High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.

Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

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.

Superconvergence of Direct Discontinuous Galerkin Methods:Eigen-structure Analysis Based on Fourier Approach

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.

Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

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.

Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

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.

A Local Macroscopic Conservative(LoMaC)Low Rank Tensor Method with the Discontinuous Galerkin Method for the Vlasov Dynamics

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.

Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods

We extend the monolithic convex limiting(MCL)methodology to nodal discontinuous Galerkin spectral-element methods(DGSEMS).The use of Legendre-Gauss-Lobatto(LGL)quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain-preserving high-resolution schemes.Compared to many other continuous and discontinuous Galerkin method variants,a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcellflux discretization.Representing a highorder spatial semi-discretization in terms of intermediate states,we performflux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains.In addition,local bounds may be imposed on scalar quantities of interest.In contrast to limiting approaches based on predictor-corrector algorithms,our MCL procedure for LGL-DGSEM yields nonlinearflux approximations that are independent of the time-step size and can be further modified to enforce entropy stability.To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations,we run simulations for challenging setups featuring strong shocks,steep density gradients,and vortex dominatedflows.

一种基于局部间断Galerkin方法的IC互连线电容提取策略

求解椭圆方程的局部间断Galerkin(LDG)方法具有精度高、并行效率高的优点,且能适用于各种网格。文章提出采用LDG方法来求解IC版图中电势分布函数满足的Laplace方程,从而给出了一个提取互连线电容的新方法。该问题的求解区域需要在矩形区域内部去掉数量不等的导体区域,在这种特殊的计算区域上,通过数值测试验证了LDG方法能达到理论的收敛阶。随着芯片制造工艺的发展,导体尺寸和间距也越来越小,给数值模拟带来新的问题。文章采用倍增网格剖分方法,大幅减小了计算单元数。对包含不同数量和形状导体的七个电路版图,用新方法提取互连线电容,得到的结果与商业工具给出的结果非常接近,表明了新方法的有效性。

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.

Solving elastic wave equations in 2D transversely isotropic media by a weighted Runge-Kutta discontinuous Galerkin method

Accurate wave propagation simulation in anisotropic media is important for forward modeling, migration and inversion. In this study, the weighted Runge-Kutta discontinuous Galerkin (RKDG) method is extended to solve the elastic wave equations in 2D transversely isotropic media. The spatial discretization is based on the numerical flux discontinuous Galerkin scheme. An explicit weighted two-step iterative Runge-Kutta method is used as time-stepping algorithm. The weighted RKDG method has good flexibility and applicability of dealing with undulating geometries and boundary conditions. To verify the correctness and effectiveness of this method, several numerical examples are presented for elastic wave propagations in vertical transversely isotropic and tilted transversely isotropic media. The results show that the weighted RKDG method is promising for solving wave propagation problems in complex anisotropic medium.

Numerical manifold method for thermo-mechanical coupling simulation of fractured rock mass

As a calculation method based on the Galerkin variation,the numerical manifold method(NMM)adopts a double covering system,which can easily deal with discontinuous deformation problems and has a high calculation accuracy.Aiming at the thermo-mechanical(TM)coupling problem of fractured rock masses,this study uses the NMM to simulate the processes of crack initiation and propagation in a rock mass under the influence of temperature field,deduces related system equations,and proposes a penalty function method to deal with boundary conditions.Numerical examples are employed to confirm the effectiveness and high accuracy of this method.By the thermal stress analysis of a thick-walled cylinder(TWC),the simulation of cracking in the TWC under heating and cooling conditions,and the simulation of thermal cracking of the SwedishÄspöPillar Stability Experiment(APSE)rock column,the thermal stress,and TM coupling are obtained.The numerical simulation results are in good agreement with the test data and other numerical results,thus verifying the effectiveness of the NMM in dealing with thermal stress and crack propagation problems of fractured rock masses.

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

A symplectic finite element method based on Galerkin discretization for solving linear systems

We propose a novel symplectic finite element method to solve the structural dynamic responses of linear elastic systems.For the dynamic responses of continuous medium structures,the traditional numerical algorithm is the dissipative algorithm and cannot maintain long-term energy conservation.Thus,a symplectic finite element method with energy conservation is constructed in this paper.A linear elastic system can be discretized into multiple elements,and a Hamiltonian system of each element can be constructed.The single element is discretized by the Galerkin method,and then the Hamiltonian system is constructed into the Birkhoffian system.Finally,all the elements are combined to obtain the vibration equation of the continuous system and solved by the symplectic difference scheme.Through the numerical experiments of the vibration response of the Bernoulli-Euler beam and composite plate,it is found that the vibration response solution and energy obtained with the algorithm are superior to those of the Runge-Kutta algorithm.The results show that the symplectic finite element method can keep energy conservation for a long time and has higher stability in solving the dynamic responses of linear elastic systems.

Fourier Continuation Discontinuous Galerkin Methods for Linear Hyperbolic Problems

Fourier continuation(FC)is an approach used to create periodic extensions of non-periodic functions to obtain highly-accurate Fourier expansions.These methods have been used in partial differential equation(PDE)-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments.Discontinuous Galerkin(DG)methods are increasingly used for solving PDEs and,as all Galerkin formulations,come with a strong framework for proving the stability and the convergence.Here we propose the use of FC in forming a new basis for the DG framework.

间断Galerkin有限元隐式算法GPU并行化研究

为了提高间断伽辽金(discontinuous Galerkin,DG)有限元方法的计算效率,围绕求解Euler方程,构建了基于图形处理器(graphics processing unit,GPU)并行加速的隐式DG算法。算法结合Roe格式进行空间离散,采用人工黏性法处理激波等间断问题,时间推进选用下上对称高斯-赛德尔(lower-upper symmetric Gauss-Seidel,LU-SGS)隐式格式。为了克服传统隐式格式固有的数据关联依赖问题,借助于本文提出的面向任意网格的单元着色分组技术,先给出了LUSGS隐式格式的并行化改造,使得隐式时间推进能按颜色组别依次并行,由于同一颜色组内算法已不存在数据关联,可以据此实现并行化。在此基础上,再结合DG算法局部紧致等特点,基于统一计算设备架构(compute unified device architecture,CUDA)编程模型,设计了依据单元的核函数,并构建了对应的线程与数据结构,给出了DG有限元隐式GPU并行算法。最后,发展的算法通过了多个二维和三维典型流动算例考核与性能测试,展示出隐式算法GPU加速的效果,且获得的计算结果能与现有的文献或实验数据接近。

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomials are used as basis functions in the assumed solution employed. Numerical examples for some selected problems are provided and the results obtained show that the Galerkin method with orthogonal polynomials as basis functions performed creditably well in terms of absolute errors obtained.

对流扩散方程的隐式全离散局部间断Galerkin方法

研究了对流扩散方程的隐式全离散局部间断Galerkin方法的稳定性和误差分析.将三阶隐式Runge-Kutta时间离散和具有广义交替数值流通量的LDG方法相结合得到全离散LDG格式,通过广义交替数值流通量,建立数值解和辅助解内积之间的关系,证明了全离散LDG格式的无条件稳定,同时引入广义Gauss-Radau投影,通过投影的逼近性质和一些基本不等式建立了数值方法的最优误差估计,最后通过数值实验验证该方法理论分析的正确性.

Nonlinear Flap-Wise Vibration Characteristics ofWind Turbine Blades Based onMulti-Scale AnalysisMethod

This work presents a novel approach to achieve nonlinear vibration response based on the Hamilton principle.We chose the 5-MW reference wind turbine which was established by the National Renewable Energy Laboratory(NREL),to research the effects of the nonlinear flap-wise vibration characteristics.The turbine wheel is simplified by treating the blade of a wind turbine as an Euler-Bernoulli beam,and the nonlinear flap-wise vibration characteristics of the wind turbine blades are discussed based on the simplification first.Then,the blade’s large-deflection flap-wise vibration governing equation is established by considering the nonlinear term involving the centrifugal force.Lastly,it is truncated by the Galerkin method and analyzed semi-analytically using the multi-scale analysis method,and numerical simulations are carried out to compare the simulation results of finite elements with the numerical simulation results using Campbell diagram analysis of blade vibration.The results indicated that the rotational speed of the impeller has a significant impact on blade vibration.When the wheel speed of 12.1 rpm and excitation amplitude of 1.23 the maximum displacement amplitude of the blade has increased from 0.72 to 3.16.From the amplitude-frequency curve,it can be seen that the multi-peak characteristic of blade amplitude frequency is under centrifugal nonlinearity.Closed phase trajectories in blade nonlinear vibration,exhibiting periodic motion characteristics,are found through phase diagrams and Poincare section diagrams.