Residual-based Compact Scheme Research Articles

A high-order time formulation of the RBC schemes for unsteady compressible Euler equations

Residual-Based Compact (RBC) schemes can approximate the compressible Euler equations with a high space-accuracy on a very compact stencil. For instance on a 2-D Cartesian mesh, the 5th- and 7th-order accuracy can be reached on a 5×5-point stencil. The time integration of the RBC schemes uses a fully implicit method of 2nd-order accuracy (Gear method) usually solved by a dual-time approach. This method is efficient for computing compressible flows in slow unsteady regimes, but for quick unsteady flows, it may be costly and not accurate enough. A new time-formulation is proposed in the present paper. Unusually, in a RBC scheme the time derivative occurs, through linear discrete operators due to compactness, not only in the main residual but also in the other two residuals (in 2-D) involved in the numerical dissipation. To extract the time derivative, a space-factorization method which preserves the high accuracy in space is developed for reducing the algebra to the direct solution of simple linear systems on the mesh lines. Then a time-integration of high accuracy is selected for the RBC schemes by comparing the efficiency of four classes of explicit methods. The new time-formulation is validated for the diagonal advection of a Gaussian shape, the rotation of a hump, the advection of a vortex for a long time and the interaction of a vortex with a shock.

An accurate finite-volume formulation of a Residual-Based Compact scheme for unsteady compressible flows

The paper discusses the design principles of a Finite-Volume Residual-Based Compact (RBC) scheme for the spatial discretization of the unsteady compressible governing equations of gas dynamics on general structured meshes. Our goal is to develop an accurate and robust approximation methodology, well suited for complex problems of industrial interest. The scheme makes use of weighted approximations that allow to ensure high accuracy while taking benefit from the structured nature of the grid. The accuracy and Cauchy stability properties of the proposed spatial approximation are discussed in detail. Numerical applications to unsteady compressible flows of increasing complexity demonstrate the advantages of the proposed formulation with respect to straightforward extensions of RBC schemes to curvilinear meshes.

On the design of high order residual-based dissipation for unsteady compressible flows

The numerical dissipation operator of residual-based compact (RBC) schemes of high accuracy is identified and analysed for hyperbolic systems of conservation laws. A necessary and sufficient condition (χ-criterion) is found that ensures dissipation in 2-D and 3-D for any order of the RBC scheme. Numerical applications of RBC schemes of order 3, 5 and 7 to a diagonal wave advection and to a converging cylindrical shock problem confirm the theoretical results.

High-order residual-based compact schemes for aerodynamics and aeroacoustics

Residual-Based Compact (RBC) schemes are reviewed and applied to realistic compressible flow problems. The principle and advantages of high-order RBC schemes are first presented. Then, an implementation in the elsA code of the 3rd-order RBC scheme is used for RANS calculations of the transonic flow over the ONERA M-6 wing and of the flow in a CME2 compressor stage. Finally, the 7th-order RBC scheme is applied to the complete Euler equations for computing a planar aeroacoustic problem and the propagation of a spinning acoustic mode in an axisymetric aeroengine inlet. Numerical results demonstrate the accuracy and efficiency of the RBC approach.

A residual-based compact scheme of optimal order for hyperbolic problems

A new fourth-order dissipative scheme on a compact 3 × 3 stencil is presented for solving 2 D hyperbolic problems. It belongs to the family of previously developed residual-based compact schemes and can be considered as optimal since it offers the maximum achievable order of accuracy on the 3 × 3-point stencil. The computation of 2 D scalar problems demonstrates the excellent accuracy and efficiency properties offered by this new RBC scheme with respect to existing second- and third-order versions.

A residual-based scheme for computing compressible flows on unstructured grids

A second-order residual-based compact scheme, initially developed for computing flows on Cartesian and curvilinear grids, is extended to general unstructured grids in a finite-volume framework. The scheme is applied to the computation of several inviscid and viscous compressible flows governed by the Euler and Navier–Stokes equations. Its efficiency and accuracy properties are compared with those of conventional second-order upwind schemes based on variable reconstruction.

High-order residual-based compact schemes for advection–diffusion problems

A residual-based compact scheme, previously developed to compute viscous compressible flows with 2nd or 3rd-order accuracy [Lerat A, Corre C. A residual-based compact scheme for the compressible Navier–Stokes equations. J Comput Phys 2001; 170(2): 642–75], is generalized to very high-orders of accuracy. Compactness is retained since for instance a 5th-order accurate dissipative approximation of a d-dimensional advection–diffusion problem can be achieved on a 5 d stencil, without requiring the linear system solutions associated with usual compact schemes. Applications to 1D and 2D model problems are presented and demonstrate that the theoretical orders of accuracy can be achieved in practice.

High-order residual-based compact schemes for compressible inviscid flows

A residual-based compact scheme, previously developed to compute d-dimensional inviscid compressible flows with third-order accuracy on a 3 d-point stencil [Lerat A, Corre C. Residual-based compact schemes for multidimensional hyperbolic system of conservation laws. Comput Fluids 2002;31:639–61], is generalized to larger stencils allowing to reach a very high order of accuracy. Compactness is retained since for instance a seventh-order accurate dissipative approximation can be achieved on a 5 d-point stencil, without requiring the linear system solutions associated with usual compact schemes. The high-order generalization is also performed for unsteady flows. Applications to model problems and unsteady inviscid flows are presented.

A residual-based compact scheme for the unsteady compressible Navier–Stokes equations

A dissipative compact scheme is developed within a dual time stepping framework for the computation of unsteady compressible flows. The design of the scheme relies on the vanishing, at steady state with respect to the dual time, of a residual that includes the physical time-derivative. High-order accuracy and numerical dissipation are obtained in a simple way through the systematic use of derivatives of this residual. The accuracy and robustness of the approach are assessed on the simple advection of an inviscid vortex and compressible mixing layer problems involving shock waves.

A Residual-Based Compact Scheme for the Compressible Navier–Stokes Equations

A simple and efficient time-dependent method is presented for solving the steady compressible Euler and Navier–Stokes equations with third-order accuracy. Owing to its residual-based structure, the numerical scheme is compact without requiring any linear algebra, and it uses a simple numerical dissipation built on the residual. The method contains no tuning parameter. Accuracy and efficiency are demonstrated for 2-D inviscid and viscous model problems. Navier–Stokes calculations are presented for a shock/boundary layer interaction, a separated laminar flow, and a transonic turbulent flow over an airfoil.