A fixed‐grid b‐spline finite element technique for fluid–structure interaction

We present a fixed‐grid finite element technique for fluid–structure interaction problems involving incompressible viscous flows and thin structures. The flow equations are discretised with isoparametric b‐spline basis functions defined on a logically Cartesian grid. In addition, the previously proposed subdivision‐stabilisation technique is used to ensure inf–sup stability. The beam equations are discretised with b‐splines and the shell equations with subdivision basis functions, both leading to a rotation‐free formulation. The interface conditions between the fluid and the structure are enforced with the Nitsche technique. The resulting coupled system of equations is solved with a Dirichlet–Robin partitioning scheme, and the fluid equations are solved with a pressure–correction method. Auxiliary techniques employed for improving numerical robustness include the level‐set based implicit representation of the structure interface on the fluid grid, a cut‐cell integration algorithm based on marching tetrahedra and the conservative data transfer between the fluid and structure discretisations. A number of verification and validation examples, primarily motivated by animal locomotion in air or water, demonstrate the robustness and efficiency of our approach. Copyright © 2013 John Wiley & Sons, Ltd.     

A high‐order mass‐lumping procedure for B‐spline collocation method with application to incompressible flow simulations

This paper presents new developments of the staggered spline collocation method for cost‐effective solution to the incompressible Navier–Stokes equations. Maximal decoupling of the velocity and the pressure is obtained by using the fractional step method of Gresho and Chan, allowing the solution to sparse elliptic problems only. In order to preserve the high‐accuracy of the B‐spline method, this fractional step scheme is used in association with a sparse approximation to the inverse of the consistent mass matrix. Such an approximation is constructed from local spline interpolation method, and represents a high‐order generalization of the mass‐lumping technique of the finite‐element method. A numerical investigation of the accuracy and the computational efficiency of the resulting semi‐consistent spline collocation schemes is presented. These schemes generate a stable and accurate unsteady Navier–Stokes solver, as assessed by benchmark computations. Copyright © 2003 John Wiley & Sons, Ltd.     

Theory and applications in stability of free‐surface time‐domain boundary element models

A method is presented for examining the stability of a free‐surface time‐domain boundary element model based on B‐splines. Effects of a non‐uniform discretization occurring in practical applications are included. It is demonstrated that instabilities may occur, even in situations where earlier stability analyses predicted the scheme to be stable. These instabilities are due to non‐uniformities in the spatial discretization, which have until now not been included in the stability analyses. Copyright © 2001 John Wiley & Sons, Ltd.     

Third‐order‐accurate semi‐implicit Runge–Kutta scheme for incompressible Navier–Stokes equations

A semi‐implicit three‐step Runge–Kutta scheme for the unsteady incompressible Navier–Stokes equations with third‐order accuracy in time is presented. The higher order of accuracy as compared to the existing semi‐implicit Runge–Kutta schemes is achieved due to one additional inversion of the implicit operator I‐τγL, which requires inversion of tridiagonal matrices when using approximate factorization method. No additional solution of the pressure‐Poisson equation or evaluation of Navier–Stokes operator is needed. The scheme is supplied with a local error estimation and time‐step control algorithm. The temporal third‐order accuracy of the scheme is proved analytically and ascertained by analysing both local and global errors in a numerical example. Copyright © 2005 John Wiley & Sons, Ltd.     

Accuracy and stability analysis of a second‐order time‐accurate loosely coupled partitioned algorithm for transient conjugate heat transfer problems

In this paper, a second‐order time‐accurate loosely coupled partitioned algorithm is presented for solving transient thermal coupling of solids and fluids, also referred to by conjugate heat transfer. The Crank–Nicolson scheme is used for time integration. The accuracy and stability of the loosely coupled solution algorithm are analyzed analytically. Based on the accuracy analysis, the design order of the time integration scheme is preserved by following a predictor (implicit)–corrector (explicit) approach. Hence, the need to perform an additional implicit solve (a subiteration) at each time step is avoided. The analytical stability analysis shows that by using the Crank–Nicolson scheme for time integration, the partitioned algorithm is unstable for large Fourier numbers, unlike the monolithic approach. Accordingly, using the stability analysis, a stability criterion is obtained for the Crank–Nicolson scheme that imposes restriction on Δt given the material properties and mesh spacings of the coupled domains. As the ratio of the thermal effusivities of the coupled domains reaches unity, the stability of the algorithm reduces. To demonstrate the applicability of the algorithm, a numerical example is considered (an unsteady conjugate natural convection in an enclosure). Copyright © 2013 John Wiley & Sons, Ltd.     

A hybrid Padé ADI scheme of higher‐order for convection–diffusion problems

A high‐order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection–diffusion problems. The scheme employs standard high‐order Padé approximations for spatial first and second derivatives in the convection‐diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22 : 87–110) are applied for the time integration. The approximate factorization imposes a second‐order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher‐order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC‐based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high‐order ADI schemes for solving unsteady convection‐diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given. Copyright © 2009 John Wiley & Sons, Ltd.     

Performance of very‐high‐order upwind schemes for DNS of compressible wall‐turbulence

The purpose of the present paper is to evaluate very‐high‐order upwind schemes for the direct numerical simulation (DNS ) of compressible wall‐turbulence. We study upwind‐biased (UW ) and weighted essentially nonoscillatory (WENO ) schemes of increasingly higher order‐of‐accuracy (J. Comp. Phys. 2000; 160 :405–452), extended up to WENO 17 (AIAA Paper 2009‐1612, 2009). Analysis of the advection–diffusion equation, both as Δx→0 (consistency), and for fixed finite cell‐Reynolds‐number Re_Δx (grid‐resolution), indicates that the very‐high‐order upwind schemes have satisfactory resolution in terms of points‐per‐wavelength (PPW ). Computational results for compressible channel flow (Re∈[180, 230]; M?_CL ∈[0.35, 1.5]) are examined to assess the influence of the spatial order of accuracy and the computational grid‐resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline O(Δt²) time‐integration and O(Δx²) discretization of the viscous terms, comparative studies of various orders‐of‐accuracy for the convective terms demonstrate that very‐high‐order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density. Copyright © 2009 John Wiley & Sons, Ltd.     

Non‐oscillatory relaxation methods for the shallow‐water equations in one and two space dimensions

In this paper, a new family of high‐order relaxation methods is constructed. These methods combine general higher‐order reconstruction for spatial discretization and higher order implicit‐explicit schemes or TVD Runge–Kutta schemes for time integration of relaxing systems. The new methods retain all the attractive features of classical relaxation schemes such as neither Riemann solvers nor characteristic decomposition are needed. Numerical experiments with the shallow‐water equations in both one and two space dimensions on flat and non‐flat topography demonstrate the high resolution and the ability of our relaxation schemes to better resolve the solution in the presence of shocks and dry areas without using either Riemann solvers or front tracking techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

A new well‐balanced Hermite weighted essentially non‐oscillatory scheme for shallow water equations

Hermite weighted essentially non‐oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high‐order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193 :115). For example, classical fifth‐order weighted essentially non‐oscillatory (WENO) reconstructions are based on a five‐cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three‐cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved. In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth‐order accurate well‐balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five‐step and fourth‐order accurate method. Besides the classical SWE, the non‐homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well‐balanced treatment of the source term involved in such equations is developed and tested. Several standard one‐dimensional test cases are used to verify the high‐order accuracy, the C‐property and the good resolution properties of the model. Copyright © 2010 John Wiley & Sons, Ltd.     

Baroclinic stability for a family of two‐level,semi‐implicit numerical methods for the 3D shallow water equations

The baroclinic stability of a family of two time‐level, semi‐implicit schemes for the 3D hydrostatic, Boussinesq Navier–Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of Casulli and Cheng (Int. J. Numer. Methods Fluids 1992; 15 :629–648), is examined in a simple 2D horizontal–vertical domain. It is demonstrated that existing mass‐conservative low‐dissipation semi‐implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two‐step predictor‐corrector schemes is proposed that have better theoretical characteristics than existing single‐step schemes. Copyright © 2006 John Wiley & Sons, Ltd.     

Selection of a staggered grid for inertia‐gravity waves in shallow water

The problem of accuracy in propagating inertia‐gravity waves on Arakawa grids is investigated. It is shown that the sole analysis of spatial discretization and the recommendation of the B‐grid for coarse resolution models and C‐grid for high resolution models must be re‐analysed when time discretization is taken into account as well. For a chosen time discretization, a coarse C‐grid is shown for example, to increase precision when using larger time‐steps (up to the stability limit) whereas the precision of the B‐grid decreases. Here, an analysis of error for different grids in function of the space–time resolutions and computational costs is presented and some recommendations on the choice of the particular staggered grid for a given application are outlined. Copyright © 2002 John Wiley & Sons, Ltd.     

Dispersion and stability analyses of the linearized two‐dimensional shallow water equations in boundary‐fitted co‐ordinates

In the present investigation, a Fourier analysis is used to study the phase and group speeds of a linearized, two‐dimensional shallow water equations, in a non‐orthogonal boundary‐fitted co‐ordinate system. The phase and group speeds for the spatially discretized equations, using the second‐order scheme in an Arakawa C grid, are calculated for grids with varying degrees of non‐orthogonality and compared with those obtained from the continuous case. The spatially discrete system is seen to be slightly dispersive, with the degree of dispersivity increasing with an decrease in the grid non‐orthogonality angle or decrease in grid resolution and this is in agreement with the conclusions reached by Sankaranarayanan and Spaulding (J. Comput. Phys., 2003; 184 : 299–320). The stability condition for the non‐orthogonal case is satisfied even when the grid non‐orthogonality angle, is as low as 30° for the Crank Nicolson and three‐time level schemes. A two‐dimensional wave deformation analysis, based on complex propagation factor developed by Leendertse (Report RM‐5294‐PR, The Rand Corp., Santa Monica, CA, 1967), is used to estimate the amplitude and phase errors of the two‐time level Crank–Nicolson scheme. There is no dissipation in the amplitude of the solution. However, the phase error is found to increase, as the grid angle decreases for a constant Courant number, and increases as Courant number increases. Copyright © 2003 John Wiley & Sons, Ltd.     

Computing non‐Newtonian fluid flow with radial basis function networks

This paper is concerned with the application of radial basis function networks (RBFNs) for solving non‐Newtonian fluid flow problems. Indirect RBFNs, which are based on an integration process, are employed to represent the solution variables; the governing differential equations are discretized by means of point collocation. To enhance numerical stability, stress‐splitting techniques are utilized. The proposed method is verified through the computation of the rectilinear and non‐rectilinear flows in a straight duct and the axisymmetric flow in an undulating tube using Newtonian, power‐law, Criminale–Ericksen–Filbey (CEF) and Oldroyd‐B models. The obtained results are in good agreement with the analytic and benchmark solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

On the practical importance of the SSP property for Runge–Kutta time integrators for some common Godunov‐type schemes

We investigate through analysis and computational experiment explicit second and third‐order strong‐stability preserving (SSP) Runge–Kutta time discretization methods in order to gain perspective on the practical necessity of the SSP property. We consider general theoretical SSP limits for these schemes and present a new optimal third‐order low‐storage SSP method that is SSP at a CFL number of 0.838. We compare results of practical preservation of the TVD property using SSP and non‐SSP time integrators to integrate a class of semi‐discrete Godunov‐type spatial discretizations. Our examples involve numerical solutions to Burgers' equation and the Euler equations. We observe that ‘well‐designed’ non‐SSP and non‐optimal SSP schemes with SSP coefficients less than one provide comparable stability when used with time steps below the standard CFL limit. Results using a third‐order non‐TVD CWENO scheme are also presented. We verify that the documented SSP methods with the number of stages greater than the order provide a useful enhanced stability region. We show by analysis and by numerical experiment that the non‐oscillatory third‐order reconstructions used in (Liu and Tadmor Numer. Math. 1998; 79 :397–425, Kurganov and Petrova Numer. Math. 2001; 88 :683–729) are in general only second‐ and first‐order accurate, respectively. Copyright © 2005 John Wiley & Sons, Ltd.     

The advantages of using high‐order finite differences schemes in laminar–turbulent transition studies

This paper presents various finite difference schemes and compare their ability to simulate instability waves in a given flow field. The governing equations for two‐dimensional, incompressible flows were solved in vorticity–velocity formulation. Four different space discretization schemes were tested, namely, a second‐order central differences, a fourth‐order central differences, a fourth‐order compact scheme and a sixth‐order compact scheme. A classic fourth‐order Runge–Kutta scheme was used in time. The influence of grid refinement in the streamwise and wall normal directions were evaluated. The results were compared with linear stability theory for the evolution of small‐amplitude Tollmien–Schlichting waves in a plane Poiseuille flow. Both the amplification rate and the wavenumber were considered as verification parameters, showing the degree of dissipation and dispersion introduced by the different numerical schemes. The results confirmed that high‐order schemes are necessary for studying hydrodynamic instability problems by direct numerical simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Some considerations for high‐order ‘incremental remap’‐based transport schemes: edges,reconstructions, and area integration

The problem of two‐dimensional tracer advection on the sphere is extremely important in modeling of geophysical fluids and has been tackled using a variety of approaches. A class of popular approaches for tracer advection include ‘incremental remap’ or cell‐integrated semi‐Lagrangian‐type schemes. These schemes achieve high‐order accuracy without the need for multistage integration in time, are capable of large time steps, and tend to be more efficient than other high‐order transport schemes when applied to a large number of tracers over a single velocity field. In this paper, the simplified flux‐form implementation of the Conservative Semi‐LAgrangian Multi‐tracer scheme (CSLAM) is reformulated using quadratic curves to approximate the upstream flux volumes and Gaussian quadrature for integrating the edge flux. The high‐order treatment of edge fluxes is motivated because of poor accuracy of the CSLAM scheme in the presence of strong nonlinear shear, such as one might observe in the midlatitudes near an atmospheric jet. Without the quadratic treatment of upstream edges, we observe at most second‐order accuracy under convergence of grid resolution, which is returned to third‐order accuracy under the improved treatment. A shallow‐water barotropic instability also reveals clear evidence of grid imprinting without the quadratic correction. Consequently, these tests reveal a problem that might arise in tracer transport near nonlinearly sheared regions of the real atmosphere, particularly near cubed‐sphere panel edges. Although CSLAM is used as the foundation for this analysis, the conclusions of this paper are applicable to the general class of incremental remap schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

In this paper, we present spectral/hp penalty least‐squares finite element formulation for the numerical solution of unsteady incompressible Navier–Stokes equations. Pressure is eliminated from Navier–Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High‐order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal‐order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space–time decoupled schemes are implemented. Second‐order accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid‐driven cavity flow at Reynolds number of 5000 and transient flow over a backward‐facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10–50). Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations

Discontinuous Galerkin (DG) methods have proven to be perfectly suited for the construction of very high‐order accurate numerical schemes on arbitrary unstructured and possibly nonconforming grids for a wide variety of applications, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods a p‐multigrid solution strategy has been developed, which is based on a semi‐implicit Runge–Kutta smoother for high‐order polynomial approximations and the implicit Backward Euler smoother for piecewise constant approximations. The effectiveness of the proposed approach is demonstrated by comparison with p‐multigrid schemes employing purely explicit smoothing operators for several 2D inviscid test cases. Copyright © 2008 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 high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.