Formal Order Of Accuracy Research Articles

On a 1D-electrostatic test problem for the PIC method

Abstract A test problem for the ‘particle-in-cell’ method is proposed which allows one to check individual errors on each stage of numerical implementation of the method. General testing usually controls only total (final) error. Using the test problem, we analyze errors of a difference method of MacCormack type and the CIC method being the most popular version of the ‘particle-in-cell’ method. It is shown that the CIC method having also the second formal order of accuracy is seriously inferior to the difference method under the same given error and same grid parameters. In particular, the results of the particle method lose their sense after approximately one calculation period, although the time interval of confidence in simulation data for the difference method is practically unlimited. The testing method proposed here is suitable for an arbitrary one-dimensional version of the ‘particle-in-cell’ method, and it allows one not only to compare numerical implementations of individual stages with each other, but also to verify available theoretical results.

A fully coupled, fully implicit simulation method for unsteady flames using Jacobian approximation and clustering

Jacobian clustering is proposed to reduce the computational cost associated with matrix operations encountered in the Newton iteration in fully coupled, fully implicit schemes for unsteady reactive flow simulations with detailed chemistry. The iterative solver is based on the Lower-Upper Symmetric Gauss-Seidel (LUSGS) algorithm and sparse matrix technique. The evaluation and sparse Lower-Upper (LU) factorization of the diagonal block of the system Jacobian are performed within clusters rather than individual cells for Computational Fluid Dynamics (CFD) simulations. Cells that are close in the state space are clustered to provide the averaged states for calculating the Jacobians of chemical source terms and transport fluxes. The cells retrieve the factorized sparse matrices from the belonging clusters to perform the necessary iterations. For the purpose of clustering, the spatial dependency of transport Jacobian in the diagonal block is eliminated. To further reduce the computational cost, the sparsity of chemical Jacobian is augmented by removing the insignificant matrix elements. The method is tested in various one-dimensional hydrocarbon flames with both the second order Crank-Nicolson scheme and a third order implicit Runge-Kutta scheme. Various chemical mechanisms with 9 to 111 species are used to test the performance of the iterative solver. Fast convergence of Newton iteration is achieved, and the formal order of accuracy is demonstrated with Jacobian clustering. The overall costs of evaluation and factorization of the block diagonal Jacobian are negligible compared to the cost of calculating transport fluxes and chemical source terms. The averaged costs of Jacobian evaluation, LU factorization and Newton iteration, all increase only linearly with the number of chemical species. The fully coupled, fully implicit Crank-Nicolson scheme with Jacobian clustering shows 4 to 42 times speedup in computational time compared to the decoupled implicit scheme with Strang operator splitting. Jacobian clustering is promising to increase the computational efficiency of high order fully coupled, fully implicit schemes for unsteady reactive flow simulations with detailed chemistry.

On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs

This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. In this way, an unconditionally stable numerical scheme is obtained, that does not suffer any CFL-type stability restriction on the maximum admissible time step. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities. A conservative scheme is then crucial to properly capture the wave propagation speed and thus to locate the discontinuity and the plateau exhibited by the solution. A novel SL technique is proposed, which is based on the integration of the governing equations over the space-time control volume which arises from the motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined with a Cauchy–Kowalevskaya procedure that provides a predictor solution within each space-time element. The one-dimensional shallow water equations (SWE) are chosen to validate the new conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly, respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative and the conservative version of the novel class of methods demonstrates that the formal order of accuracy is achieved and numerical evidences about the conservation property are shown. The AP property for the corresponding relaxation system is also investigated.

A Numerical Analysis of Six Physics‐Dynamics Coupling Schemes for Atmospheric Models

AbstractSix strategies to couple the dynamical core with physical parameterizations in atmospheric models are analyzed from a numerical perspective. Thanks to a suitably designed theoretical framework featuring a high level of abstraction, the truncation error analysis and the linear stability study are carried out under weak assumptions. Indeed, second‐order conditions are derived which are not influenced either by the specific formulation of the governing equations, nor by the number of parameterizations, nor by the structural design and implementation details of the time‐stepping methods. The theoretical findings are verified on two idealized test beds. Particularly, a hydrostatic model in isentropic coordinates is used for vertical slice simulations of a moist airflow past an isolated mountain. Self‐convergence tests show that the sensitivity of the prognostic variables to the coupling scheme may vary. For those variables (e.g., momentum) whose evolution is mainly driven by the dry dynamics, the truncation error associated with the dynamical core dominates and hides the error due to the coupling. In contrast, the coupling error of moist variables (e.g., the precipitation rate) emerges gradually as the spatio‐temporal resolution increases. Eventually, each coupling scheme tends toward the formal order of accuracy, upon a careful treatment of the grid cell condensation. Indeed, the well‐established saturation adjustment may cap the convergence rate to first order. A prognostic formulation of the condensation and evaporation process is derived from first principles. This solution is shown effective to alleviate the convergence issues in our experiments. Potential implications for a complete forecasting system are discussed.

High order central WENO-Implicit-Explicit Runge Kutta schemes for the BGK model on general polygonal meshes

In this work, a family of high order accurate Central Weighted ENO (CWENO) finite volume schemes for the solution of nonlinear kinetic equation of relaxation type is presented. After discretization of the velocity space by using a discrete ordinate approach, the space reconstruction is realized by integration over conformal arbitrary shaped closed space control volumes in a CWENO fashion. Compared to other WENO methods on unstructured meshes, in the method here presented, the total stencil size is the minimum possible and the linear weights can be arbitrarily chosen. These two aspects make their use for kinetic equations and the practical implementation on general unstructured meshes particularly interesting. The full discretization is then obtained by combining the previous phase-space approximation with an Implicit-Explicit Runge Kutta high order time discretization which guarantees stability, accuracy and preservation of the asymptotic state. In particular, to guarantee in the finite volume framework space accuracy higher than two, a new class of IMEX methods has been set into place and its properties have been studied. The formal order of accuracy is numerically measured for different regimes, computational performances of the proposed class of methods are tested on several standard two dimensional benchmark problems for kinetic equations. The novel methods are finally applied to a prototype engineering problem consisting in a supersonic flow around a NACA 0012 airfoil. In our computations we employ up to ≈325 millions of degrees of freedom and 256 GB of RAM run on 128 cores with Fortran-MPI providing evidence that the above schemes are suitable for implementation on parallel distributed memory supercomputers.

Globally constraint-preserving FR/DG scheme for Maxwell's equations at all orders

Computational electrodynamics (CED), the numerical solution of Maxwell's equations, plays an incredibly important role in several problems in science and engineering. High accuracy solutions are desired, and the discontinuous Galerkin (DG) method is one of the better ways of delivering high accuracy in numerical CED. Maxwell's equations have a pair of involution constraints for which mimetic schemes that globally satisfy the constraints at a discrete level are highly desirable. Balsara and Käppeli (2019) presented a von Neumann stability analysis of globally constraint-preserving DG schemes for CED up to fourth order. That paper was focused on developing the theory and documenting the superior dissipation and dispersion of DGTD schemes in media with constant permittivity and permeability. In this paper we present working DGTD schemes for CED that go up to fifth order of accuracy and analyze their performance when permittivity and permeability vary strongly in space.Our DGTD schemes achieve constraint preservation by collocating the electric displacement and magnetic induction as well as their higher order modes in the faces of the mesh. Our first finding is that at fourth and higher orders of accuracy, one has to evolve some zone-centered modes in addition to the face-centered modes. It is well-known that the limiting step in DG schemes causes a reduction of the optimal accuracy of the scheme; though the schemes still retain their formal order of accuracy with WENO-type limiters. In this paper, we document simulations where permittivity and permeability vary by almost an order of magnitude without requiring any limiting of the DG scheme. This very favorable second finding ensures that DGTD schemes retain optimal accuracy even in the presence of large spatial variations in permittivity and permeability. We also study the conservation of electromagnetic energy in these problems. Our third finding shows that the electromagnetic energy is conserved very well even when permittivity and permeability vary strongly in space; as long as the conductivity is zero.

A high-order Discontinuous Galerkin Method with mesh refinement for optimal control

A high-order Discontinuous Galerkin (DG) finite element time-stepping method is applied for the numerical solution of optimal control problems within the framework of Pontryagin’s Maximum Principle. The method constitutes an efficient and versatile alternative to the well-known Pseudospectral (PS) methods. The two main advantages of DG in comparison with the PS methods are: the local nature of the piecewise polynomial solution and the straightforward implementation of element-wise mesh and polynomial refinement if required. Two types of non-linear optimal control problems were analysed: continuous and bang–bang time-solutions. In the case of bang–bang optimal control problems, anh-refinement strategy was developed to achieve agreement between the observed and the formal order of accuracy. The paper also deals with sub-optimal control problems where: (i) time-step is fixed and non-infinitesimal; (ii) the control has two modes (on/off); (iii) the control command is only applied at the beginning of each time-step; and iv) the number of switching instants is large and not known a priori.

Numerical study on the convergence to steady state solutions of a new class of high order WENO schemes

A new class of high order weighted essentially non-oscillatory (WENO) schemes (Zhu and Qiu, 2016, [50]) is applied to solve Euler equations with steady state solutions. It is known that the classical WENO schemes (Jiang and Shu, 1996, [23]) might suffer from slight post-shock oscillations. Even though such post-shock oscillations are small enough in magnitude and do not visually affect the essentially non-oscillatory property, they are truly responsible for the residue to hang at a truncation error level instead of converging to machine zero. With the application of this new class of WENO schemes, such slight post-shock oscillations are essentially removed and the residue can settle down to machine zero in steady state simulations. This new class of WENO schemes uses a convex combination of a quartic polynomial with two linear polynomials on unequal size spatial stencils in one dimension and is extended to two dimensions in a dimension-by-dimension fashion. By doing so, such WENO schemes use the same information as the classical WENO schemes in Jiang and Shu (1996) [23] and yield the same formal order of accuracy in smooth regions, yet they could converge to steady state solutions with very tiny residue close to machine zero for our extensive list of test problems including shocks, contact discontinuities, rarefaction waves or their interactions, and with these complex waves passing through the boundaries of the computational domain.

Semi-implicit iterative methods for low Mach number turbulent reacting flows: Operator splitting versus approximate factorization

Two formally second-order accurate, semi-implicit, iterative methods for the solution of scalar transport–reaction equations are developed for Direct Numerical Simulation (DNS) of low Mach number turbulent reacting flows. The first is a monolithic scheme based on a linearly implicit midpoint method utilizing an approximately factorized exact Jacobian of the transport and reaction operators. The second is an operator splitting scheme based on the Strang splitting approach. The accuracy properties of these schemes, as well as their stability, cost, and the effect of chemical mechanism size on relative performance, are assessed in two one-dimensional test configurations comprising an unsteady premixed flame and an unsteady nonpremixed ignition, which have substantially different Damköhler numbers and relative stiffness of transport to chemistry. All schemes demonstrate their formal order of accuracy in the fully-coupled convergence tests. Compared to a (non-)factorized scheme with a diagonal approximation to the chemical Jacobian, the monolithic, factorized scheme using the exact chemical Jacobian is shown to be both more stable and more economical. This is due to an improved convergence rate of the iterative procedure, and the difference between the two schemes in convergence rate grows as the time step increases. The stability properties of the Strang splitting scheme are demonstrated to outpace those of Lie splitting and monolithic schemes in simulations at high Damköhler number; however, in this regime, the monolithic scheme using the approximately factorized exact Jacobian is found to be the most economical at practical CFL numbers. The performance of the schemes is further evaluated in a simulation of a three-dimensional, spatially evolving, turbulent nonpremixed planar jet flame.

On the freestream preservation of high-order conservative flux-reconstruction schemes

The appropriate procedure for constructing the symmetric conservative metric is presented with which both the freestream preservation and global conservation properties are satisfied in the high-order conservative flux-reconstruction scheme on a three-dimensional stationary-curvilinear grid. A freestream preservation test is conducted, and the symmetric conservative metric constructed by the appropriate procedure preserves the freestream regardless of the order of shape functions, while other metrics cannot always preserve the freestream. Also a convecting vortex is computed on three-dimensional wavy grids, and the formal order of accuracy is achieved when the symmetric conservative metric is appropriately constructed, while it is not when they are inappropriately constructed. In addition, although the sufficient condition for the freestream preservation with the nonconservative (cross product form) metric was reported in the previous study to be that the order of solution polynomial has to be greater than or equal to the twice of the order of a shape function, a special case is newly found in the present study: when the Radau polynomial is used for the correction function, the freestream is preserved even if the solution order is lower than the known condition. Using the properties of Legendre polynomials, the mechanism for this special case is analytically explained, considering the cancellation of aliasing errors.

Numerical assessments of high-order accurate shock capturing schemes: Kelvin–Helmholtz type vortical structures in high-resolutions

This paper investigates performance of extensions of the state-of-the-art high-resolution shock capturing schemes by solving hyperbolic conservation laws in gas dynamics. Such numerical schemes used for the integration of compressible flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, several joint solvers are developed within the framework of the reconstruction and flux-splitting approaches using the underlying MUSCL and WENO frameworks. The numerical assessments include testing and evaluation of various interpolation procedures, flux-limiters, Riemann solvers, flux-splitting schemes as well as their formal order of accuracy. A three-stage optimal TVD Runge–Kutta time stepping is employed for temporal integration. The modular development of these joint solvers provides an ease in characterizing the solution procedures. The performances of these high-resolution solvers are compared for several carefully selected two-dimensional Riemann problems including shock and rarefaction waves as well as joint discontinuities. Based on solutions obtained by all forms of five-point stencil schemes, we demonstrate that the reconstruction based WENO scheme with Roe solver is more accurate than all the versions of the flux-splitting WENO solvers tested in this study. We also show that results are highly dependent on the choice of the flux limiter. Performing benchmark quality high-resolution computations, it is shown that the Euler equations discretized by the fifth-order WENO scheme produce solutions which convect vorticity and create small-scale vortical flow structures which are usually associated with the high Reynolds number viscous flows. Surprisingly, it is found that these Kelvin–Helmholtz instability like vortical structures are not captured in any form of the third-order five-point stencil schemes.

On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes

AbstractIn this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian meshes. We compare them in terms of accuracy, performance for smooth and shocked solutions, and efficiency in CPU timing. For linear systems both schemes are high order accurate, however for nonlinear systems, analysis and numerical simulation results verify that one of them (Class A) is only second order accurate, while the other (Class B) is high order accurate. The WENO scheme in Class A is easier to implement and costs less than that in Class B. Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes, despite of the difference in their formal order of accuracy. The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.

A conservative nodal variational multiscale method for Lagrangian shock hydrodynamics

A new method based on a continuous, piece-wise linear approximation of the equations for Lagrangian shock hydrodynamics is presented. Numerical instabilities are controlled by a stabilizing operator derived using the paradigm of the variational multiscale analysis. Encouraging numerical comparisons with existing methods in the case of quadrilateral and hexahedral elements indicate that the proposed method is capable of preventing hourglass patterns in the solution, while maintaining accuracy in regions of smooth flow. The proposed approach satisfies Galilean invariance properties and hinges upon the interpretation of the Lagrangian shock hydrodynamics equations as a system of nonlinear wave equations. A specific implementation in terms of a predictor/multi-corrector version of the mid-point time integrator guarantees global conservation of mass, momentum, and total energy for each iterate. Stability and formal order of accuracy are investigated applying the von Neumann analysis to the linearized shock hydrodynamics equations in one dimension. This approach yields tight bounds for stable time-step estimation, formal second-order accuracy of the method in time and space, and valuable indications on the choice of the most appropriate values for the stabilization parameters present in the formulation.

A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

In this article, a conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method. In a first instance, piecewise polynomials of degree N are used as test functions as well as to represent the data in each element at the beginning of a time step. The time evolution of these data and the flux computation, however, are then done with a different set of piecewise polynomials of degree M ⩾ N , which are reconstructed from the underlying polynomials of degree N. This approach yields a general, unified framework that contains as two special cases classical high order finite volume (FV) schemes ( N = 0 ) as well as the usual discontinuous Galerkin (DG) method ( N = M ) . In the first case, the polynomial is reconstructed from cell averages, for the latter, the reconstruction reduces to the identity operator. A completely new class of numerical schemes is generated by choosing N ≠ 0 and M > N . The reconstruction operator is implemented for arbitrary polynomial degrees N and M on unstructured triangular and tetrahedral meshes in two and three space dimensions. To provide a high order accurate one-step time integration of the same formal order of accuracy as the spatial discretization operator, the (reconstructed) polynomial data of degree M are evolved in time locally inside each element using a new local continuous space–time Galerkin method. As a result of this approach, we obtain, as a high order accurate predictor, space–time polynomials for the vector of conserved variables and for the physical fluxes and source terms, which then can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes. This feature is particularly important for DG schemes in three space dimensions, where the cost of numerical quadrature may become prohibitively expensive for very high orders of accuracy. Numerical convergence studies of all members of the new general class of proposed schemes are shown up to sixth-order of accuracy in space and time on unstructured two- and three-dimensional meshes for two very prominent nonlinear hyperbolic systems, namely for the Euler equations of compressible gas dynamics and the equations of ideal magnetohydrodynamics (MHD). The results indicate that the new class of intermediate schemes ( N ≠ 0 , M > N ) is computationally more efficient than classical finite volume or DG schemes. Finally, a large set of interesting test cases is solved on unstructured meshes, where the proposed new time stepping approach is applied to the equations of ideal and relativistic MHD as well as to nonlinear elasticity, using a standard high order WENO finite volume discretization in space to cope with discontinuous solutions.

OPTIMIZED ARTIFICIAL DISSIPATION TERMS FOR ENHANCED STABILITY LIMITS

Finite difference approximations for the convection equation are developed, which exhibit enhanced stability limits for explicit Runge–Kutta integration. Stability limits are increased by adding artificial dissipation terms, which are optimized to yield greatest stable time steps. For the artificial dissipation terms, symmetric finite difference approximations of even-order derivatives are used with differencing stencils equal to the convective stencils. The spatial discretization inclusive of the added dissipation term is shown to be consistent with a first derivative. The formal order of accuracy in space is decreased by one order, while the order of time integration is not affected. As a result, the time step limits of originally stable Runge–Kutta integration is increased, for some combinations of spatial discretization and time integration by a factor of two. Algorithms, which are unstable without damping are stabilized. The dispersion properties of the algorithms are not influenced by the proposed damping terms. Spectral analysis of the algorithms show very low dissipation error for dimensionless wave numbers k Δ x < 0.5. Stability conditions based on von Neumann stability analysis are given for the proposed schemes for explicit Runge–Kutta time integration of orders up to ten.

Higher order weak linearizations of stochastically driven nonlinear oscillators

We present derivative-free weak and strong solutions of stochastically driven nonlinear oscillators of engineering interest using higher order forms of the locally transversal linearization (LTL) method. Unlike strong solutions, weak stochastic solutions attempt to predict only mathematical expectations of functions of the true solution and are obtainable with much less computational effort. The linearized equations corresponding to the higher order implicit LTL schemes are arrived at using backward Euler–Maruyama and Newmark expansions in order to conditionally replace nonlinear drift and multiplicative diffusion vector fields. We also briefly describe alterations through which explicit forms of such higher order linearizations are obtained. In the weak forms, the Gaussian stochastic integrals, appearing in the linearized solutions, are replaced by random variables with simpler and discrete probability distributions. The resulting local approximations to the true solution are of significantly higher formal order of accuracy, as reflected through local error estimates. Numerical illustrations are presently provided for the Duffing and Van der Pol oscillators driven by additive and multiplicative noises, which are indicative of the numerical accuracy, computational speed and algorithmic simplicity.

Finite difference approximations of first derivatives for two-dimensional grid singularities

Explicit finite difference approximations of first derivatives are developed for two-dimensional three-way and five-way grid singularities. The schemes for three-way singularities also apply for six-way singularities. The development objectives are such that the finite difference schemes are non-dissipative and compatible with regular difference approximations for convection dominated phenomena. Special emphasis is given to the directional dependency of numerical phase speeds and stability limits for explicit time integration. For that purpose, free coefficients are used to minimize the isotropy error of the numerical phase velocity. It is shown that numerical phase speeds and stability limits are almost isotropic. Hence, employment of the proposed schemes is nearly independent of the local orientation of the differencing stencil. The time step limits are greater than those of regular finite differences, indicating that the overall time step of a computation is not reduced by employing the proposed schemes. The developed schemes for three-way singularities are of second, fourth and sixth order of accuracy, whereas the five-way difference approximations are second and fourth order accurate, respectively. The spectral characteristics, the formal order of accuracy and the stability limits of the proposed schemes imply that they can favorably be used in combination with regular difference approximations of the same order of accuracy on block-structured grids.

Evaluation of geometric conservation law using pressure‐based fluid solver and moving grid technique

The geometric conservation law (GCL) is an important concept for moving grid techniques because it directly regulates the treatments of the fluid flow and grid movement. With the grid movement at every time instant, the Jacobian, associated with the volume of each element in curvilinear co‐ordinates, needs to be updated in a conservative manner. In this study, alternative GCL schemes for evaluating the Jacobian have been investigated in the context of a pressure‐based Navier‐Stokes solver, utilizing moving grid and the first‐order implicit time stepping procedure as well as the PISO scheme. GCL‐based on first and second‐order, implicit as well as time‐averaged, time integration schemes were considered. Accuracy and conservative properties were tested on steady‐state, laminar flow inside a 2D channel and time dependent, turbulent flow around a 3D elastic wing; both treated with moving grid techniques. It seems that the formal order of accuracy is not a decisive indicator. Instead, the speed of grid movement and the interplay between the flow solver and the GCL treatments make a more noticeable impact.

Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grids

AbstractThe discontinuous Galerkin (DG) and spectral volume (SV) methods are two recently developed high‐order methods for hyperbolic conservation laws capable of handling unstructured grids. In this paper, their overall performance in terms of efficiency, accuracy and memory requirement is evaluated using a 2D scalar conservation laws and the 2D Euler equations. To measure their accuracy, problems with analytical solutions are used. Both methods are also used to solve problems with strong discontinuities to test their ability in discontinuity capturing. Both the DG and SV methods are capable of achieving their formal order of accuracy while the DG method has a lower error magnitude and takes more memory. They are also similar in efficiency. The SV method appears to have a higher resolution for discontinuities because the data limiting can be done at the sub‐element level. Copyright © 2004 John Wiley & Sons, Ltd.

Finite-volume compact schemes on staggered grids

Compact finite-difference schemes have been recently used in several Direct Numerical Simulations of turbulent flows, since they can achieve high-order accuracy and high resolution without exceedingly increasing the size of the computational stencil. The development of compact finite-volume schemes is more involved, due to the appearance of surface and volume integrals. While Pereira et al. [J. Comput. Phys. 167 (2001)] and Smirnov et al. [AIAA Paper, 2546, 2001] focused on collocated grids, in this paper we use the staggered grid arrangement. Compact schemes can be tuned to achieve very high resolution for a given formal order of accuracy. We develop and test high-resolution schemes by following a procedure proposed by Lele [J. Comput. Phys. 103 (1992)] which, to the best of our knowledge, has not yet been applied to compact finite-volume methods on staggered grids. Results from several one- and two-dimensional simulations for the scalar transport and Navier–Stokes equations are presented, showing that the proposed method is capable to accurately reproduce complex steady and unsteady flows.