Towards the solution reconstruction,one of the main steps in Godunov type finite volume scheme,a class of integrated linear reconstruction(ILR)methods has been developed recently,from which the advantages such as para...Towards the solution reconstruction,one of the main steps in Godunov type finite volume scheme,a class of integrated linear reconstruction(ILR)methods has been developed recently,from which the advantages such as parameters free and maximum principle preserving can be observed.It is noted that only time-dependent problems are considered in the previous study on ILR,while the steady state problems play an important role in applications such as optimal design of vehicle shape.In this paper,focusing on the steady Euler equations,we will extend the study of ILR to the steady state problems.The numerical framework to solve the steady Euler equations consists of a Newton iteration for the linearization,and a geometric multigrid solver for the derived linear system.It is found that even for a shock free problem,the convergence of residual towards the machine precision can not be obtained by directly using the ILR.With the lack of the differentiability of reconstructed solution as a partial explanation,a simple Laplacian smoothing procedure is introduced in the method as a postprocessing technique,which dramatically improves the convergence to steady state.To prevent the numerical oscillations around the discontinuity,an efficient WENO reconstruction based on secondary reconstruction is employed.It is shown that the extra two operations for ILR are very efficient.Several numerical examples are presented to show the effectiveness of the proposed scheme for the steady state problems.展开更多
In this paper,a class of compact higher-order gas-kinetic schemes(GKS)with spectrallike resolution will be presented.Based on the high-order gas evolution model,both the flux function and conservative flow variables ...In this paper,a class of compact higher-order gas-kinetic schemes(GKS)with spectrallike resolution will be presented.Based on the high-order gas evolution model,both the flux function and conservative flow variables in GKS can be evaluated explicitly from the time-accurate gas distribution function at a cell interface.As a result,inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step.The flow variable update and slope update are coming from the same physical solution at the cell interface.This strategy needs time accurate solution at a cell interface,which cannot be achieved by the Riemann problem based flow solvers,even though they can also provide the interface flux functions and interface flow variables.Instead,in order to update the slopes in the Riemann-solver based schemes,such as HWENO,there are additional governing equations for slopes or equivalent degrees of freedom inside each cell.In GKS,only a single time accurate gas evolution model is needed at the cell interface for updating cell averaged flow variables through interface fluxes and updating the cell averaged slopes through the interface flow variables.Based on both cell averaged values and their slopes,compact 6th-order and 8th-order linear and nonlinear reconstructions can be developed.As analyzed in this paper,the local linear compact reconstruction without limiter can achieve a spectral-like resolution at large wavenumber than the well-established compact scheme of Lele with globally coupled flow variables and their derivatives.For nonlinear gas dynamic evolution,in order to avoid spurious oscillation in discontinuous region,the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction.Consequently discontinuous solutions can be captured through the 6th-order and 8th-order compact WENO-type nonlinear reconstruction.In GKS,the time evolution solution of the gas distribution function at a cell interface is based on an integral solution of the kinetic model equation,which covers a physical process from an initial non-equilibrium state to a final equilibrium one.Since the initial non-equilibrium state is obtained based on the nonlinear WENO-Z reconstruction,and the equilibrium state is basically constructed from the linear symmetric reconstruction,the GKS evolution models unifies the nonlinear and linear reconstructions in a gas relaxation process in the determination of a time-dependent gas distribution function.This property gives GKS great advantages in capturing both discontinuous shock waves and the linear aero-acoustic waves in a single computation due to its dynamical adaptation of non-equilibrium and equilibrium states in different flow regions.This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions.Compared with discontinuous Galerkin(DG)scheme,the current compact GKS uses the same local and compact stencil,achieves the 6th-order and 8th-order accuracy,uses a much larger time step with CFL number≥0.3,has the robustness as a 2nd-order scheme,and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process.The nonlinear reconstruction in the compact GKS is solely based on the WENO-Z technique.At the same time,the current scheme solves the Navier-Stokes equations automatically due to combined inviscid and viscous flux terms from a single time evolution gas distribution function at a cell interface.Due to the use of multi-stage multi-derivative(MSMD)time-stepping technique,for achieving a 4th-order time accuracy,the current scheme uses only two stages instead of four in the traditional Runge-Kutta method.As a result,the current GKS becomes much more efficient than the corresponding same order DG method.A variety of numerical tests are presented to validate the compact 6th and 8th-order GKS.The current scheme presents a state-of-art numerical solutions under a wide range of flow conditions,i.e.,strong shock discontinuity,shear instability,aero-acoustic wave propagation,and NS solutions.It promotes the development of high-order scheme to a new level of maturity.The success of the current scheme crucially depends on the high-order gas evolution model,which cannot be achieved by any other approach once the 1st-order Riemann flux function is still used in the development of high-order numerical algorithms.展开更多
基金supported by FDCT of the Macao S.A.R.(0082/2020/A2)National Natural Science Foundation of China(Nos.11922120 and 11871489)+4 种基金the Multi-Year Research Grant(No.2020-00265-FST)of University of Macao,and a grant from Department of Science and Technology of Guangdong Province(No.2020B1212030001)The research of Ruo Li was supported by the National Key R&D Program of China,Project Number 2020YFA0712000.The research of Xucheng Meng is partially supported by the National Natural Science Foundation of China(No.12101057)the Scientific Research Fund of Beijing Normal University(No.28704-111032105)the Start-up Research Fund from BNU-HKBU United International College(No.R72021112)。
文摘Towards the solution reconstruction,one of the main steps in Godunov type finite volume scheme,a class of integrated linear reconstruction(ILR)methods has been developed recently,from which the advantages such as parameters free and maximum principle preserving can be observed.It is noted that only time-dependent problems are considered in the previous study on ILR,while the steady state problems play an important role in applications such as optimal design of vehicle shape.In this paper,focusing on the steady Euler equations,we will extend the study of ILR to the steady state problems.The numerical framework to solve the steady Euler equations consists of a Newton iteration for the linearization,and a geometric multigrid solver for the derived linear system.It is found that even for a shock free problem,the convergence of residual towards the machine precision can not be obtained by directly using the ILR.With the lack of the differentiability of reconstructed solution as a partial explanation,a simple Laplacian smoothing procedure is introduced in the method as a postprocessing technique,which dramatically improves the convergence to steady state.To prevent the numerical oscillations around the discontinuity,an efficient WENO reconstruction based on secondary reconstruction is employed.It is shown that the extra two operations for ILR are very efficient.Several numerical examples are presented to show the effectiveness of the proposed scheme for the steady state problems.
基金The current research is supported by Hong Kong research grant council(16206617)and National Science Foundation of China(11772281,91852114,11701038).
文摘In this paper,a class of compact higher-order gas-kinetic schemes(GKS)with spectrallike resolution will be presented.Based on the high-order gas evolution model,both the flux function and conservative flow variables in GKS can be evaluated explicitly from the time-accurate gas distribution function at a cell interface.As a result,inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step.The flow variable update and slope update are coming from the same physical solution at the cell interface.This strategy needs time accurate solution at a cell interface,which cannot be achieved by the Riemann problem based flow solvers,even though they can also provide the interface flux functions and interface flow variables.Instead,in order to update the slopes in the Riemann-solver based schemes,such as HWENO,there are additional governing equations for slopes or equivalent degrees of freedom inside each cell.In GKS,only a single time accurate gas evolution model is needed at the cell interface for updating cell averaged flow variables through interface fluxes and updating the cell averaged slopes through the interface flow variables.Based on both cell averaged values and their slopes,compact 6th-order and 8th-order linear and nonlinear reconstructions can be developed.As analyzed in this paper,the local linear compact reconstruction without limiter can achieve a spectral-like resolution at large wavenumber than the well-established compact scheme of Lele with globally coupled flow variables and their derivatives.For nonlinear gas dynamic evolution,in order to avoid spurious oscillation in discontinuous region,the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction.Consequently discontinuous solutions can be captured through the 6th-order and 8th-order compact WENO-type nonlinear reconstruction.In GKS,the time evolution solution of the gas distribution function at a cell interface is based on an integral solution of the kinetic model equation,which covers a physical process from an initial non-equilibrium state to a final equilibrium one.Since the initial non-equilibrium state is obtained based on the nonlinear WENO-Z reconstruction,and the equilibrium state is basically constructed from the linear symmetric reconstruction,the GKS evolution models unifies the nonlinear and linear reconstructions in a gas relaxation process in the determination of a time-dependent gas distribution function.This property gives GKS great advantages in capturing both discontinuous shock waves and the linear aero-acoustic waves in a single computation due to its dynamical adaptation of non-equilibrium and equilibrium states in different flow regions.This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions.Compared with discontinuous Galerkin(DG)scheme,the current compact GKS uses the same local and compact stencil,achieves the 6th-order and 8th-order accuracy,uses a much larger time step with CFL number≥0.3,has the robustness as a 2nd-order scheme,and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process.The nonlinear reconstruction in the compact GKS is solely based on the WENO-Z technique.At the same time,the current scheme solves the Navier-Stokes equations automatically due to combined inviscid and viscous flux terms from a single time evolution gas distribution function at a cell interface.Due to the use of multi-stage multi-derivative(MSMD)time-stepping technique,for achieving a 4th-order time accuracy,the current scheme uses only two stages instead of four in the traditional Runge-Kutta method.As a result,the current GKS becomes much more efficient than the corresponding same order DG method.A variety of numerical tests are presented to validate the compact 6th and 8th-order GKS.The current scheme presents a state-of-art numerical solutions under a wide range of flow conditions,i.e.,strong shock discontinuity,shear instability,aero-acoustic wave propagation,and NS solutions.It promotes the development of high-order scheme to a new level of maturity.The success of the current scheme crucially depends on the high-order gas evolution model,which cannot be achieved by any other approach once the 1st-order Riemann flux function is still used in the development of high-order numerical algorithms.
