Considering droplet phenomena at low Mach numbers,large differences in the magnitude of the occurring characteristic waves are presented.As acoustic phenomena often play a minor role in such applications,classical exp...Considering droplet phenomena at low Mach numbers,large differences in the magnitude of the occurring characteristic waves are presented.As acoustic phenomena often play a minor role in such applications,classical explicit schemes which resolve these waves suffer from a very restrictive timestep restriction.In this work,a novel scheme based on a specific level set ghost fluid method and an implicit-explicit(IMEX)flux splitting is proposed to overcome this timestep restriction.A fully implicit narrow band around the sharp phase interface is combined with a splitting of the convective and acoustic phenomena away from the interface.In this part of the domain,the IMEX Runge-Kutta time discretization and the high order discontinuous Galerkin spectral element method are applied to achieve high accuracies in the bulk phases.It is shown that for low Mach numbers a significant gain in computational time can be achieved compared to a fully explicit method.Applica-tions to typical droplet dynamic phenomena validate the proposed method and illustrate its capabilities.展开更多
We investigate strong stability preserving(SSP)implicit-explicit(IMEX)methods for partitioned systems of differential equations with stiff and nonstiff subsystems.Conditions for order p and stage order q=p are derived...We investigate strong stability preserving(SSP)implicit-explicit(IMEX)methods for partitioned systems of differential equations with stiff and nonstiff subsystems.Conditions for order p and stage order q=p are derived,and characterization of SSP IMEX methods is provided following the recent work by Spijker.Stability properties of these methods with respect to the decoupled linear system with a complex parameter,and a coupled linear system with real parameters are also investigated.Examples of methods up to the order p=4 and stage order q—p are provided.Numerical examples on six partitioned test systems confirm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration,and they are also suitable for preserving the accuracy in the stiff limit or preserving the positivity of the numerical solution for large stepsizes.展开更多
A fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts(upwind gSBP)schemes in space and implicit-explicit Runge-Kutta(IME...A fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts(upwind gSBP)schemes in space and implicit-explicit Runge-Kutta(IMEX-RK)schemes in time.Hereby,advection terms are discretized explicitly,while diffusion terms are solved implicitly.In this context,specific combinations of space and time discretizations enjoy enhanced stability properties.In fact,if the first-and second-derivative upwind gSBP operators fulfill a compatibility condition,the allowable time step size is independent of grid refinement,although the advective terms are discretized explicitly.In one space dimension it is shown that upwind gSBP schemes represent a general framework including standard discontinuous Galerkin(DG)schemes on a global level.While previous work for DG schemes has demonstrated that the combination of upwind advection fluxes and the central-type first Bassi-Rebay(BR1)scheme for diffusion does not allow for grid-independent stable time steps,the current work shows that central advection fluxes are compatible with BR1 regarding enhanced stability of IMEX time stepping.Furthermore,unlike previous discrete energy stability investigations for DG schemes,the present analysis is based on the discrete energy provided by the corresponding SBP norm matrix and yields time step restrictions independent of the discretization order in space,since no finite-element-type inverse constants are involved.Numerical experiments are provided confirming these theoretical findings.展开更多
In this paper,we introduce an extension of a splitting method for singularly perturbed equations,the socalled RS-IMEX splitting[Kaiser et al.,Journal of Scientific Computing,70(3),1390–1407],to deal with the fully co...In this paper,we introduce an extension of a splitting method for singularly perturbed equations,the socalled RS-IMEX splitting[Kaiser et al.,Journal of Scientific Computing,70(3),1390–1407],to deal with the fully compressible Euler equations.The straightforward application of the splitting yields sub-equations that are,due to the occurrence of complex eigenvalues,not hyperbolic.A modification,slightly changing the convective flux,is introduced that overcomes this issue.It is shown that the splitting gives rise to a discretization that respects the low-Mach number limit of the Euler equations;numerical results using finite volume and discontinuous Galerkin schemes show the potential of the discretization.展开更多
In this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains,taking Hermite functions,sinc functions,and rational Chebyshev...In this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains,taking Hermite functions,sinc functions,and rational Chebyshev polynomials as basis functions.The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme,being of particular interest the treatment of three-dimensional Sylvester equations that we make.The resulting method is easy to understand and express,and can be implemented in a transparent way by means of a few lines of code.We test numerically the three choices of basis functions,showing the convenience of this new approach,especially when rational Chebyshev polynomials are considered.展开更多
High-order discretizations of partial differential equations(PDEs)necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner.Implicit-explicit(IMEX)int...High-order discretizations of partial differential equations(PDEs)necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner.Implicit-explicit(IMEX)integration based on general linear methods(GLMs)offers an attractive solution due to their high stage and method order,as well as excellent stability properties.The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly.This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel.The first approach is based on diagonally implicit multi-stage integration methods(DIMSIMs)of types 3 and 4.The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy.Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge-Kutta methods.展开更多
This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either...This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.展开更多
Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been dev...Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit （VSIMEX） linear multistep methods for time-dependent PDEs. Families of order-p, pstep VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers＇ equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.展开更多
The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,wh...The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.展开更多
We have introduced a fully second order IMplicit/EXplicit(IMEX)time integration technique for solving the compressible Euler equations plus nonlinear heat conduction problems(also known as the radiation hydrodynamics ...We have introduced a fully second order IMplicit/EXplicit(IMEX)time integration technique for solving the compressible Euler equations plus nonlinear heat conduction problems(also known as the radiation hydrodynamics problems)in Kadioglu et al.,J.Comp.Physics[22,24].In this paper,we study the implications when this method is applied to the incompressible Navier-Stokes(N-S)equations.The IMEX method is applied to the incompressible flow equations in the following manner.The hyperbolic terms of the flow equations are solved explicitly exploiting the well understood explicit schemes.On the other hand,an implicit strategy is employed for the non-hyperbolic terms.The explicit part is embedded in the implicit step in such a way that it is solved as part of the non-linear function evaluation within the framework of the Jacobian-Free Newton Krylov(JFNK)method[8,29,31].This is done to obtain a self-consistent implementation of the IMEX method that eliminates the potential order reduction in time accuracy due to the specific operator separation.We employ a simple yet quite effective fractional step projection methodology(similar to those in[11,19,21,30])as our preconditioner inside the JFNK solver.We present results from several test calculations.For each test,we show second order time convergence.Finally,we present a study for the algorithm performance of the JFNK solver with the new projection method based preconditioner.展开更多
基金support provided by the Deutsche Forschun-gsgemeinschaft(DFG,German Research Foundation)through the project GRK 2160/1“Droplet Interaction Technologies”and through the project no.457811052
文摘Considering droplet phenomena at low Mach numbers,large differences in the magnitude of the occurring characteristic waves are presented.As acoustic phenomena often play a minor role in such applications,classical explicit schemes which resolve these waves suffer from a very restrictive timestep restriction.In this work,a novel scheme based on a specific level set ghost fluid method and an implicit-explicit(IMEX)flux splitting is proposed to overcome this timestep restriction.A fully implicit narrow band around the sharp phase interface is combined with a splitting of the convective and acoustic phenomena away from the interface.In this part of the domain,the IMEX Runge-Kutta time discretization and the high order discontinuous Galerkin spectral element method are applied to achieve high accuracies in the bulk phases.It is shown that for low Mach numbers a significant gain in computational time can be achieved compared to a fully explicit method.Applica-tions to typical droplet dynamic phenomena validate the proposed method and illustrate its capabilities.
文摘We investigate strong stability preserving(SSP)implicit-explicit(IMEX)methods for partitioned systems of differential equations with stiff and nonstiff subsystems.Conditions for order p and stage order q=p are derived,and characterization of SSP IMEX methods is provided following the recent work by Spijker.Stability properties of these methods with respect to the decoupled linear system with a complex parameter,and a coupled linear system with real parameters are also investigated.Examples of methods up to the order p=4 and stage order q—p are provided.Numerical examples on six partitioned test systems confirm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration,and they are also suitable for preserving the accuracy in the stiff limit or preserving the positivity of the numerical solution for large stepsizes.
文摘A fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts(upwind gSBP)schemes in space and implicit-explicit Runge-Kutta(IMEX-RK)schemes in time.Hereby,advection terms are discretized explicitly,while diffusion terms are solved implicitly.In this context,specific combinations of space and time discretizations enjoy enhanced stability properties.In fact,if the first-and second-derivative upwind gSBP operators fulfill a compatibility condition,the allowable time step size is independent of grid refinement,although the advective terms are discretized explicitly.In one space dimension it is shown that upwind gSBP schemes represent a general framework including standard discontinuous Galerkin(DG)schemes on a global level.While previous work for DG schemes has demonstrated that the combination of upwind advection fluxes and the central-type first Bassi-Rebay(BR1)scheme for diffusion does not allow for grid-independent stable time steps,the current work shows that central advection fluxes are compatible with BR1 regarding enhanced stability of IMEX time stepping.Furthermore,unlike previous discrete energy stability investigations for DG schemes,the present analysis is based on the discrete energy provided by the corresponding SBP norm matrix and yields time step restrictions independent of the discretization order in space,since no finite-element-type inverse constants are involved.Numerical experiments are provided confirming these theoretical findings.
文摘In this paper,we introduce an extension of a splitting method for singularly perturbed equations,the socalled RS-IMEX splitting[Kaiser et al.,Journal of Scientific Computing,70(3),1390–1407],to deal with the fully compressible Euler equations.The straightforward application of the splitting yields sub-equations that are,due to the occurrence of complex eigenvalues,not hyperbolic.A modification,slightly changing the convective flux,is introduced that overcomes this issue.It is shown that the splitting gives rise to a discretization that respects the low-Mach number limit of the Euler equations;numerical results using finite volume and discontinuous Galerkin schemes show the potential of the discretization.
文摘In this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains,taking Hermite functions,sinc functions,and rational Chebyshev polynomials as basis functions.The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme,being of particular interest the treatment of three-dimensional Sylvester equations that we make.The resulting method is easy to understand and express,and can be implemented in a transparent way by means of a few lines of code.We test numerically the three choices of basis functions,showing the convenience of this new approach,especially when rational Chebyshev polynomials are considered.
基金funded by awards NSF CCF1613905,NSF ACI1709727,AFOSR DDDAS FA9550-17-1-0015the Computational Science Laboratory at Virginia Tech.
文摘High-order discretizations of partial differential equations(PDEs)necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner.Implicit-explicit(IMEX)integration based on general linear methods(GLMs)offers an attractive solution due to their high stage and method order,as well as excellent stability properties.The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly.This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel.The first approach is based on diagonally implicit multi-stage integration methods(DIMSIMs)of types 3 and 4.The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy.Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge-Kutta methods.
基金supported by the NSF under Grant DMS-2208391sponsored by the NSF under Grant DMS-1753581.
文摘This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
基金supported by an NSERC Canada Postgraduate Scholarshipsupported by a grant from NSERC Canada
文摘Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit （VSIMEX） linear multistep methods for time-dependent PDEs. Families of order-p, pstep VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers＇ equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.
基金the National Numerical Wind Tunnel Project(No.NNW2018-ZT4A08)the National Natural Science Foundation of China(Nos.11872323 and 11971025)the Natural Science Foundation of Fujian Province(No.2019J06002)。
文摘The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.
基金by a contractor of the U.S.Government under Contract No.DEAC07-05ID14517.
文摘We have introduced a fully second order IMplicit/EXplicit(IMEX)time integration technique for solving the compressible Euler equations plus nonlinear heat conduction problems(also known as the radiation hydrodynamics problems)in Kadioglu et al.,J.Comp.Physics[22,24].In this paper,we study the implications when this method is applied to the incompressible Navier-Stokes(N-S)equations.The IMEX method is applied to the incompressible flow equations in the following manner.The hyperbolic terms of the flow equations are solved explicitly exploiting the well understood explicit schemes.On the other hand,an implicit strategy is employed for the non-hyperbolic terms.The explicit part is embedded in the implicit step in such a way that it is solved as part of the non-linear function evaluation within the framework of the Jacobian-Free Newton Krylov(JFNK)method[8,29,31].This is done to obtain a self-consistent implementation of the IMEX method that eliminates the potential order reduction in time accuracy due to the specific operator separation.We employ a simple yet quite effective fractional step projection methodology(similar to those in[11,19,21,30])as our preconditioner inside the JFNK solver.We present results from several test calculations.For each test,we show second order time convergence.Finally,we present a study for the algorithm performance of the JFNK solver with the new projection method based preconditioner.
