We present a computational gas dynamics method based on the Spectral Deferred Corrections (SDC) time integration technique and the Piecewise Parabolic Method (PPM) finite volume method. The PPM framework is used to de...We present a computational gas dynamics method based on the Spectral Deferred Corrections (SDC) time integration technique and the Piecewise Parabolic Method (PPM) finite volume method. The PPM framework is used to define edge-averaged quantities, which are then used to evaluate numerical flux functions. The SDC technique is used to integrate solution in time. This kind of approach was first taken by Anita et al in [1]. However, [1] is problematic when it is implemented to certain shock problems. Here we propose significant improvements to [1]. The method is fourth order (both in space and time) for smooth flows, and provides highly resolved discontinuous solutions. We tested the method by solving variety of problems. Results indicate that the fourth order of accuracy in both space and time has been achieved when the flow is smooth. Results also demonstrate the shock capturing ability of the method.展开更多
In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-depe...In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.展开更多
In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time ...In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.展开更多
Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple the...Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.展开更多
This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start wi...This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.展开更多
In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th...In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th order negative-norm estimate of the semi- discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction （SDC） method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of （O（hk＋1） in L2 norm and improve the LDG solution from （O（hk＋1） to （O（h2k＋1） with the accuracy enhancement post-processing technique.展开更多
In this paper,we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spec...In this paper,we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods.In both of the two high order semi-implicit time integration methods,the convective flux is treated explicitly and the viscous and heat fluxes are treated implicitly.The remarkable benefits of such semi-implicit temporal discretizations are that they can not only overcome the stringent time step restriction compared with time explicit methods,but also avoid the construction of the large Jacobian matrix as is done for fully implicit methods,thus are relatively easy to implement.To save computing time as well as capture the flow structures of interest accurately,a local mesh refinement(h-adaptive)technique,in which we present detailed criteria for selecting candidate elements and complete strategies to refine and coarsen them,is also applied for the Navier-Stokes equations.Numerical experiments are provided to illustrate the high order accuracy,efficiency and capabilities of the semi-implicit schemes in combination with adaptive local discontinuous Galerkin methods for the Navier-Stokes equations.展开更多
In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)...In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.展开更多
In this paper,we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks.Optimal error estimates for the density and filament polarization in different norms a...In this paper,we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks.Optimal error estimates for the density and filament polarization in different norms are established.We use a semi-implicit spectral deferred correction time method for time discretization,which allows a relative large time step and avoids computation of a Jacobian matrix.Numerical experiments are presented to verify the theoretical analysis and to show the capability for simulations of action wave formation.展开更多
文摘We present a computational gas dynamics method based on the Spectral Deferred Corrections (SDC) time integration technique and the Piecewise Parabolic Method (PPM) finite volume method. The PPM framework is used to define edge-averaged quantities, which are then used to evaluate numerical flux functions. The SDC technique is used to integrate solution in time. This kind of approach was first taken by Anita et al in [1]. However, [1] is problematic when it is implemented to certain shock problems. Here we propose significant improvements to [1]. The method is fourth order (both in space and time) for smooth flows, and provides highly resolved discontinuous solutions. We tested the method by solving variety of problems. Results indicate that the fourth order of accuracy in both space and time has been achieved when the flow is smooth. Results also demonstrate the shock capturing ability of the method.
文摘In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.
基金supported by NSFC(Grant No.11601490).Research of Y.Xu is supported by NSFC(Grant No.12071455).
文摘In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.
基金supported by the NSFC Grant no.12271492the Natural Science Foundation of Henan Province of China Grant no.222300420550+1 种基金supported by the NSFC Grant no.12271498the National Key R&D Program of China Grant no.2022YFA1005202/2022YFA1005200.
文摘Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.
基金Research of R.Guo is supported by NSFC grant No.11601490Research of Y.Xu is supported by NSFC grant No.11722112,91630207.
文摘This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.
文摘In this paper, we present a local discontinuous Galerkin （LDG） method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k ＋ 1 in L2 norm and present the （2k＋1）-th order negative-norm estimate of the semi- discrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction （SDC） method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of （O（hk＋1） in L2 norm and improve the LDG solution from （O（hk＋1） to （O（h2k＋1） with the accuracy enhancement post-processing technique.
基金National Numerical Windtunnel Project NNW2019ZT4-B08, NSFC grant No. 12071455.
文摘In this paper,we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods.In both of the two high order semi-implicit time integration methods,the convective flux is treated explicitly and the viscous and heat fluxes are treated implicitly.The remarkable benefits of such semi-implicit temporal discretizations are that they can not only overcome the stringent time step restriction compared with time explicit methods,but also avoid the construction of the large Jacobian matrix as is done for fully implicit methods,thus are relatively easy to implement.To save computing time as well as capture the flow structures of interest accurately,a local mesh refinement(h-adaptive)technique,in which we present detailed criteria for selecting candidate elements and complete strategies to refine and coarsen them,is also applied for the Navier-Stokes equations.Numerical experiments are provided to illustrate the high order accuracy,efficiency and capabilities of the semi-implicit schemes in combination with adaptive local discontinuous Galerkin methods for the Navier-Stokes equations.
基金Research of R.Guo is supported by NSFC grant No.11601490Research of Y.Xu is supported by NSFC grant No.11371342,11626253,91630207.
文摘In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.
基金supported by National Natural Science Foundation of China(Grant Nos.11801569 and 11571367)Natural Science Foundation of Shandong Province(CN)(Grant Nos.ZR2018BA011 and ZR2019MA015)+1 种基金the Fundamental Research Funds for the Central Universities(Grant Nos.18CX02021A and 18CX05003A)National Science Foundation of USA(Grant No.DMS-1818467).
文摘In this paper,we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks.Optimal error estimates for the density and filament polarization in different norms are established.We use a semi-implicit spectral deferred correction time method for time discretization,which allows a relative large time step and avoids computation of a Jacobian matrix.Numerical experiments are presented to verify the theoretical analysis and to show the capability for simulations of action wave formation.
