Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of un...Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of unsteady flows.To enhance computational efficiency,we propose the Implicit-Explicit Two-Step Runge-Kutta(IMEX-TSRK)time-stepping discretization methods for unsteady flows,and develop a novel adaptive algorithm that correctly partitions spatial regions to apply implicit or explicit methods.The novel adaptive IMEX-TSRK schemes effectively handle the numerical stiffness of the small grid size and improve computational efficiency.Compared to implicit and explicit Runge-Kutta(RK)schemes,the IMEX-TSRK methods achieve the same order of accuracy with fewer first derivative calculations.Numerical case tests demonstrate that the IMEX-TSRK methods maintain numerical stability while enhancing computational efficiency.Specifically,in high Reynolds number flows,the computational efficiency of the IMEX-TSRK methods surpasses that of explicit RK schemes by more than one order of magnitude,and that of implicit RK schemes several times over.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
In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear conve...In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-stepis only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.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
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
This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that...This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that the value of any European contingent claim should satisfy, where the asset price obeys the SVJ model. This equation is numerically solved by using the implicit- explicit backward difference method and time semi-discretization. In order to explain the validity of our method, the stability of time semi-discretization scheme is also proved. Finally, we use a simulation example to illustrate the efficiency of the method.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
The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial...The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial singularity.In the numerical simulations of solutions with initial singularity,variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence,whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied.In this paper,we revisit the variable time-step implicit-explicit two-step backward differentiation formula(IMEX BDF2)scheme to solve the parabolic integro-differential equations(PIDEs)with initial singularity.We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios r_(k):=T_(k)/T_(k-1)<r_(max)=4.8645(k≥3)and a much mild requirement on the first ratio,i.e.r_(2)>0.This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy,i.e.the graded mesh t_(k)=T(k/N)^(γ).In this situation,the convergence order of O(N^(-min(2,γα))is achieved,where N denotes the total number of mesh points andαindicates the regularity of the exact solution.This is,the optimal convergence will be achieved by taking%γ_(opt)=2/α.Numerical examples are provided to demonstrate our theoretical analysis.展开更多
We present a high-resolution relaxation scheme for a multi-class Lighthill-Whitham-Richards (MCLWR) traffic flow model. This scheme is based on high-order reconstruction for spatial discretization and an implicit-expl...We present a high-resolution relaxation scheme for a multi-class Lighthill-Whitham-Richards (MCLWR) traffic flow model. This scheme is based on high-order reconstruction for spatial discretization and an implicit-explicit Runge-Kutta method for time integration. The resulting method retains the simplicity of the relaxation schemes. There is no need to involve Riemann solvers and characteristic decomposition. Even the computation of the eigenvalues is not required. This makes the scheme particularly well suited for the MCLWR model in which the analytical expressions of the eigenvalues are difficult to obtain for more than four classes of road users. The numerical results illustrate the effectiveness of the presented method.展开更多
We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not...We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.展开更多
In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and dec...In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and decoupled,which greatly improve the computational efficiency.In order to resolve the time level mismatch of the discretization process,a special time marching method with high-order accuracy is constructed.Under the condition of slight time step constraints,the optimal error estimates of this method are given.Moreover,the theoretical results are verified by numerical experiments.Real simulations show the patterns of spots,rings,stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.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
基金supported by the National Natural Science Foundation of China(No.92252201)the Fundamental Research Funds for the Central Universitiesthe Academic Excellence Foundation of Beihang University(BUAA)for PhD Students。
文摘Efficient and accurate simulation of unsteady flow presents a significant challenge that needs to be overcome in computational fluid dynamics.Temporal discretization method plays a crucial role in the simulation of unsteady flows.To enhance computational efficiency,we propose the Implicit-Explicit Two-Step Runge-Kutta(IMEX-TSRK)time-stepping discretization methods for unsteady flows,and develop a novel adaptive algorithm that correctly partitions spatial regions to apply implicit or explicit methods.The novel adaptive IMEX-TSRK schemes effectively handle the numerical stiffness of the small grid size and improve computational efficiency.Compared to implicit and explicit Runge-Kutta(RK)schemes,the IMEX-TSRK methods achieve the same order of accuracy with fewer first derivative calculations.Numerical case tests demonstrate that the IMEX-TSRK methods maintain numerical stability while enhancing computational efficiency.Specifically,in high Reynolds number flows,the computational efficiency of the IMEX-TSRK methods surpasses that of explicit RK schemes by more than one order of magnitude,and that of implicit RK schemes several times over.
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the NSFC grant 11871428the Nature Science Research Program for Colleges and Universities of Jiangsu Province grant 20KJB110011Qiang Zhang:Research supported by the NSFC grant 11671199。
文摘In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-stepis only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.
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This paper considers pricing European options under the well-known of SVJ model of Bates and related computational methods. According to the no-arbitrage principle, we first derive a partial differential equation that the value of any European contingent claim should satisfy, where the asset price obeys the SVJ model. This equation is numerically solved by using the implicit- explicit backward difference method and time semi-discretization. In order to explain the validity of our method, the stability of time semi-discretization scheme is also proved. Finally, we use a simulation example to illustrate the efficiency of the method.
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.
基金supported by the NSFC(Grant No.12171376)the Fundamental Research Funds for the Central Universities(Grant No.2042021kf0050)and WHU-2022-SYJS-0002+1 种基金Yana Di is partially supported by the NSFC(Grant Nos.12271048,12171042)the Guangdong Key Laboratory(Grant No.2022B1212010006).
文摘The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes.However,it may not be readily applicable to problems exhibiting an initial singularity.In the numerical simulations of solutions with initial singularity,variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence,whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied.In this paper,we revisit the variable time-step implicit-explicit two-step backward differentiation formula(IMEX BDF2)scheme to solve the parabolic integro-differential equations(PIDEs)with initial singularity.We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios r_(k):=T_(k)/T_(k-1)<r_(max)=4.8645(k≥3)and a much mild requirement on the first ratio,i.e.r_(2)>0.This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy,i.e.the graded mesh t_(k)=T(k/N)^(γ).In this situation,the convergence order of O(N^(-min(2,γα))is achieved,where N denotes the total number of mesh points andαindicates the regularity of the exact solution.This is,the optimal convergence will be achieved by taking%γ_(opt)=2/α.Numerical examples are provided to demonstrate our theoretical analysis.
基金Project supported by the Aoxiang Project and the Scientific and Technological Innovation Foundation of Northwestern Polytechnical University, China (No 2007KJ01011)
文摘We present a high-resolution relaxation scheme for a multi-class Lighthill-Whitham-Richards (MCLWR) traffic flow model. This scheme is based on high-order reconstruction for spatial discretization and an implicit-explicit Runge-Kutta method for time integration. The resulting method retains the simplicity of the relaxation schemes. There is no need to involve Riemann solvers and characteristic decomposition. Even the computation of the eigenvalues is not required. This makes the scheme particularly well suited for the MCLWR model in which the analytical expressions of the eigenvalues are difficult to obtain for more than four classes of road users. The numerical results illustrate the effectiveness of the presented method.
基金Open Access funding provided by Universita degli Studi di Verona.
文摘We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.
基金supported by National Natural Science Foundation of China(Grant No.11801569)Natural Science Foundation of Shandong Province(CN)(Grant No.ZR2021MA001)the Fundamental Research Funds for the Central Universities(Grant Nos.22CX03025A and 22CX03020A).
文摘In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and decoupled,which greatly improve the computational efficiency.In order to resolve the time level mismatch of the discretization process,a special time marching method with high-order accuracy is constructed.Under the condition of slight time step constraints,the optimal error estimates of this method are given.Moreover,the theoretical results are verified by numerical experiments.Real simulations show the patterns of spots,rings,stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.
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.
