Linear stability of WENO schemes coupled with explicit Runge–Kutta schemes

This paper focuses on the results of the linear stability analysis of the finite‐difference weighted essentially non‐oscillatory (WENO) schemes with optimal weights. The standard WENO schemes between the third and 11th order, the order‐optimised WENO schemes of the sixth and eighth order and the bandwidth‐optimised WENO schemes of the third and fourth order are considered. Several explicit Runge–Kutta schemes including the recently published strong stability‐preserving explicit Runge–Kutta schemes are considered for time discretisation. The stability limits as well as dissipation and dispersion properties dependent on the Courant–Friedrichs–Lewy number are presented for a hyperbolic model equation. The different combinations of space and time discretisation schemes are compared in terms of their accuracy and efficiency. For a parabolic model equation, the viscous term is discretised with high‐order central differences. The stability limits for the parabolic problem are presented as well. Numerical results of linear test cases are shown. Copyright © 2011 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.     

A comparative study of high‐order variable‐property segregated algorithms for unsteady low Mach number flows

In this paper, five different algorithms are presented for the simulation of low Mach flows with large temperature variations, based on second‐order central‐difference or fourth‐order compact spatial discretization and a pressure projection‐type method. A semi‐implicit three‐step Runge–Kutta/Crank–Nicolson or second‐order iterative scheme is used for time integration. The different algorithms solve the coupled set of governing scalar equations in a decoupled segregate manner. In the first algorithm, a temperature equation is solved and density is calculated from the equation of state, while the second algorithm advances the density using the differential form of the equation of state. The third algorithm solves the continuity equation and the fourth algorithm solves both the continuity and enthalpy equation in conservative form. An iterative decoupled algorithm is also proposed, which allows the computation of the fully coupled solution. All five algorithms solve the momentum equation in conservative form and use a constant‐ or variable‐coefficient Poisson equation for the pressure. The efficiency of the fourth‐order compact scheme and the performances of the decoupling algorithms are demonstrated in three flow problems with large temperature variations: non‐Boussinesq natural convection, channel flow instability, flame–vortex interaction. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical solution of the incompressible Navier–Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered‐mesh grid system. The convection terms are discretized with a fifth‐order‐accurate upwind compact difference approximation, the viscous terms are discretized with a sixth‐order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth‐order difference approximation on a cell‐centered mesh. Time advancement uses a three‐stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Two‐dimensional compact finite difference immersed boundary method

We present a compact finite differences method for the calculation of two‐dimensional viscous flows in biological fluid dynamics applications. This is achieved by using body‐forces that allow for the imposition of boundary conditions in an immersed moving boundary that does not coincide with the computational grid. The unsteady, incompressible Navier–Stokes equations are solved in a Cartesian staggered grid with fourth‐order Runge–Kutta temporal discretization and fourth‐order compact schemes for spatial discretization, used to achieve highly accurate calculations. Special attention is given to the interpolation schemes on the boundary of the immersed body. The accuracy of the immersed boundary solver is verified through grid convergence studies. Validation of the method is done by comparison with reference experimental results. In order to demonstrate the application of the method, 2D small insect hovering flight is calculated and compared with available experimental and computational results. Copyright © 2009 John Wiley & Sons, Ltd.     

Fifth‐order Hermitian schemes for computational linear aeroacoustics

We develop a class of fifth‐order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth‐order accuracy. A one‐dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy–Kovaleskaya (the ‘Δ‐P5 scheme’); the second one relies on a semi‐discrete form and evolves in time the discrete unknowns by using a five‐stage Runge–Kutta method (the ‘RGK‐P5 scheme’). Although the RGK‐P5 scheme shares the same local spatial interpolator with the Δ‐P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi‐dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210 :133–170; J. Comput. Phys. 2006; 217 :530–562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the ‘reconstruction stage’, we define a fifth‐order local spatial interpolator based on an upwind stencil. In the ‘decomposition stage’, we decompose the time derivatives into simple wave contributions. In the ‘evolution stage’, we use these fluctuations to update either by a Cauchy–Kovaleskaya procedure or by a five‐stage Runge–Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the ‘evolution stage’, two fifth‐order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK‐P5 appears as the best compromise between simplicity and accuracy, while the Δ‐P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright © 2007 John Wiley & Sons, Ltd.     

Combined compact difference method for solving the incompressible Navier–Stokes equations

This paper presents a numerical method for solving the two‐dimensional unsteady incompressible Navier–Stokes equations in a vorticity–velocity formulation. The method is applicable for simulating the nonlinear wave interaction in a two‐dimensional boundary layer flow. It is based on combined compact difference schemes of up to 12th order for discretization of the spatial derivatives on equidistant grids and a fourth‐order five‐ to six‐alternating‐stage Runge–Kutta method for temporal integration. The spatial and temporal schemes are optimized together for the first derivative in a downstream direction to achieve a better spectral resolution. In this method, the dispersion and dissipation errors have been minimized to simulate physical waves accurately. At the same time, the schemes can efficiently suppress numerical grid‐mesh oscillations. The results of test calculations on coarse grids are in good agreement with the linear stability theory and comparable with other works. The accuracy and the efficiency of the current code indicate its potential to be extended to three‐dimensional cases in which full boundary layer transition happens. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of central differencing and low‐dissipation weights in a weighted compact nonlinear scheme

This paper proposes WCNS‐CU‐Z, a weighted compact nonlinear scheme, that incorporates adapted central difference and low‐dissipative weights together with concepts of the adaptive central‐upwind sixth‐order weighted essentially non‐oscillatory scheme (WENO‐CU) and WENO‐Z schemes. The newly developed WCNS‐CU‐Z is a high‐resolution scheme, because interpolation of this scheme employs a central stencil constructed by upwind and downwind stencils. The smoothness indicator of the downwind stencil is calculated using the entire central stencil, and the downwind stencil is stopped around the discontinuity for stability. Moreover, interpolation of the sixth‐order WCNS‐CU‐Z exhibits sufficient accuracy in the smooth region through use of low‐dissipative weights. The sixth‐order WCNS‐CU‐Zs are implemented with a robust linear difference formulation (R‐WCNS‐CU6‐Z), and the resolution and robustness of this scheme were evaluated. These evaluations showed that R‐WCNS‐CU6‐Z is capable of achieving a higher resolution than the seventh‐order classical robust weighted compact nonlinear scheme and can provide a crisp result in terms of discontinuity. Among the schemes tested, R‐WCNS‐CU6‐Z has been shown to be robust, and variable interpolation type R‐WCNS‐CU6‐Z (R‐WCNS‐CU6‐Z‐V) provides a stable computation by modifying the first‐order interpolation when negative density or negative pressure arises after nonlinear interpolation. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical assessment of a shock‐detecting sensor for low dissipative high‐order simulation of shock–vortex interactions

Considering the importance of high‐order schemes implementation for the simulation of shock‐containing turbulent flows, the present work involves the assessment of a shock‐detecting sensor for filtering of high‐order compact finite‐difference schemes for simulation of this type of flows. To accomplish this, a sensor that controls the amount of numerical dissipation is applied to a sixth‐order compact scheme as well as a fourth‐order two‐register Runge–Kutta method for numerical simulation of various cases including inviscid and viscous shock–vortex and shock–mixing‐layer interactions. Detailed study is performed to investigate the performance of the sensor, that is, the effect of control parameters employed in the sensor are investigated in the long‐time integration. In addition, the effects of nonlinear weighting factors controlling the value of the second‐order and high‐order filters in fine and coarse non‐uniform grids are investigated. The results indicate the accuracy of the nonlinear filter along with the promising performance of the shock‐detecting sensor, which would pave the way for future simulations of turbulent flows containing shocks. Copyright © 2014 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.     

Flux‐difference splitting‐based upwind compact schemes for the incompressible Navier–Stokes equations

Third‐order and fifth‐order upwind compact finite difference schemes based on flux‐difference splitting are proposed for solving the incompressible Navier–Stokes equations in conjunction with the artificial compressibility (AC) method. Since the governing equations in the AC method are hyperbolic, flux‐difference splitting (FDS) originally developed for the compressible Euler equations can be used. In the present upwind compact schemes, the split derivatives for the convective terms at grid points are linked to the differences of split fluxes between neighboring grid points, and these differences are computed by using FDS. The viscous terms are approximated with a sixth‐order central compact scheme. Comparisons with 2D benchmark solutions demonstrate that the present compact schemes are simple, efficient, and high‐order accurate. Copyright © 2008 John Wiley & Sons, Ltd.     

Fourth‐ and tenth‐order compact finite difference solutions of perturbed circular vortex flows

In this study, high‐order compact finite difference calculations are reported for 2D unsteady incompressible circular vortex flow in primitive variable formulation. The fourth‐order Runge–Kutta temporal discretization is used together with fourth‐ or tenth‐order compact spatial discretization. Dependent on the perturbation initially imposed, the solutions display a tripole, triangular or square vortex. The comparison of the predictions with the detailed spectral calculations of Kloosterziel and Carnevale (J. Fluid Mech. 1999; 388 :217–257) shows that the vorticity fields are very well captured. The spectral resolution of the present method was quantified from the decomposition of the vorticity distribution in its azimuthal components and compared with reported spectral results. Using identical grid resolution to the reference results yields negligible differences in the main features of the flow. The perturbation amplitude and its first harmonic are virtually identical to the reference results for both fourth‐ or tenth‐order spatial discretization, as theoretically expected but seldom a posteriori verified. The differences between the two spatial discretizations appear only for coarser grids, favouring the tenth‐order discretization. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

High‐order compact finite difference schemes for the vorticity–divergence representation of the spherical shallow water equations

This work is devoted to the application of the super compact finite difference method (SCFDM) and the combined compact finite difference method (CCFDM) for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence, and height. The fourth‐order compact, the sixth‐order and eighth‐order SCFDM, and the sixth‐order and eighth‐order CCFDM schemes are used for the spatial differencing. To advance the solution in time, a semi‐implicit Runge–Kutta method is used. In addition, to control the nonlinear instability, an eighth‐order compact spatial filter is employed. For the numerical solution of the elliptic equations in the problem, a direct hybrid method, which consists of a high‐order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate, is utilized. The accuracy and convergence rate for all methods are verified against exact analytical solutions. Qualitative and quantitative assessments of the results for an unstable barotropic mid‐latitude zonal jet employed as an initial condition are addressed. It is revealed that the sixth‐order and eighth‐order CCFDMs and SCFDMs lead to a remarkable improvement of the solution over the fourth‐order compact method. Copyright © 2015 John Wiley & Sons, Ltd.     

On the discontinuous Galerkin computation of N‐waves focusing

Sonic boom focusing phenomenon can be predicted using the solution to the nonlinear Tricomi equation which is a hybrid (hyperbolic‐elliptic) second‐order partial differential equation. In this paper, the hyperbolic conservation law form is derived, which is valid in the entire domain. In this manner, the presence of two regions where the equation behaves differently (hyperbolic in the upper and elliptic in the lower half‐plane) is avoided. On the upper boundary, a new mixed boundary condition for the acoustic pressure is employed. The discretization is carried out using a discontinuous Galerkin (DG) method combined with a Runge–Kutta total‐variation diminishing scheme. The results show the accuracy of DG methods to solve problems involving sharp gradients and discontinuities. Comparisons with analytical results for the linear case, and other numerical results using classical explicit and compact finite difference schemes and weighted essentially non‐oscillatory schemes are included. Copyright © 2010 John Wiley & Sons, Ltd.     

Application of point implicit Runge–Kutta methods to inviscid and laminar flow problems using AUSM and AUSM+ upwinding

Explicit Runge–Kutta methods preconditioned by a pointwise matrix valued preconditioner can significantly improve the convergence rate to approximate steady state solutions of laminar flows. This has been shown for central discretisation schemes and Roe upwinding. Since the first-order approximation to the inviscid flux assuming constant weighting of the dissipative terms is given by the absolute value of the Roe matrix, the construction of the preconditioner is rather simple compared to other upwind techniques. However, in this article we show that similar improvements in the convergence rates can also be obtained for the AUSM⁺ scheme. Following the ideas for the central and Roe schemes, the preconditioner is obtained by a first-order approximation to the derivative of the convective flux. Viscous terms are included into the preconditioner considering a thin shear layer approximation. A complete derivation of the derivative terms is shown. In numerical examples, we demonstrate the improved convergence rates when compared with a standard explicit Runge–Kutta method accelerated with local time stepping.     

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.     

Higher‐order‐accurate numerical method for temporal stability simulations of Rayleigh‐Bénard‐Poiseuille flows

For Rayleigh‐Bénard‐Poiseuille flows, thermal stratification resulting from a wall‐normal temperature gradient together with an opposing gravitational field can lead to buoyancy‐driven instability. Moreover, for sufficiently large Reynolds numbers, viscosity‐driven instability can occur. Two higher‐order‐accurate methods based on the full and linearized Navier‐Stokes equations were developed for investigating the temporal stability of such flows. The new methods employ a spectral discretization in the homogeneous directions. In the wall‐normal direction, the convective and viscous terms are discretized with fifth‐order‐accurate biased and fourth‐order‐accurate central compact finite differences. A fourth‐order‐accurate explicit Runge‐Kutta method is employed for time integration. To validate the methods, the primary instability was investigated for different combinations of the Reynolds and Rayleigh number. The results from these primary stability investigations are consistent with linear stability theory results from the literature with respect to both the onset of the instability and the dependence of the temporal growth rate on the wave angle. For the cases with buoyancy‐driven instability, strong linear growth is observed for a broad range of spanwise wavenumbers. The largest growth rates are obtained for a wave angle of 90°. For the cases with viscosity‐driven instability, the linear growth rates are lower and the first mode to experience nonlinear growth is a higher harmonic with half the wavelength of the fundamental.     

From h to p efficiently: optimal implementation strategies for explicit time‐dependent problems using the spectral/hp element method

We investigate the relative performance of a second‐order Adams–Bashforth scheme and second‐order and fourth‐order Runge–Kutta schemes when time stepping a 2D linear advection problem discretised using a spectral/hp element technique for a range of different mesh sizes and polynomial orders. Numerical experiments explore the effects of short (two wavelengths) and long (32 wavelengths) time integration for sets of uniform and non‐uniform meshes. The choice of time‐integration scheme and discretisation together fixes a CFL limit that imposes a restriction on the maximum time step, which can be taken to ensure numerical stability. The number of steps, together with the order of the scheme, affects not only the runtime but also the accuracy of the solution. Through numerical experiments, we systematically highlight the relative effects of spatial resolution and choice of time integration on performance and provide general guidelines on how best to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role played by higher polynomial orders in reducing CPU time while preserving accuracy becomes more evident, especially for uniform meshes, compared with what has been typically considered when studying this type of problem.© 2014. The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons, Ltd.     

Semi‐analytical method for departure point determination

A new method for departure point determination on Cartesian grids, the semi‐analytical upwind path line tracing (SUT) method, is presented and compared to two typical departure point determination methods used in semi‐Lagrangian advection schemes, the Euler method and the four‐step Runge–Kutta method. Rigorous comparisons of the three methods were conducted for a severely curving hypothetical flow field and for advective transport in the rotation of a Gaussian concentration hill. The SUT method was shown to have equivalent accuracy to the Runge–Kutta method but with significantly improved computational efficiency. Depending on the case being simulated, the SUT method provides either far greater or equivalent computational efficiency and more certain accuracy than the Euler method. Copyright © 2004 John Wiley & Sons, Ltd.