A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations

We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary order for the spatial discretization and a third‐order Runge–Kutta scheme to advance in time. The local elements are coupled together by numerical fluxes, evaluated using the HLLC Riemann solver. We apply the model to test cases involving smooth flows and demonstrate the exponentially fast convergence with regard to polynomial order. We also illustrate that even for results of ‘engineering accuracy’ the computational efficiency increases with increasing order of the model and time of integration. The model is found to be robust in the presence of shocks where Gibbs oscillations can be suppressed by slope limiting. Copyright 2004 John Wiley & Sons, Ltd.     

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convectiondiffusion problems) and the KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.     

Implicit time accurate simulation of unsteady flow

Implicit time integration was studied in the context of unsteady shock‐boundary layer interaction flow. With an explicit second‐order Runge–Kutta scheme, a reference solution to compare with the implicit second‐order Crank–Nicolson scheme was determined. The time step in the explicit scheme is restricted by both temporal accuracy as well as stability requirements, whereas in the A‐stable implicit scheme, the time step has to obey temporal resolution requirements and numerical convergence conditions. The non‐linear discrete equations for each time step are solved iteratively by adding a pseudo‐time derivative. The quasi‐Newton approach is adopted and the linear systems that arise are approximately solved with a symmetric block Gauss–Seidel solver. As a guiding principle for properly setting numerical time integration parameters that yield an efficient time accurate capturing of the solution, the global error caused by the temporal integration is compared with the error resulting from the spatial discretization. Focus is on the sensitivity of properties of the solution in relation to the time step. Numerical simulations show that the time step needed for acceptable accuracy can be considerably larger than the explicit stability time step; typical ratios range from 20 to 80. At large time steps, convergence problems that are closely related to a highly complex structure of the basins of attraction of the iterative method may occur. Copyright © 2001 John Wiley & Sons, Ltd.     

A new weighted upwind finite volume element method based on non‐standard covolume for time‐dependent convection–diffusion problems

In modern numerical simulation of problems in energy resources and environmental science, it is important to develop efficient numerical methods for time‐dependent convection–diffusion problems. On the basis of nonstandard covolume grids, we propose a new kind of high‐order upwind finite volume element method for the problems. We first prove the stability and mass conservation in the discrete forms of the scheme. Optimal second‐order error estimate in L₂‐norm in spatial step is then proved strictly. The scheme is effective for avoiding numerical diffusion and nonphysical oscillations and has second‐order accuracy. Numerical experiments are given to verify the performance of the scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

The third‐order polynomial method for two‐dimensional convection and diffusion

Using the upstream polynomial approximation a series of accurate two‐dimensional explicit numerical schemes is developed for the solution of the convection–diffusion equation. A third‐order polynomial approximation (TOP) of the convection term and a consistent second‐order approximation of the diffusion term are combined in a single‐step flux‐difference algorithm. Stability analysis confirms that the TOP‐12 scheme satisfies the CFL condition for two dimensions. Using smaller and narrower flux stencils compared to algorithms of similar accuracy, the TOP‐12 scheme is more efficient in terms of computations per single node. Numerical tests and comparison with other well‐known algorithms show a high performance of the developed schemes. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

A γ‐DGBGK scheme for compressible multi‐fluids

The paper presents a Discontinuous Galerkin γ‐BGK (γ‐DGBGK) method for compressible multicomponent flow simulations by coupling the discontinuous Galerkin method with a γ‐BGK scheme based on WENO limiters. In this γ‐DGBGK method, the construction of the flux in the DG method is based on the kinetic scheme which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous terms in the flux formulation at cell interfaces. WENO limiters are used to obtain uniform high‐order accuracy and sharp non‐oscillatory shock transition, and time accuracy obtained by integration for the flux function at the cell interface. Numerical examples in one and two space dimensions are presented to illustrate the robust and accuracy of the present scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐order Petrov–Galerkin finite element method for the classical Boussinesq wave model

A high‐order Petrov–Galerkin finite element scheme is presented to solve the one‐dimensional depth‐integrated classical Boussinesq equations for weakly non‐linear and weakly dispersive waves. Finite elements are used both in the space and the time domains. The shape functions are bilinear in space–time, whereas the weighting functions are linear in space and quadratic in time, with C⁰‐continuity. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weighting functions. An implicit, conditionally stable, one‐step predictor–corrector time integration scheme results. The accuracy and stability of the non‐linear discrete equations are investigated by means of a local Taylor series expansion. A linear spectral analysis is used for the full characterization of the predictor–corrector inner iterations. Based on the order of the analytical terms of the Boussinesq model and on the order of the numerical discretization, it is concluded that the scheme is fourth‐order accurate in terms of phase velocity. The dissipation term is third order only affecting the shortest wavelengths. A numerical convergence analysis showed a second‐order convergence rate in terms of both element size and time step. Four numerical experiments are addressed and their results are compared with analytical solutions or experimental data available in the literature: the propagation of a solitary wave, the oscillation of a flat bottom closed basin, the oscillation of a non‐flat bottom closed basin, and the propagation of a periodic wave over a submerged bar. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

A second-order upwind finite-volume method for the Euler solution on unstructured triangular meshes

A scheme for the numerical solution of the two-dimensional (2D) Euler equations on unstructured triangular meshes has been developed. The basic first-order scheme is a cell-centred upwind finite-volume scheme utilizing Roe's approximate Riemann solver. To obtain second-order accuracy, a new gradient based on the weighted average of Barth and Jespersen's three-point support gradient model is used to reconstruct the cell interface values. Characteristic variables in the direction of local pressure gradient are used in the limiter to minimize the numerical oscillation around solution discontinuities. An Approximate LU (ALU) factorization scheme originally developed for structured grid methods is adopted for implicit time integration and shows good convergence characterisitics in the test. To eliminate the data dependency which prohibits vectorization in the inversion process, a black-gray-white colouring and numbering technique on unstructured triangular meshes is developed for the ALU factorization scheme. This results in a high degree of vectorization of the final code. Numerical experiments on transonic Ringleb flow, transonic channel flow with circular bump, supersonic shock reflection flow and subsonic flow over multielement aerofoils are calculated to validate the methodology.     

Two‐dimensional flow past circular cylinders using finite volume lattice Boltzmann formulation

In this article, an extension to the total variation diminishing finite volume formulation of the lattice Boltzmann equation method on unstructured meshes was presented. The quadratic least squares procedure is used for the estimation of first‐order and second‐order spatial gradients of the particle distribution functions. The distribution functions were extrapolated quadratically to the virtual upwind node. The time integration was performed using the fourth‐order Runge–Kutta procedure. A grid convergence study was performed in order to demonstrate the order of accuracy of the present scheme. The formulation was validated for the benchmark two‐dimensional, laminar, and unsteady flow past a single circular cylinder. These computations were then investigated for the low Mach number simulations. Further validation was performed for flow past two circular cylinders arranged in tandem and side‐by‐side. Results of these simulations were extensively compared with the previous numerical data. Copyright © 2011 John Wiley & Sons, Ltd.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

This paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

Verification of Euler/Navier–Stokes codes using the method of manufactured solutions

The method of manufactured solutions is used to verify the order of accuracy of two finite‐volume Euler and Navier–Stokes codes. The Premo code employs a node‐centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central differences for the diffusion terms, thus yielding a numerical scheme that is formally second‐order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and Navier–Stokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of confidence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given. Copyright © 2004 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.     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

A simple high‐resolution advection scheme

A simple, robust, mass‐conserving numerical scheme for solving the linear advection equation is described. The scheme can estimate peak solution values accurately even in regions where spatial gradients are high. Such situations present a severe challenge to classical numerical algorithms. Attention is restricted to the case of pure advection in one and two dimensions since this is where past numerical problems have arisen. The authors' scheme is of the Godunov type and is second‐order in space and time. The required cell interface fluxes are obtained by MUSCL interpolation and the exact solution of a degenerate Riemann problem. Second‐order accuracy in time is achieved via a Runge–Kutta predictor–corrector sequence. The scheme is explicit and expressed in finite volume form for ease of implementation on a boundary‐conforming grid. Benchmark test problems in one and two dimensions are used to illustrate the high‐spatial accuracy of the method and its applicability to non‐uniform grids. Copyright © 2007 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.     

Numerical and experimental study of driven flow in a polar cavity

A numerical and an experimental study of the flow of an incompressible fluid in a polar cavity is presented. The experiments included flow visualization, in two perpendicular planes, and quantitative measurements of the velocity field by a laser Doppler anemometer. Measurements were done for two ranges of Reynolds numbers; about 60 and about 350. The stream function-vorticity form of the governing equations was approximated by upwind or central finite-differences. Both types of finite-difference approximations were solved by a multi-grid method. Numerical solutions were computed on a sequence of grids and the relative accuracy of the solutions was studied. Our most accurate numerical solutions had an estimated error of 0.1 per cent and 1 per cent for Re = 60 and Re = 350, respectively. It was also noted that the solution to the second order finite difference equations was more accurate, compared to the solution to the first order equations, only if fine enough meshes were used. The possibility of using extrapolations to improve accuracy was also considered. Extrapolated solutions were found to be valid only if solutions computed on fine enough meshes were used. The numerical and the experimental results were found to be in very good agreement.     

A Boltzmann based model for open channel flows

A finite volume, Boltzmann Bhatnagar–Gross–Krook (BGK) numerical model for one‐ and two‐dimensional unsteady open channel flows is formulated and applied. The BGK scheme satisfies the entropy condition and thus prevents unphysical shocks. In addition, the van Leer limiter and the collision term ensure that the BGK scheme admits oscillation‐free solutions only. The accuracy and efficiency of the BGK scheme are demonstrated through the following examples: (i) strong shock waves, (ii) extreme expansion waves, (iii) a combination of strong shock waves and extreme expansion waves, and (iv) one‐ and two‐dimensional dam break problems. These test cases are performed for a variety of Courant numbers (C_r), with the only condition being C_r≤1. All the computational results are free of spurious oscillations and unphysical shocks (i.e., expansion shocks). In addition, comparisons of numerical tests with measured data from dam break laboratory experiments show good agreement for C_r≤0.6. This reduction in the stability domain is due to the explicit integration of the friction term. Furthermore, BGK schemes are easily extended to multidimensional problems and do not require characteristic decomposition. The proposed scheme is second‐order in both space and time when the external forces are zero and second‐order in space but first‐order in time when the external forces are non‐zero. However, since all the test cases presented are either for zero or small values of external forces, the results tend to maintain second‐order accuracy. In problems where the external forces become significant, it is possible to improve the order of accuracy of the scheme in time by, for example, applying the Runge–Kutta method in the integration of the external forces. Copyright © 2001 John Wiley & Sons, Ltd.