimensional array in GPU is given and computational efficiency of two‐dimensional memory allocation is compared with one‐dimensional memory allocation. The effectiveness of the present simulation model is verified through several typical numerical examples. Numerical tests show that significant speedups of the GPU program over the CPU serial version can be obtained, and Davis's method in conjunction with MUSCL and HLLC schemes is accurate and robust for simulating granular avalanche flows with shock waves. As an application example, a case with a teardrop‐shaped hydraulic jump in Johnson and Gray's granular jet experiment is reproduced by using specific friction coefficients given in the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

The Lagrange–Galerkin method for the two‐dimensional shallow water equations on adaptive grids

The weak Lagrange–Galerkin finite element method for the two‐dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non‐linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self‐adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange–Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second‐order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant–Friedrich–Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange–Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid‐point rule and a general family of Runge–Kutta schemes. Results for the two‐dimensional advection equation with and without time‐dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two‐dimensional shallow water equations on a non‐linear soliton wave are presented to illustrate the power and flexibility of this strategy. Copyright © 2000 John Wiley & Sons, Ltd.     

Central schemes for open‐channel flow

The resolution of the Saint‐Venant equations for modelling shock phenomena in open‐channel flow by using the second‐order central schemes of Nessyahu and Tadmor (NT) and Kurganov and Tadmor (KT) is presented. The performances of the two schemes that we have extended to the non‐homogeneous case and that of the classical first‐order Lax–Friedrichs (LF) scheme in predicting dam‐break and hydraulic jumps in rectangular open channels are investigated on the basis of different numerical and physical conditions. The efficiency and robustness of the schemes are tested by comparing model results with analytical or experimental solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

Raviart–Thomas and Brezzi–Douglas–Marini finite‐element approximations of the shallow‐water equations

An analysis of the discrete shallow‐water equations using the Raviart–Thomas and Brezzi–Douglas–Marini finite elements is presented. For inertia–gravity waves, the discrete formulations are obtained and the dispersion relations are computed in order to quantify the dispersive nature of the schemes on two meshes made up of equilateral and biased triangles. A linear algebra approach is also used to ascertain the possible presence of spurious modes arising from the discretization. The geostrophic balance is examined and the smallest representable vortices are characterized on both structured and unstructured meshes. Numerical solutions of two test problems to simulate gravity and Rossby modes are in good agreement with the analytical results. Copyright © 2007 John Wiley & Sons, Ltd.