The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms.In this work,we consider the extension of a recently proposed non-overlapping domain ...The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms.In this work,we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique,modified upwind differences with explicitimplicit coupling,the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy.Moreover,for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictor-corrector or stabilized schemes.These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.展开更多
In this paper,we present a unified finite volume method preserving discrete maximum principle(DMP)for the conjugate heat transfer problems with general interface conditions.We prove the existence of the numerical solu...In this paper,we present a unified finite volume method preserving discrete maximum principle(DMP)for the conjugate heat transfer problems with general interface conditions.We prove the existence of the numerical solution and the DMP-preserving property.Numerical experiments show that the nonlinear iteration numbers of the scheme in[24]increase rapidly when the interfacial coefficients decrease to zero.In contrast,the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero,which reveals that the unified scheme is more robust than the scheme in[24].The accuracy and DMP-preserving property of the scheme are also veri ed in the numerical experiments.展开更多
In this paper,we construct a new cell-centered nonlinear finite volume scheme that preserves the extremum principle for heterogeneous anisotropic diffusion equation on distorted meshes.We introduce a new nonlinear app...In this paper,we construct a new cell-centered nonlinear finite volume scheme that preserves the extremum principle for heterogeneous anisotropic diffusion equation on distorted meshes.We introduce a new nonlinear approach to construct the conservative flux,that is,a linear second order flux is firstly given and a nonlinear conservative flux is then constructed by using an adaptive method and a nonlinear weighted method.Our new scheme does not need to use the convex combination of the cell-center unknowns to approximate the auxiliary unknowns,so it can deal with the problem with general discontinuous coefficients.Numerical results show that our new scheme performs more robust than some existing schemes on highly distorted meshes.展开更多
For a new nonlinear iterative method named as Picard-Newton(P-N)iterative method for the solution of the time-dependent reaction-diffusion systems,which arise in non-equilibrium radiation diffusion applications,two ti...For a new nonlinear iterative method named as Picard-Newton(P-N)iterative method for the solution of the time-dependent reaction-diffusion systems,which arise in non-equilibrium radiation diffusion applications,two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented.The non-equilibrium radiation diffusion problems with flux limiter are considered,which appends pesky complexity and nonlinearity to the diffusion coef-ficient.Numerical results are presented to demonstrate that compared with Picard method,for a desired accuracy,significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.展开更多
In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and co...In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and conservative gradient discretization method for spatial discretization.The method is based on a new cellcentered meshes,and it is locally conservative.It has smaller truncation error than the classical finite volume method on uniform meshes.We use the framework of the gradient discretization method to analyze the stability and convergence.The numerical experiments show that the new method has second-order convergence.Moreover,it is more accurate than the classical finite volume method in flux error,L2 error and L¥error.展开更多
The carbuncle phenomenon has been regarded as a spurious solution produced by most of contact-preserving methods.The hybrid method of combining high resolution flux with more dissipative solver is an attractive attemp...The carbuncle phenomenon has been regarded as a spurious solution produced by most of contact-preserving methods.The hybrid method of combining high resolution flux with more dissipative solver is an attractive attempt to cure this kind of non-physical phenomenon.In this paper,a matrix-based stability analysis for 2-D Euler equations is performed to explore the cause of instability of numerical schemes.By combining the Roe with HLL flux in different directions and different flux components,we give an interesting explanation to the linear numerical instability.Based on such analysis,some hybrid schemes are compared to illustrate different mechanisms in controlling shock instability.Numerical experiments are presented to verify our analysis results.The conclusion is that the scheme of restricting directly instability source is more stable than other hybrid schemes.展开更多
A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It...A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It does not stir numerical oscillation,while permits large time step length,and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes,the Crank-Nicolson(CN)scheme and the backward difference formula second-order(BDF2)scheme.By developing a new reasoning technique,we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers,and prove rigorously the TLCD scheme is uniquely solvable,unconditionally stable,and has second-order convergence in both s-pace and time.Numerical tests verify the theoretical results,and illustrate its superiority over the CN and BDF2 schemes.展开更多
Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to g...Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to generalize the vertex-centred EXCMG method to cell-centeredfinite volume(FV)methods for diffusion equations with strongly discontinuous and anisotropic coefficients,since a non-nested hierarchy of grid nodes are used in the cell-centered discretization.For cell-centered FV schemes,the vertex values(auxiliary unknowns)need to be approximated by cell-centered ones(primary unknowns).One of the novelties is to propose a new gradient transfer(GT)method of interpolating vertex unknowns with cell-centered ones,which is easy to implement and applicable to general diffusion tensors.The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method,and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients.Numerical experiments are presented to demonstrate the high efficiency of the proposed method.展开更多
In this paper,a nonlinear finite volume scheme preserving the discrete maximum principle for the anisotropic diffusion equation on distorted meshes is described.We prove the coercivity of the scheme under some constra...In this paper,a nonlinear finite volume scheme preserving the discrete maximum principle for the anisotropic diffusion equation on distorted meshes is described.We prove the coercivity of the scheme under some constraints on the cell deformation and the diffusion coefficient.Numerical results show that the scheme is indeed coercive and satisfies the discrete maximum principle,and the accuracy of this scheme is remarkably better than that of an existing scheme preserving the discrete maximum principle on random triangular meshes.展开更多
We apply the monotonicity correction to thefinite element method for the anisotropic diffusion problems,including linear and quadraticfinite elements on triangular meshes.When formulating thefinite element schemes,we ...We apply the monotonicity correction to thefinite element method for the anisotropic diffusion problems,including linear and quadraticfinite elements on triangular meshes.When formulating thefinite element schemes,we need to calculate the integrals on every triangular element,whose results are the linear combination of the two-point pairs.Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs.We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of thefinite element equations.Finally,the original stiffness matrix can be transformed into a nonlinear M-matrix,and the corrected schemes have the positivity-preserving property.We also give the monotonicity correction to the time derivative term for the time-dependent problems.Numerical experiments show that the correctedfinite element method has monotonicity and maintains the convergence order of the original schemes in H1-norm and L2-norm,respectively.展开更多
This paper presents a second-order direct arbitrary Lagrangian Eulerian(ALE)method for compressible flow in two-dimensional cylindrical geometry.This algorithm has half-face fluxes and a nodal velocity solver,which ca...This paper presents a second-order direct arbitrary Lagrangian Eulerian(ALE)method for compressible flow in two-dimensional cylindrical geometry.This algorithm has half-face fluxes and a nodal velocity solver,which can ensure the compatibility between edge fluxes and the nodal flow intrinsically.In two-dimensional cylindrical geometry,the control volume scheme and the area-weighted scheme are used respectively,which are distinguished by the discretizations for the source term in the momentum equation.The two-dimensional second-order extensions of these schemes are constructed by employing the monotone upwind scheme of conservation law(MUSCL)on unstructured meshes.Numerical results are provided to assess the robustness and accuracy of these new schemes.展开更多
In this paper,we correct the finite volume element methods for diffusion equations on general triangular and quadrilateral meshes.First,we decompose the numerical fluxes of original schemes into two parts,i.e.,the pri...In this paper,we correct the finite volume element methods for diffusion equations on general triangular and quadrilateral meshes.First,we decompose the numerical fluxes of original schemes into two parts,i.e.,the principal part with a twopoint flux structure and the defective part.And then with the help of local extremums,we transform the original numerical fluxes into nonlinear numerical fluxes,which can be expressed as a nonlinear combination of two-point fluxes.It is proved that the corrected schemes satisfy the discrete strong extremum principle without restrictions on the diffusion coefficient and meshes.Numerical results indicate that the corrected schemes not only satisfy the discrete strong extremum principle but also preserve the convergence order of the original finite volume element methods.展开更多
In this paper,we construct a global repair technique for the finite element scheme of anisotropic diffusion equations to enforce the repaired solutions satisfying the discrete maximum principle.It is an extension of t...In this paper,we construct a global repair technique for the finite element scheme of anisotropic diffusion equations to enforce the repaired solutions satisfying the discrete maximum principle.It is an extension of the existing local repair technique.Both of the repair techniques preserve the total energy and are easy to be implemented.The numerical experiments show that these repair techniques do not destroy the accuracy of the finite element scheme,and the computational cost of the global repair technique is cheaper than the local repair technique when the diffusion tensors are highly anisotropic.展开更多
A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term wit...A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle.The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion.Moreover,this manipulation can nearly keep the same accuracy as the original scheme.We prove the existence of the numerical solution for this nonlinear scheme theoretically.Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.展开更多
We construct a nonlinear monotone finite volume scheme for threedimensional diffusion equation on tetrahedral meshes.Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme,we ...We construct a nonlinear monotone finite volume scheme for threedimensional diffusion equation on tetrahedral meshes.Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme,we present a new efficient eliminating method.The scheme has only cell-centered unknowns and can deal with discontinuous or tensor diffusion coefficient problems on distorted meshes rigorously.The numerical results illustrate that the resulting scheme can preserve positivity on distorted tetrahedral meshes,and also show that our scheme appears to be approximate second-order accuracy for solution.展开更多
基金the National Natural Science Foundation of China(No.10571017)supported in part by the National Natural Science Foundation of China(No.60533020)supported in part by NSF DMS 0712744
文摘The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms.In this work,we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique,modified upwind differences with explicitimplicit coupling,the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy.Moreover,for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictor-corrector or stabilized schemes.These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.
基金National Natural Science Foundation of China(11971069,12071045)Foundation of CAEP(CX20210042)Science Challenge Project(No.TZ2016002).
文摘In this paper,we present a unified finite volume method preserving discrete maximum principle(DMP)for the conjugate heat transfer problems with general interface conditions.We prove the existence of the numerical solution and the DMP-preserving property.Numerical experiments show that the nonlinear iteration numbers of the scheme in[24]increase rapidly when the interfacial coefficients decrease to zero.In contrast,the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero,which reveals that the unified scheme is more robust than the scheme in[24].The accuracy and DMP-preserving property of the scheme are also veri ed in the numerical experiments.
基金This work was supported by the National Natural Science Foundation of China(Nos.11971069,12071045)the Scientific Research Foundation of NUAA No.YAH21109.
文摘In this paper,we construct a new cell-centered nonlinear finite volume scheme that preserves the extremum principle for heterogeneous anisotropic diffusion equation on distorted meshes.We introduce a new nonlinear approach to construct the conservative flux,that is,a linear second order flux is firstly given and a nonlinear conservative flux is then constructed by using an adaptive method and a nonlinear weighted method.Our new scheme does not need to use the convex combination of the cell-center unknowns to approximate the auxiliary unknowns,so it can deal with the problem with general discontinuous coefficients.Numerical results show that our new scheme performs more robust than some existing schemes on highly distorted meshes.
基金supported by the Basic Research Project of National Defence(B1520110011)the Foundation of CAEP(2010A0202010),the Foundation of National Key Laboratory of Science and Technology on Computational Physics。
文摘For a new nonlinear iterative method named as Picard-Newton(P-N)iterative method for the solution of the time-dependent reaction-diffusion systems,which arise in non-equilibrium radiation diffusion applications,two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented.The non-equilibrium radiation diffusion problems with flux limiter are considered,which appends pesky complexity and nonlinearity to the diffusion coef-ficient.Numerical results are presented to demonstrate that compared with Picard method,for a desired accuracy,significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.
基金supported by the National Natural Science Foundation of China(No.11971069),NSAF(No.U1630249)and Science Challenge Project(No.TZ2016002).
文摘In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and conservative gradient discretization method for spatial discretization.The method is based on a new cellcentered meshes,and it is locally conservative.It has smaller truncation error than the classical finite volume method on uniform meshes.We use the framework of the gradient discretization method to analyze the stability and convergence.The numerical experiments show that the new method has second-order convergence.Moreover,it is more accurate than the classical finite volume method in flux error,L2 error and L¥error.
基金supported by the National Natural Science Foundation of China(11071025)the Foundation of CAEP(2010A0202010)the Foundation of National Key Laboratory of Science and Technology Computation Physics and the Defense Industrial Technology Development Program(B1520110011).
文摘The carbuncle phenomenon has been regarded as a spurious solution produced by most of contact-preserving methods.The hybrid method of combining high resolution flux with more dissipative solver is an attractive attempt to cure this kind of non-physical phenomenon.In this paper,a matrix-based stability analysis for 2-D Euler equations is performed to explore the cause of instability of numerical schemes.By combining the Roe with HLL flux in different directions and different flux components,we give an interesting explanation to the linear numerical instability.Based on such analysis,some hybrid schemes are compared to illustrate different mechanisms in controlling shock instability.Numerical experiments are presented to verify our analysis results.The conclusion is that the scheme of restricting directly instability source is more stable than other hybrid schemes.
基金This work is supported by the National Natural Science Foundation of China(11871112,11971069,11971071,U1630249)Yu Min Foundation and the Foundation of LCP.
文摘A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied.The scheme is constructed with two-layer coupled discretization(TLCD)at each time step.It does not stir numerical oscillation,while permits large time step length,and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes,the Crank-Nicolson(CN)scheme and the backward difference formula second-order(BDF2)scheme.By developing a new reasoning technique,we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers,and prove rigorously the TLCD scheme is uniquely solvable,unconditionally stable,and has second-order convergence in both s-pace and time.Numerical tests verify the theoretical results,and illustrate its superiority over the CN and BDF2 schemes.
基金supported by Science Challenge Project(Grant No.TZ2016002)the National Natural Science Foundation of China(Grant Nos.41874086 and 11971069)+1 种基金173 Program of China(Grant No.2020-JCJQ-ZD-029)the Excellent Youth Foundation of Hunan Province of China(Grant No.2018JJ1042).
文摘Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to generalize the vertex-centred EXCMG method to cell-centeredfinite volume(FV)methods for diffusion equations with strongly discontinuous and anisotropic coefficients,since a non-nested hierarchy of grid nodes are used in the cell-centered discretization.For cell-centered FV schemes,the vertex values(auxiliary unknowns)need to be approximated by cell-centered ones(primary unknowns).One of the novelties is to propose a new gradient transfer(GT)method of interpolating vertex unknowns with cell-centered ones,which is easy to implement and applicable to general diffusion tensors.The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method,and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients.Numerical experiments are presented to demonstrate the high efficiency of the proposed method.
基金supported by National Natural Science Foundation of China(Grant Nos.12071045 and 11971069)the Foundation of CAEP(China Academy of Engineering Physics)(Grant No.CX20210042)the Foundation of LCP(Laboratory of Computational Physics).
文摘In this paper,a nonlinear finite volume scheme preserving the discrete maximum principle for the anisotropic diffusion equation on distorted meshes is described.We prove the coercivity of the scheme under some constraints on the cell deformation and the diffusion coefficient.Numerical results show that the scheme is indeed coercive and satisfies the discrete maximum principle,and the accuracy of this scheme is remarkably better than that of an existing scheme preserving the discrete maximum principle on random triangular meshes.
基金supported by the Science Challenge Project(No.TZ2016002)the National Science Foundation of China(No.12071177,No.11971069).
文摘We apply the monotonicity correction to thefinite element method for the anisotropic diffusion problems,including linear and quadraticfinite elements on triangular meshes.When formulating thefinite element schemes,we need to calculate the integrals on every triangular element,whose results are the linear combination of the two-point pairs.Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs.We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of thefinite element equations.Finally,the original stiffness matrix can be transformed into a nonlinear M-matrix,and the corrected schemes have the positivity-preserving property.We also give the monotonicity correction to the time derivative term for the time-dependent problems.Numerical experiments show that the correctedfinite element method has monotonicity and maintains the convergence order of the original schemes in H1-norm and L2-norm,respectively.
基金Project supported by the National Natural Science Foundation of China(U1630249,11971071,11971069,11871113)the Science Challenge Project(JCKY2016212A502)the Foundation of Laboratory of Computation Physics.
文摘This paper presents a second-order direct arbitrary Lagrangian Eulerian(ALE)method for compressible flow in two-dimensional cylindrical geometry.This algorithm has half-face fluxes and a nodal velocity solver,which can ensure the compatibility between edge fluxes and the nodal flow intrinsically.In two-dimensional cylindrical geometry,the control volume scheme and the area-weighted scheme are used respectively,which are distinguished by the discretizations for the source term in the momentum equation.The two-dimensional second-order extensions of these schemes are constructed by employing the monotone upwind scheme of conservation law(MUSCL)on unstructured meshes.Numerical results are provided to assess the robustness and accuracy of these new schemes.
基金partially supported by the National Science Foundation of China(No.12071177,No.12126307,No.11971069)the Science Challenge Project(No.TZ2016002).
文摘In this paper,we correct the finite volume element methods for diffusion equations on general triangular and quadrilateral meshes.First,we decompose the numerical fluxes of original schemes into two parts,i.e.,the principal part with a twopoint flux structure and the defective part.And then with the help of local extremums,we transform the original numerical fluxes into nonlinear numerical fluxes,which can be expressed as a nonlinear combination of two-point fluxes.It is proved that the corrected schemes satisfy the discrete strong extremum principle without restrictions on the diffusion coefficient and meshes.Numerical results indicate that the corrected schemes not only satisfy the discrete strong extremum principle but also preserve the convergence order of the original finite volume element methods.
基金supported by General Program of Science and Technology Development Project of Beijing Municipal Education Commission KM201310011006,Program of the Young People of Outstanding Ability for the Construction of the Teachers Procession YETP1445,Major Research Plan of the National Natural Science Foundation of China 91130015,National Natural Science Foundation of China 61201113,11101013,11401015.The second author is supported by the National Nature Science Foundation of China 11171036.
文摘In this paper,we construct a global repair technique for the finite element scheme of anisotropic diffusion equations to enforce the repaired solutions satisfying the discrete maximum principle.It is an extension of the existing local repair technique.Both of the repair techniques preserve the total energy and are easy to be implemented.The numerical experiments show that these repair techniques do not destroy the accuracy of the finite element scheme,and the computational cost of the global repair technique is cheaper than the local repair technique when the diffusion tensors are highly anisotropic.
基金supported by the Postdoctoral Science Foundation of China(No.2017M620689)the National Science Foundation of China(Nos.11571048,11401034)the CAEP developing fund of science and technology(No.2014A0202009).
文摘A nonlinear finite volume element scheme for anisotropic diffusion problems on general triangular meshes is proposed.Starting with a standard linear conforming finite volume element approximation,a corrective term with respect to the flux jumps across element boundaries is added to make the scheme satisfy the discrete maximum principle.The new scheme is free of the anisotropic non-obtuse angle condition which is a severe restriction on the grids for problems with anisotropic diffusion.Moreover,this manipulation can nearly keep the same accuracy as the original scheme.We prove the existence of the numerical solution for this nonlinear scheme theoretically.Numerical results and a grid convergence study are presented for both continuous and discontinuous anisotropic diffusion problems.
基金The authors thank two reviewers for their numerous constructive comments and suggestions that helped to improve the paper significantly.This work is partially supported by NSAF(No.U1430101)the National Natural Science Foundation of China(91330106,11571047,11571048,11401034)+2 种基金China Postdoctoral Science Foundation(20110490328)the natural science foundation of Shandong Province(ZR2012AM019,ZR2013AM023,ZR2014AM013)Independent innovation foundation of Shandong University(2012TS018).
文摘We construct a nonlinear monotone finite volume scheme for threedimensional diffusion equation on tetrahedral meshes.Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme,we present a new efficient eliminating method.The scheme has only cell-centered unknowns and can deal with discontinuous or tensor diffusion coefficient problems on distorted meshes rigorously.The numerical results illustrate that the resulting scheme can preserve positivity on distorted tetrahedral meshes,and also show that our scheme appears to be approximate second-order accuracy for solution.