In this paper,we design a new error estimator and give a posteriori error analysis for a poroelasticity model.To better overcome“locking phenomenon”on pressure and displacement,we proposed a new error estimators bas...In this paper,we design a new error estimator and give a posteriori error analysis for a poroelasticity model.To better overcome“locking phenomenon”on pressure and displacement,we proposed a new error estimators based on multiphysics discontinuous Galerkin method for the poroelasticity model.And we prove the upper and lower bound of the proposed error estimators,which are numerically demonstrated to be computationally very efficient.Finally,we present numerical examples to verify and validate the efficiency of the proposed error estimators,which show that the adaptive scheme can overcome“locking phenomenon”and greatly reduce the computation cost.展开更多
We present the approaches to implementing the k-√k L turbulence model within the framework of the high-order discontinuous Galerkin(DG)method.We use the DG discretization to solve the full Reynolds-averaged Navier-St...We present the approaches to implementing the k-√k L turbulence model within the framework of the high-order discontinuous Galerkin(DG)method.We use the DG discretization to solve the full Reynolds-averaged Navier-Stokes equations.In order to enhance the robustness of approaches,some effective techniques are designed.The HWENO(Hermite weighted essentially non-oscillatory)limiting strategy is adopted for stabilizing the turbulence model variable k.Modifications have been made to the model equation itself by using the auxiliary variable that is always positive.The 2nd-order derivatives of velocities required in computing the von Karman length scale are evaluated in a way to maintain the compactness of DG methods.Numerical results demonstrate that the approaches have achieved the desirable accuracy for both steady and unsteady turbulent simulations.展开更多
The primary impediments impeding the implementation of high-order methods in simulating viscous flow over complex configurations are robustness and convergence.These challenges impose significant constraints on comput...The primary impediments impeding the implementation of high-order methods in simulating viscous flow over complex configurations are robustness and convergence.These challenges impose significant constraints on computational efficiency,particularly in the domain of engineering applications.To address these concerns,this paper proposes a robust implicit high-order discontinuous Galerkin(DG)method for solving compressible Navier-Stokes(NS)equations on arbitrary grids.The method achieves a favorable equilibrium between computational stability and efficiency.To solve the linear system,an exact Jacobian matrix solving strategy is employed for preconditioning and matrix-vector generation in the generalized minimal residual(GMRES)method.This approach mitigates numerical errors in Jacobian solution during implicit calculations and facilitates the implementation of an adaptive Courant-Friedrichs-Lewy(CFL)number increasing strategy,with the aim of improving convergence and robustness.To further enhance the applicability of the proposed method for intricate grid distortions,all simulations are performed in the reference domain.This practice significantly improves the reversibility of the mass matrix in implicit calculations.A comprehensive analysis of various parameters influencing computational stability and efficiency is conducted,including CFL number,Krylov subspace size,and GMRES convergence criteria.The computed results from a series of numerical test cases demonstrate the promising results achieved by combining the DG method,GMRES solver,exact Jacobian matrix,adaptive CFL number,and reference domain calculations in terms of robustness,convergence,and accuracy.These analysis results can serve as a reference for implicit computation in high-order calculations.展开更多
In this article,we propose and research a first-order,linearized discontinuous Galerkin method for the approximation of the hydrodynamic and sediment transport model.The method is decoupled and fully discrete,and is s...In this article,we propose and research a first-order,linearized discontinuous Galerkin method for the approximation of the hydrodynamic and sediment transport model.The method is decoupled and fully discrete,and is shown to be unconditionally stable.Furthermore,error estimates are proved.Finally,the theoretical analysis is confirmed by numerical examples.展开更多
Elastic beams resting on an elastic foundation are frequently encountered in civil,mechanical,aeronautical,and other engineering disciplines,and the analysis of static and dynamic deflections is one of the essential r...Elastic beams resting on an elastic foundation are frequently encountered in civil,mechanical,aeronautical,and other engineering disciplines,and the analysis of static and dynamic deflections is one of the essential requirements related to various applications.The Galerkin method is a classical mathematical method for solving differential equations without a closed-form solution with a wide range of applications in engineering and scientific fields.In this study,a demonstration is presented to solve the nonlinear differential equation by transforming it into a series of nonlinear algebraic equations with the Galerkin method for asymptotic solutions in series,and the nonlinear deformation of beams resting on the nonlinear foundation is successfully solved as an example.The approximate solutions based on trigonometric functions are utilized,and the nonlinear algebraic equations are solved both numerically and iteratively.Although widely used in linear problems,it is worth reminding that the Galerkin method also provides an effective approach in dealing with increasingly complex nonlinear equations in practical applications with the aid of powerful tools for symbolic manipulation of nonlinear algebraic equations.展开更多
In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal e...In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.展开更多
In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al...In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.展开更多
In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the diver...In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.展开更多
In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system....In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system.The LoMaC property refers to the exact local conservation of macroscopic mass,momentum,and energy at the discrete level.The recently developed LoMaC low rank tensor algorithm(arXiv:2207.00518)simultaneously evolves the macroscopic conservation laws of mass,momentum,and energy using the kinetic flux vector splitting;then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables.This paper is a generalization of our previous work,but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term.The algorithm is developed in a similar fashion as that for a finite difference scheme,by observing that the DG method can be viewed equivalently in a nodal fashion.With the nodal DG method,assuming a tensorized computational grid,one will be able to(i)derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms,and(ii)define a weighted inner product space based on the nodal DG grid points.The algorithm can be extended to the high dimensional problems by hierarchical Tucker(HT)decomposition of solution tensors and a corresponding conservative projection algorithm.In a similar spirit,the algorithm can be extended to DG methods on nodal points of an unstructured mesh,or to other types of discretization,e.g.,the spectral method in velocity direction.Extensive numerical results are performed to showcase the efficacy of the method.展开更多
This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis techniq...This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.展开更多
In this paper,a fast element-free Galerkin(FEFG)method for three-dimensional(3D)elasticity problems is established.The FEFG method is a combination of the improved element-free Galerkin(IEFG)method and the dimension s...In this paper,a fast element-free Galerkin(FEFG)method for three-dimensional(3D)elasticity problems is established.The FEFG method is a combination of the improved element-free Galerkin(IEFG)method and the dimension splitting method(DSM).By using the DSM,a 3D problem is converted to a series of 2D ones,and the IEFG method with a weighted orthogonal function as the basis function and the cubic spline function as the weight function is applied to simulate these 2D problems.The essential boundary conditions are treated by the penalty method.The splitting direction uses the finite difference method(FDM),which can combine these 2D problems into a discrete system.Finally,the system equation of the 3D elasticity problem is obtained.Some specific numerical problems are provided to illustrate the effectiveness and advantages of the FEFG method for 3D elasticity by comparing the results of the FEFG method with those of the IEFG method.The convergence and relative error norm of the FEFG method for elasticity are also studied.展开更多
In this paper, an improved interpolating moving least-square （IIMLS） method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the II...In this paper, an improved interpolating moving least-square （IIMLS） method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square （MLS） approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin （IIEFG） method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin （EFG） method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method.展开更多
An automatically adaptive element free method is presented to analyze the seismic response of liquefiable soils. The method is based on the element free Galerkin method （EFGM） and the fission procedure that is part ...An automatically adaptive element free method is presented to analyze the seismic response of liquefiable soils. The method is based on the element free Galerkin method （EFGM） and the fission procedure that is part of h-refinement, indicated by error estimation. In the proposed method, a posteriori error estimate procedure that depends on the energy norm of stress and the T-Belytschko （TB） stress recovery scheme is incorporated. The effective cyclic elasto-plastic constitutive model is used to describe the nonlinear behavior of the saturated soil. The governing equations are established by u-p formulation. The proposed method can effectively avoid the volumetric locking due to large deformation that usually occurs in numerical computations using the finite element method （FEM）. The efficiency of the proposed method is demonstrated by evaluating the seismic response of an embankment and comparing it to results obtained through FEM. It is shown that the proposed method provides an accurate seismic analysis of saturated soil that includes the effects of liquefaction .展开更多
The main goal of this paper is to investigate natural convective heat transfer and flow characteristics of non-Newtonian nanofluid streaming between two infinite vertical flat plates in the presence of magnetic field ...The main goal of this paper is to investigate natural convective heat transfer and flow characteristics of non-Newtonian nanofluid streaming between two infinite vertical flat plates in the presence of magnetic field and thermal radiation.Initially,a similarity transformation is used to convert momentum and energy conservation equations in partial differential forms into non-linear ordinary differential equations (ODE) applying meaningful boundary conditions.In order to obtain the non-linear ODEs analytically,Galerkin method (GM) is employed.Subsequently,the ODEs are also solved by a reliable numerical solution.In order to test the accuracy,precision and reliability of the analytical method,results of the analytical analysis are compared with the numerical results.With respect to the comparisons,fairly good compatibilities with insignificant errors are observed.Eventually,the impacts of effective parameters including magnetic and radiation parameters and nanofluid volume fraction on the velocity,skin friction coefficient and Nusselt number distributions are comprehensively described.Based on the results,it is revealed that with increasing the role of magnetic force,velocity profile,skin friction coefficient and thermal performance descend.Radiation parameter has insignificant influence on velocity profile while it obviously has augmentative and decreasing effects on skin friction and Nusselt number,respectively.展开更多
This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using prec...This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.展开更多
A novel class of weighted essentially nonoscillatory （WENO） schemes based on Hermite polynomi- als, termed as HWENO schemes, is developed and applied as limiters for high order discontinuous Galerkin （DG） method o...A novel class of weighted essentially nonoscillatory （WENO） schemes based on Hermite polynomi- als, termed as HWENO schemes, is developed and applied as limiters for high order discontinuous Galerkin （DG） method on triangular grids. The developed HWENO methodology utilizes high-order derivative information to keep WENO re- construction stencils in the von Neumann neighborhood. A simple and efficient technique is also proposed to enhance the smoothness of the existing stencils, making higher-order scheme stable and simplifying the reconstruction process at the same time. The resulting HWENO-based limiters are as compact as the underlying DG schemes and therefore easy to implement. Numerical results for a wide range of flow conditions demonstrate that for DG schemes of up to fourth order of accuracy, the designed HWENO limiters can simul- taneously obtain uniform high order accuracy and sharp, es- sentially non-oscillatory shock transition.展开更多
In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squar...In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares （CVMLS） approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin （EFG） method and the ICVMLS approximation, the improved complex variable element-free Galerkin （ICVEFG） method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFC method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.展开更多
In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-f...In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-free Galerkin （ICVEFG） method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square （CVMLS） approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.展开更多
In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved c...In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin（ICVEFG） method is presented for two-dimensional（2D） elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.展开更多
An h-adaptive method is developed for high-order discontinuous Galerkin methods(DGM)to solve the laminar compressible Navier-Stokes(N-S)equations on unstructured mesh.The vorticity is regarded as the indicator of adap...An h-adaptive method is developed for high-order discontinuous Galerkin methods(DGM)to solve the laminar compressible Navier-Stokes(N-S)equations on unstructured mesh.The vorticity is regarded as the indicator of adaptivity.The elements where the vorticity is larger than a pre-defined upper limit are refined,and those where the vorticity is smaller than a pre-defined lower limit are coarsened if they have been refined.A high-order geometric approximation of curved boundaries is adopted to ensure the accuracy.Numerical results indicate that highly accurate numerical results can be obtained with the adaptive method at relatively low expense.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12371393 and 11971150)Natural Science Foundation of Henan(Grant No.242300421047).
文摘In this paper,we design a new error estimator and give a posteriori error analysis for a poroelasticity model.To better overcome“locking phenomenon”on pressure and displacement,we proposed a new error estimators based on multiphysics discontinuous Galerkin method for the poroelasticity model.And we prove the upper and lower bound of the proposed error estimators,which are numerically demonstrated to be computationally very efficient.Finally,we present numerical examples to verify and validate the efficiency of the proposed error estimators,which show that the adaptive scheme can overcome“locking phenomenon”and greatly reduce the computation cost.
基金supported by the National Natural Science Foundation of China(Grant Nos.92252201 and 11721202)the Fundamental Research Funds for the Central Universities.
文摘We present the approaches to implementing the k-√k L turbulence model within the framework of the high-order discontinuous Galerkin(DG)method.We use the DG discretization to solve the full Reynolds-averaged Navier-Stokes equations.In order to enhance the robustness of approaches,some effective techniques are designed.The HWENO(Hermite weighted essentially non-oscillatory)limiting strategy is adopted for stabilizing the turbulence model variable k.Modifications have been made to the model equation itself by using the auxiliary variable that is always positive.The 2nd-order derivatives of velocities required in computing the von Karman length scale are evaluated in a way to maintain the compactness of DG methods.Numerical results demonstrate that the approaches have achieved the desirable accuracy for both steady and unsteady turbulent simulations.
基金supported by the National Natural Science Foundation of China(Grant No.12102247)the Technology Development Program(Grant No.JCKY2022110C119).
文摘The primary impediments impeding the implementation of high-order methods in simulating viscous flow over complex configurations are robustness and convergence.These challenges impose significant constraints on computational efficiency,particularly in the domain of engineering applications.To address these concerns,this paper proposes a robust implicit high-order discontinuous Galerkin(DG)method for solving compressible Navier-Stokes(NS)equations on arbitrary grids.The method achieves a favorable equilibrium between computational stability and efficiency.To solve the linear system,an exact Jacobian matrix solving strategy is employed for preconditioning and matrix-vector generation in the generalized minimal residual(GMRES)method.This approach mitigates numerical errors in Jacobian solution during implicit calculations and facilitates the implementation of an adaptive Courant-Friedrichs-Lewy(CFL)number increasing strategy,with the aim of improving convergence and robustness.To further enhance the applicability of the proposed method for intricate grid distortions,all simulations are performed in the reference domain.This practice significantly improves the reversibility of the mass matrix in implicit calculations.A comprehensive analysis of various parameters influencing computational stability and efficiency is conducted,including CFL number,Krylov subspace size,and GMRES convergence criteria.The computed results from a series of numerical test cases demonstrate the promising results achieved by combining the DG method,GMRES solver,exact Jacobian matrix,adaptive CFL number,and reference domain calculations in terms of robustness,convergence,and accuracy.These analysis results can serve as a reference for implicit computation in high-order calculations.
基金Supported by the National Natural Science Foundation of China(Grant No.41930643)the Natural Science Foundation of Henan Province(Grant No.232300420109).
文摘In this article,we propose and research a first-order,linearized discontinuous Galerkin method for the approximation of the hydrodynamic and sediment transport model.The method is decoupled and fully discrete,and is shown to be unconditionally stable.Furthermore,error estimates are proved.Finally,the theoretical analysis is confirmed by numerical examples.
基金supported by the National Natural Science Foundation of China(Grant No.11672142)support through the Technology Innovation 2025 Program(Grant No.2019B10122)of the Municipality of Ningbo,Zhejiang,China.
文摘Elastic beams resting on an elastic foundation are frequently encountered in civil,mechanical,aeronautical,and other engineering disciplines,and the analysis of static and dynamic deflections is one of the essential requirements related to various applications.The Galerkin method is a classical mathematical method for solving differential equations without a closed-form solution with a wide range of applications in engineering and scientific fields.In this study,a demonstration is presented to solve the nonlinear differential equation by transforming it into a series of nonlinear algebraic equations with the Galerkin method for asymptotic solutions in series,and the nonlinear deformation of beams resting on the nonlinear foundation is successfully solved as an example.The approximate solutions based on trigonometric functions are utilized,and the nonlinear algebraic equations are solved both numerically and iteratively.Although widely used in linear problems,it is worth reminding that the Galerkin method also provides an effective approach in dealing with increasingly complex nonlinear equations in practical applications with the aid of powerful tools for symbolic manipulation of nonlinear algebraic equations.
基金supported by the NSF under Grant DMS-1818467Simons Foundation under Grant 961585.
文摘In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.
基金supported by the National Science Foundation grant DMS-1818998.
文摘In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.
基金supported by the NSFC Grant 11901555,12271499the Cyrus Tang Foundationsupported by the NSFC Grant 11871448 and 12126604.
文摘In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.
基金supported by the NSF(Grant Nos.the NSF-DMS-1818924 and 2111253)the Air Force Office of Scientific Research FA9550-22-1-0390 and Department of Energy DE-SC0023164+1 种基金supported by the NSF(Grant Nos.NSF-DMS-1830838 and NSF-DMS-2111383)the Air Force Office of Scientific Research FA9550-22-1-0390.
文摘In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system.The LoMaC property refers to the exact local conservation of macroscopic mass,momentum,and energy at the discrete level.The recently developed LoMaC low rank tensor algorithm(arXiv:2207.00518)simultaneously evolves the macroscopic conservation laws of mass,momentum,and energy using the kinetic flux vector splitting;then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables.This paper is a generalization of our previous work,but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term.The algorithm is developed in a similar fashion as that for a finite difference scheme,by observing that the DG method can be viewed equivalently in a nodal fashion.With the nodal DG method,assuming a tensorized computational grid,one will be able to(i)derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms,and(ii)define a weighted inner product space based on the nodal DG grid points.The algorithm can be extended to the high dimensional problems by hierarchical Tucker(HT)decomposition of solution tensors and a corresponding conservative projection algorithm.In a similar spirit,the algorithm can be extended to DG methods on nodal points of an unstructured mesh,or to other types of discretization,e.g.,the spectral method in velocity direction.Extensive numerical results are performed to showcase the efficacy of the method.
基金supported by the National Natural Science Foundation of China(Grant Nos.11871428 and 12071214)the Natural Science Foundation for Colleges and Universities of Jiangsu Province of China(Grant No.20KJB110011)+1 种基金supported by the National Science Foundation(Grant No.DMS-1620335)and the Simons Foundation(Grant No.637716)supported by the National Natural Science Foundation of China(Grant Nos.11871428 and 12272347).
文摘This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.
基金supported by the National Natural Science Foundation of China(Grant Nos.52004169 and 11571223).
文摘In this paper,a fast element-free Galerkin(FEFG)method for three-dimensional(3D)elasticity problems is established.The FEFG method is a combination of the improved element-free Galerkin(IEFG)method and the dimension splitting method(DSM).By using the DSM,a 3D problem is converted to a series of 2D ones,and the IEFG method with a weighted orthogonal function as the basis function and the cubic spline function as the weight function is applied to simulate these 2D problems.The essential boundary conditions are treated by the penalty method.The splitting direction uses the finite difference method(FDM),which can combine these 2D problems into a discrete system.Finally,the system equation of the 3D elasticity problem is obtained.Some specific numerical problems are provided to illustrate the effectiveness and advantages of the FEFG method for 3D elasticity by comparing the results of the FEFG method with those of the IEFG method.The convergence and relative error norm of the FEFG method for elasticity are also studied.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11171208)the Shanghai Leading Academic Discipline Project, China (Grant No. S30106)
文摘In this paper, an improved interpolating moving least-square （IIMLS） method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square （MLS） approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin （IIEFG） method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin （EFG） method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method.
基金the National Natural Science Foundation of China Under Grant No. 50779003and50639010Natural Science Foundation of Liaoning Proviance Under Grant No. 20071085
文摘An automatically adaptive element free method is presented to analyze the seismic response of liquefiable soils. The method is based on the element free Galerkin method （EFGM） and the fission procedure that is part of h-refinement, indicated by error estimation. In the proposed method, a posteriori error estimate procedure that depends on the energy norm of stress and the T-Belytschko （TB） stress recovery scheme is incorporated. The effective cyclic elasto-plastic constitutive model is used to describe the nonlinear behavior of the saturated soil. The governing equations are established by u-p formulation. The proposed method can effectively avoid the volumetric locking due to large deformation that usually occurs in numerical computations using the finite element method （FEM）. The efficiency of the proposed method is demonstrated by evaluating the seismic response of an embankment and comparing it to results obtained through FEM. It is shown that the proposed method provides an accurate seismic analysis of saturated soil that includes the effects of liquefaction .
文摘The main goal of this paper is to investigate natural convective heat transfer and flow characteristics of non-Newtonian nanofluid streaming between two infinite vertical flat plates in the presence of magnetic field and thermal radiation.Initially,a similarity transformation is used to convert momentum and energy conservation equations in partial differential forms into non-linear ordinary differential equations (ODE) applying meaningful boundary conditions.In order to obtain the non-linear ODEs analytically,Galerkin method (GM) is employed.Subsequently,the ODEs are also solved by a reliable numerical solution.In order to test the accuracy,precision and reliability of the analytical method,results of the analytical analysis are compared with the numerical results.With respect to the comparisons,fairly good compatibilities with insignificant errors are observed.Eventually,the impacts of effective parameters including magnetic and radiation parameters and nanofluid volume fraction on the velocity,skin friction coefficient and Nusselt number distributions are comprehensively described.Based on the results,it is revealed that with increasing the role of magnetic force,velocity profile,skin friction coefficient and thermal performance descend.Radiation parameter has insignificant influence on velocity profile while it obviously has augmentative and decreasing effects on skin friction and Nusselt number,respectively.
文摘This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.
基金supported by the National Basic Research Program of China (2009CB724104)the National Natural Science Foundation of China (90716010)
文摘A novel class of weighted essentially nonoscillatory （WENO） schemes based on Hermite polynomi- als, termed as HWENO schemes, is developed and applied as limiters for high order discontinuous Galerkin （DG） method on triangular grids. The developed HWENO methodology utilizes high-order derivative information to keep WENO re- construction stencils in the von Neumann neighborhood. A simple and efficient technique is also proposed to enhance the smoothness of the existing stencils, making higher-order scheme stable and simplifying the reconstruction process at the same time. The resulting HWENO-based limiters are as compact as the underlying DG schemes and therefore easy to implement. Numerical results for a wide range of flow conditions demonstrate that for DG schemes of up to fourth order of accuracy, the designed HWENO limiters can simul- taneously obtain uniform high order accuracy and sharp, es- sentially non-oscillatory shock transition.
基金supported by the National Natural Science Foundation of China (Grant No.11026223)the Shanghai Leading Academic Discipline Project,China (Grant No.S30106)the Innovation Fund Project for Graduate Student of Shanghai University,China (Grant No.SHUCX112359)
文摘In this paper, the improved complex variable moving least-squares （ICVMLS） approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares （CVMLS） approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin （EFG） method and the ICVMLS approximation, the improved complex variable element-free Galerkin （ICVEFG） method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFC method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11171208)the Shanghai Leading Academic Discipline Project, China (Grant No. S30106)the Innovation Fund Project for Graduate Student of Shanghai University,China (Grant No. SHUCX112359)
文摘In this paper, based on the element-free Galerkin （EFG） method and the improved complex variable moving least- square （ICVMLS） approximation, a new meshless method, which is the improved complex variable element-free Galerkin （ICVEFG） method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square （CVMLS） approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11171208 and U1433104)
文摘In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin（ICVEFG） method is presented for two-dimensional（2D） elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.
基金supported by the National Natural Science Foundation of China(11272152)
文摘An h-adaptive method is developed for high-order discontinuous Galerkin methods(DGM)to solve the laminar compressible Navier-Stokes(N-S)equations on unstructured mesh.The vorticity is regarded as the indicator of adaptivity.The elements where the vorticity is larger than a pre-defined upper limit are refined,and those where the vorticity is smaller than a pre-defined lower limit are coarsened if they have been refined.A high-order geometric approximation of curved boundaries is adopted to ensure the accuracy.Numerical results indicate that highly accurate numerical results can be obtained with the adaptive method at relatively low expense.
