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.

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.

Modified Burgers' equation by the local discontinuous Galerkin method

In this paper,we present the local discontinuous Galerkin method for solving Burgers＇ equation and the modified Burgers＇ equation.We describe the algorithm formulation and practical implementation of the local discontinuous Galerkin method in detail.The method is applied to the solution of the one-dimensional viscous Burgers＇ equation and two forms of the modified Burgers＇ equation.The numerical results indicate that the method is very accurate and efficient.

Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations

In the current work, we extend the local discontinuous Galerkin method to a more general application system. The Burgers and coupled Burgers equations are solved by the local discontinuous Galerkin method. Numerical experiments are given to verify the efficiency and accuracy of our method. Moreover the numerical results show that the method can approximate sharp fronts accurately with minimal oscillation.

New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method

Currently, many studies on the local discontinuous Galerkin method focus on the Cartesian grid with low computational e ciency and poor adaptability to complex shapes. A new immersed boundary method is presented, and this method employs the adaptive Cartesian grid to improve the adaptability to complex shapes and the immersed boundary to increase computational e ciency. The new immersed boundary method employs different boundary cells(the physical cell and ghost cell) to impose the boundary condition and the reconstruction algorithm of the ghost cell is the key for this method. The classical model elliptic equation is used to test the method. This method is tested and analyzed from the viewpoints of boundary cell type, error distribution and accuracy. The numerical result shows that the presented method has low error and a good rate of the convergence and works well in complex geometries. The method has good prospect for practical application research of the numerical calculation research.

Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System

Boussinesq type equations have been widely studied to model the surface water wave.In this paper,we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system,the BBM-BBM system,the Bona-Smith system,etc.We propose local discontinuous Galerkin(LDG)methods,with carefully chosen numerical fluxes,to numerically solve this abcd Boussinesq system.The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a,b,c,d.Numerical experiments are shown to test the convergence rates,and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.

Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

In this paper, we study the classical Allen-Cahn equations and investigate the maximum- principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen- Cahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its flexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to dem-onstrate the energy decay property. Moreover, due to the stiff source given in the equation, we use the conservative modified exponential Runge-Kutta methods and thus can use rela-tively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

Local Discontinuous Galerkin Method for the Time-Fractional KdV Equation with the Caputo-Fabrizio Fractional Derivative

This paper studies the time-fractional Korteweg-de Vries (KdV) equations with Caputo-Fabrizio fractional derivatives. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. Stability and convergence are demonstrated by a specific choice of numerical fluxes. Finally, the efficiency and accuracy of the scheme are verified by numerical experiments.

LOCAL DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC INTERFACE PROBLEMS

In this paper,the minimal dissipation local discontinuous Galerkin method is studied to solve the elliptic interface problems in two-dimensional domains.The interface may be arbitrary smooth curves.It is shown that the error estimates in L;-norm for the solution and the flux are O（h;|log h|）and O(h|log h|;),respectively.In numerical experiments,the successive substitution iterative methods are used to solve the LDG schemes.Numerical results verify the efficiency and accuracy of the method.

LOCAL DISCONTINUOUS GALERKIN METHOD FOR RADIAL POROUS FLOW WITH DISPERSION AND ADSORPTION

Based on the local discontinuous Galerkin methods for time-dependent convection-diffusion systems newly developed by Corkburn and Shu,according to the form of the generalized convection-diffusion equations which model the radial porous flow with dispersion and adsorption,a local discontinuous Galerkin method for radial porous flow with dispersion and adsorption was developed,a high order accurary new scheme for radial porous flow is obtained.The presented method was applied to the numerical tests of two cases of radial porous,i.e., the convection-dispersion flow and the convection-dispersion-adsorption flow,the corresponding parts of the numerical results are in good agreement with the published solutions,so the presented method is reliable.Reckoning of the computational cost also shows that the method is practicable.

Meshless Local Discontinuous Petrov-Galerkin Method with Application to Blasting Problems

A meshiess local discontinuous Petrov-Galerkin （MLDPG） method based on the local symmetric weak form （LSWF） is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin （RKDG） method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional nonoverlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method （MM）, and it can handle the problems with extremely complicated physics and geometries easily. Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes.

Local Discontinuous Galerkin Methods with Novel Basis for Fractional Diffusion Equations with Non-smooth Solutions

In this paper,we develop novel local discontinuous Galerkin(LDG)methods for fractional diffusion equations with non-smooth solutions.We consider such problems,for which the solutions are not smooth at boundary,and therefore the traditional LDG methods with piecewise polynomial solutions suffer accuracy degeneracy.The novel LDG methods utilize a solution information enriched basis,simulate the problem on a paired special mesh,and achieve optimal order of accuracy.We analyze the L2 stability and optimal error estimate in L2-norm.Finally,numerical examples are presented for validating the theoretical conclusions.

Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second‑Order Elliptic Problems on Cartesian Grids

This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.

A Local Discontinuous Galerkin Method with Generalized Alternating Fluxes for 2D Nonlinear Schrödinger Equations

In this paper,we consider the local discontinuous Galerkin method with generalized alter-nating numerical fluxes for two-dimensional nonlinear Schrödinger equations on Carte-sian meshes.The generalized fluxes not only lead to a smaller magnitude of the errors,but can guarantee an energy conservative property that is useful for long time simulations in resolving waves.By virtue of generalized skew-symmetry property of the discontinuous Galerkin spatial operators,two energy equations are established and stability results con-taining energy conservation of the prime variable as well as auxiliary variables are shown.To derive optimal error estimates for nonlinear Schrödinger equations,an additional energy equation is constructed and two a priori error assumptions are used.This,together with properties of some generalized Gauss-Radau projections and a suitable numerical initial condition,implies optimal order of k+1.Numerical experiments are given to demonstrate the theoretical results.

A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations

For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.

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。

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.

Local Discontinuous Galerkin Methods with Decoupled Implicit-Explicit Time Marching for the Growth-Mediated Autochemotactic Pattern Formation Model

In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and decoupled,which greatly improve the computational efficiency.In order to resolve the time level mismatch of the discretization process,a special time marching method with high-order accuracy is constructed.Under the condition of slight time step constraints,the optimal error estimates of this method are given.Moreover,the theoretical results are verified by numerical experiments.Real simulations show the patterns of spots,rings,stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.

Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.

Caputo型时间分数阶变系数扩散方程的局部间断Galerkin方法

提出一种带有Caputo导数的时间分数阶变系数扩散方程的数值解法.方程的解在初始时刻附近通常具有弱正则性,采用非一致网格上的L1公式离散时间分数阶导数,并使用局部间断Galerkin(local discontinuous Galerkin,LDG)方法离散空间导数,给出方程的全离散格式.基于离散的分数阶Gronwall不等式,证明了格式的数值稳定性和收敛性,且所得结果关于α是鲁棒的,即当α→1^(-)时不会发生爆破.最后,通过数值算例验证理论分析的结果.