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
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
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 s...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.展开更多
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 materi...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.展开更多
<div style="text-align:justify;"> In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the...<div style="text-align:justify;"> In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that <em>L</em><sup>2 </sup>norms error estimates can reach to order <em>k</em> + 1 by using time discretization methods. </div>展开更多
In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we...In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms,respectively.Thus,one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper.In addition,the L^(2)-norm error estimates are also proved for two schemes,whose numerical fuxes for the linear dispersive term are both dissipative type.More precisely,when choosing the approximation space with the piecewise kth degree polynomials,the error estimate provides the kth order of convergence rate in L^(2)-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term.Furthermore,the(k+1∕2)th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term.Moreover,a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations.Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.展开更多
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 the...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.展开更多
This paper develops and analyzes a new family of dual-wind discontinuous Galerkin(DG)methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations.The new DG methods are designed using...This paper develops and analyzes a new family of dual-wind discontinuous Galerkin(DG)methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations.The new DG methods are designed using the DG fnite element discrete calculus framework of[17]that defnes discrete diferential operators to replace continuous differential operators when discretizing a partial diferential equation(PDE).The proposed methods,which are non-monotone,utilize a dual-winding methodology and a new skewsymmetric DG derivative operator that,when combined,eliminate the need for choosing indeterminable penalty constants.The relationship between these new methods and the local DG methods proposed in[38]for Hamilton-Jacobi equations as well as the generalized-monotone fnite diference methods proposed in[13]and corresponding DG methods proposed in[12]for fully nonlinear second order PDEs is also examined.Admissibility and stability are established for the proposed dual-wind DG methods.The stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes.Numerical experiments are provided to gauge the performance of the new methods.展开更多
In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear conve...In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-stepis only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.展开更多
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 o...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.展开更多
The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its...The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its higher order derivatives to reduce cost and degree of difficulty in computing derivatives. Alternative methods for estimating the element size of large aspect ratio and smooth artificial viscosity are proposed to further improve robustness and accuracy of the model. Several benchmark tests are conducted, ranging from subsonic to hypersonic flows involving strong shocks. Instead of adjusting empirical parameters to achieve optimum results for each case, all tests use a constant parameter for the model with reasonable success, indicating excellent robustness of the method. The model is only limited to third-order accuracy for smooth flows. This limitation may be relaxed by using a switch or a wall function. Overall, the model is a good candidate for compressible flows with potentials of further improvement.展开更多
In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is use...In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is used to study the asymptotic behaviour of Kuramoto-Sivashinsky equation and to construct the bifurcation diagrams.展开更多
Higher order accuracy is one of the well-known beneficial properties of the discontinu-ous Galerkin(DG)method.Furthermore,many studies have demonstrated the supercon-vergence property of the semi-discrete DG method.On...Higher order accuracy is one of the well-known beneficial properties of the discontinu-ous Galerkin(DG)method.Furthermore,many studies have demonstrated the supercon-vergence property of the semi-discrete DG method.One can take advantage of this super-convergence property by post-processing techniques to enhance the accuracy of the DG solution.The smoothness-increasing accuracy-conserving(SIAC)filter is a popular post-processing technique introduced by Cockburn et al.(Math.Comput.72(242):577-606,2003).It can raise the convergence rate of the DG solution(with a polynomial of degree k)from order k+1 to order 2k+1 in the L2 norm.This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction.The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features,such as extra smoothness.Second,we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving(or even improving)its ability to enhance the accuracy of the DG solution.We prove the superconvergence error estimate of the new SIAC filters.Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters.展开更多
This paper is dedicated to the development of numerical analysis for high-order methods solving partial differential equations on scattered point clouds.We build a novel geometric error analysis framework by estimatin...This paper is dedicated to the development of numerical analysis for high-order methods solving partial differential equations on scattered point clouds.We build a novel geometric error analysis framework by estimating the error in the approximation of the Riemann metric tensor.The innovative framework serves as a fundamental tool for analyzing discontinuous Galerkin methods applied to the Laplace-Beltrami operator on possibly discontinuous geometry.We provide numerical examples of patchy surfaces reconstructed from point clouds to support our theoretical findings.展开更多
In this paper,we shall carry out the L^(2)-norm stability analysis of the Runge-Kutta discontinuous Galerkin(RKDG)methods on rectangle meshes when solving a linear constant-coefficient hyperbolic equation.The matrix t...In this paper,we shall carry out the L^(2)-norm stability analysis of the Runge-Kutta discontinuous Galerkin(RKDG)methods on rectangle meshes when solving a linear constant-coefficient hyperbolic equation.The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis.This extension looks easy for most cases;however,there are a few troubles with obtaining good stability results under a standard CFL condition,especially,for those Q^(k)-elements with lower degree k as stated in the one-dimensional case.To overcome this difficulty,we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose.In addition,an optimal error estimate on Q^(k)-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau(GGR)projection.展开更多
In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and dec...In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and decoupled,which greatly improve the computational efficiency.In order to resolve the time level mismatch of the discretization process,a special time marching method with high-order accuracy is constructed.Under the condition of slight time step constraints,the optimal error estimates of this method are given.Moreover,the theoretical results are verified by numerical experiments.Real simulations show the patterns of spots,rings,stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.展开更多
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 ...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.展开更多
This paper proposes a robust and efficient oscillation-eliminating discontinuous Galerkin(OEDG)method for solving multicomponent chemically reacting flows,which is an extension and application of the recent work[M.Pen...This paper proposes a robust and efficient oscillation-eliminating discontinuous Galerkin(OEDG)method for solving multicomponent chemically reacting flows,which is an extension and application of the recent work[M.Peng,Z.Sun,and K.Wu,Math.Comput.,2024,doi.org/10.1090/mcom/3998].Following recently developed high-order bound-preserving discontinuous Galerkin method in[J.Du and Y.Yang,J.Comput.Phys.,469(2022),111548],we incorporate an OE procedure after each Runge-–Kutta time stage to suppress spurious oscillations.The OE procedure is defined by the solution operator of a damping equation,which can be analytically solved without requiring discretization,making its implementation straightforward,non-intrusive,and efficient.Through careful design of the damping coefficients,the proposed OEDG method not only achieves the essentially non-oscillatory(ENO)property without compromising accuracy but also preserves the conservative property—an indispensable aspect of the bound-preserving technique introduced in[J.Du and Y.Yang,J.Comput.Phys.,469(2022),111548].The effectiveness and robustness of the OEDG method are demonstrated through a series of one-and two-dimensional numerical tests on the compressible Euler and Navier–Stokes equations for chemically reacting flows.These results highlight the method's capability to handle complex flow dynamics while maintaining stability and high-order accuracy.展开更多
In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least s...In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least squares(IMLS)method,which enhances the efficiency and stability of the numerical solution.The numerical solution formulas are derived using the improved element-free Galerkin(IEFG)method.We introduce the solid isotropic microstructures with penalization(SIMP)model to formulate a mathematical model for topology opti-mization,which effectively penalizes intermediate densities.The optimization problem is defined with the numerical solution formula and volume fraction as constraints.The objective function,which is the minimum value of flexibility,is optimized iteratively using the optimization criterion method to update the design variables efficiently and converge to an optimal solution.Sensitivity analysis is performed using the adjoint method,which provides accurate and efficient gradient information for the optimization algorithm.We validate the proposed framework through a series of numerical examples,including clamped beam,cantilever beam,and simply supported beam made of orthotropic materials.The convergence of the objective function is demonstrated by increasing the number of iterations.Additionally,the stability of the iterative process is analyzed by examining the fluctuation law of the volume fraction.By adjusting the parameters to an appropriate range,we achieve the final optimization results of the IEFG method without the checkerboard phenomenon.Comparative studies between the Element-Free Galerkin(EFG)and IEFG methods reveal that both methods yield consistent optimization results under identical parameter settings.However,the IEFG method significantly reduces computational time,highlighting its efficiency and suitability for orthotropic materials.展开更多
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 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The work of J.Sun and Y.Xing is partially sponsored by NSF grant DMS-1753581.
文摘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.
基金Jie Du is supported by the National Natural Science Foundation of China under Grant Number NSFC 11801302Tsinghua University Initiative Scientific Research Program+1 种基金Eric Chung is supported by Hong Kong RGC General Research Fund(Projects 14304217 and 14302018)The third author is supported by the NSF grant DMS-1818467.
文摘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.
文摘<div style="text-align:justify;"> In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that <em>L</em><sup>2 </sup>norms error estimates can reach to order <em>k</em> + 1 by using time discretization methods. </div>
基金This work was supported by the National Numerical Windtunnel Project NNW2019ZT4-B08Science Challenge Project TZZT2019-A2.3the National Natural Science Foundation of China Grant no.11871449.
文摘In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms,respectively.Thus,one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper.In addition,the L^(2)-norm error estimates are also proved for two schemes,whose numerical fuxes for the linear dispersive term are both dissipative type.More precisely,when choosing the approximation space with the piecewise kth degree polynomials,the error estimate provides the kth order of convergence rate in L^(2)-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term.Furthermore,the(k+1∕2)th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term.Moreover,a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations.Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.
文摘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.
基金The work of this author was partially supported by the NSF Grant DMS-1620168.
文摘This paper develops and analyzes a new family of dual-wind discontinuous Galerkin(DG)methods for stationary Hamilton-Jacobi equations and their vanishing viscosity regularizations.The new DG methods are designed using the DG fnite element discrete calculus framework of[17]that defnes discrete diferential operators to replace continuous differential operators when discretizing a partial diferential equation(PDE).The proposed methods,which are non-monotone,utilize a dual-winding methodology and a new skewsymmetric DG derivative operator that,when combined,eliminate the need for choosing indeterminable penalty constants.The relationship between these new methods and the local DG methods proposed in[38]for Hamilton-Jacobi equations as well as the generalized-monotone fnite diference methods proposed in[13]and corresponding DG methods proposed in[12]for fully nonlinear second order PDEs is also examined.Admissibility and stability are established for the proposed dual-wind DG methods.The stability results are shown to hold independent of the scaling of the stabilizer allowing for choices that go beyond the Godunov barrier for monotone schemes.Numerical experiments are provided to gauge the performance of the new methods.
基金the NSFC grant 11871428the Nature Science Research Program for Colleges and Universities of Jiangsu Province grant 20KJB110011Qiang Zhang:Research supported by the NSFC grant 11671199。
文摘In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-stepis only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.
文摘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.
基金Project supported by the National Natural Science Foundation of China(No.11402016)
文摘The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its higher order derivatives to reduce cost and degree of difficulty in computing derivatives. Alternative methods for estimating the element size of large aspect ratio and smooth artificial viscosity are proposed to further improve robustness and accuracy of the model. Several benchmark tests are conducted, ranging from subsonic to hypersonic flows involving strong shocks. Instead of adjusting empirical parameters to achieve optimum results for each case, all tests use a constant parameter for the model with reasonable success, indicating excellent robustness of the method. The model is only limited to third-order accuracy for smooth flows. This limitation may be relaxed by using a switch or a wall function. Overall, the model is a good candidate for compressible flows with potentials of further improvement.
文摘In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is used to study the asymptotic behaviour of Kuramoto-Sivashinsky equation and to construct the bifurcation diagrams.
基金Funding for this work was partially supported by the National Natural Science Foundation of China(NSFC)under Grant no.11801062.
文摘Higher order accuracy is one of the well-known beneficial properties of the discontinu-ous Galerkin(DG)method.Furthermore,many studies have demonstrated the supercon-vergence property of the semi-discrete DG method.One can take advantage of this super-convergence property by post-processing techniques to enhance the accuracy of the DG solution.The smoothness-increasing accuracy-conserving(SIAC)filter is a popular post-processing technique introduced by Cockburn et al.(Math.Comput.72(242):577-606,2003).It can raise the convergence rate of the DG solution(with a polynomial of degree k)from order k+1 to order 2k+1 in the L2 norm.This paper first investigates general basis functions used to construct the SIAC filter for superconvergence extraction.The generic basis function framework relaxes the SIAC filter structure and provides flexibility for more intricate features,such as extra smoothness.Second,we study the distribution of the basis functions and propose a new SIAC filter called compact SIAC filter that significantly reduces the support size of the original SIAC filter while preserving(or even improving)its ability to enhance the accuracy of the DG solution.We prove the superconvergence error estimate of the new SIAC filters.Numerical results are presented to confirm the theoretical results and demonstrate the performance of the new SIAC filters.
基金supported by National Natural Science Foundation of China(Grant Nos.12001194 and 12471402)the National Science Foundation of Hunan Province(Grant No.2024JJ5413)+1 种基金supported by the Andrew Sisson Fund,Dyason Fellowship,and the Faculty Science Researcher Development Grant of the University of Melbournesupported by National Natural Science Foundation of China(Grant No.12071244).
文摘This paper is dedicated to the development of numerical analysis for high-order methods solving partial differential equations on scattered point clouds.We build a novel geometric error analysis framework by estimating the error in the approximation of the Riemann metric tensor.The innovative framework serves as a fundamental tool for analyzing discontinuous Galerkin methods applied to the Laplace-Beltrami operator on possibly discontinuous geometry.We provide numerical examples of patchy surfaces reconstructed from point clouds to support our theoretical findings.
基金supported by the NSFC(Grant No.12301513)the Natural Science Foundation of Jiangsu Province,China(Grant No.BK20230374)+1 种基金the Natural Science Foundation of Jiangsu Higher Education Institutions of China(Grant No.23KJB110019)supported by the NSFC(Grant No.12071214).
文摘In this paper,we shall carry out the L^(2)-norm stability analysis of the Runge-Kutta discontinuous Galerkin(RKDG)methods on rectangle meshes when solving a linear constant-coefficient hyperbolic equation.The matrix transferring process based on temporal differences of stage solutions still plays an important role to achieve a nice energy equation for carrying out the energy analysis.This extension looks easy for most cases;however,there are a few troubles with obtaining good stability results under a standard CFL condition,especially,for those Q^(k)-elements with lower degree k as stated in the one-dimensional case.To overcome this difficulty,we make full use of the commutative property of the spatial DG derivative operators along two directions and set up a new proof line to accomplish the purpose.In addition,an optimal error estimate on Q^(k)-elements is also presented with a revalidation on the supercloseness property of generalized Gauss-Radau(GGR)projection.
基金supported by National Natural Science Foundation of China(Grant No.11801569)Natural Science Foundation of Shandong Province(CN)(Grant No.ZR2021MA001)the Fundamental Research Funds for the Central Universities(Grant Nos.22CX03025A and 22CX03020A).
文摘In this paper,two fully-discrete local discontinuous Galerkin(LDG)methods are applied to the growth-mediated autochemotactic pattern formation model in self-propelling bacteria.The numerical methods are linear and decoupled,which greatly improve the computational efficiency.In order to resolve the time level mismatch of the discretization process,a special time marching method with high-order accuracy is constructed.Under the condition of slight time step constraints,the optimal error estimates of this method are given.Moreover,the theoretical results are verified by numerical experiments.Real simulations show the patterns of spots,rings,stripes as well as inverted spots because of the interplay of chemotactic drift and growth rate of the cells.
文摘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.
基金supported by National Key R&D Program of China(Grant No.2021YFA0719200)supported by the Shenzhen Science and Technology Program(Grant No.RCJC20221008092757098)supported by the Postdoctoral Science Foundation of China(Grant No.2024M751284)。
文摘This paper proposes a robust and efficient oscillation-eliminating discontinuous Galerkin(OEDG)method for solving multicomponent chemically reacting flows,which is an extension and application of the recent work[M.Peng,Z.Sun,and K.Wu,Math.Comput.,2024,doi.org/10.1090/mcom/3998].Following recently developed high-order bound-preserving discontinuous Galerkin method in[J.Du and Y.Yang,J.Comput.Phys.,469(2022),111548],we incorporate an OE procedure after each Runge-–Kutta time stage to suppress spurious oscillations.The OE procedure is defined by the solution operator of a damping equation,which can be analytically solved without requiring discretization,making its implementation straightforward,non-intrusive,and efficient.Through careful design of the damping coefficients,the proposed OEDG method not only achieves the essentially non-oscillatory(ENO)property without compromising accuracy but also preserves the conservative property—an indispensable aspect of the bound-preserving technique introduced in[J.Du and Y.Yang,J.Comput.Phys.,469(2022),111548].The effectiveness and robustness of the OEDG method are demonstrated through a series of one-and two-dimensional numerical tests on the compressible Euler and Navier–Stokes equations for chemically reacting flows.These results highlight the method's capability to handle complex flow dynamics while maintaining stability and high-order accuracy.
基金supported by the Graduate Student Scientific Research Innovation Project through Research Innovation Fund for Graduate Students in Shanxi Province(Project No.2024KY648).
文摘In this paper,we develop an advanced computational framework for the topology optimization of orthotropic materials using meshless methods.The approximation function is established based on the improved moving least squares(IMLS)method,which enhances the efficiency and stability of the numerical solution.The numerical solution formulas are derived using the improved element-free Galerkin(IEFG)method.We introduce the solid isotropic microstructures with penalization(SIMP)model to formulate a mathematical model for topology opti-mization,which effectively penalizes intermediate densities.The optimization problem is defined with the numerical solution formula and volume fraction as constraints.The objective function,which is the minimum value of flexibility,is optimized iteratively using the optimization criterion method to update the design variables efficiently and converge to an optimal solution.Sensitivity analysis is performed using the adjoint method,which provides accurate and efficient gradient information for the optimization algorithm.We validate the proposed framework through a series of numerical examples,including clamped beam,cantilever beam,and simply supported beam made of orthotropic materials.The convergence of the objective function is demonstrated by increasing the number of iterations.Additionally,the stability of the iterative process is analyzed by examining the fluctuation law of the volume fraction.By adjusting the parameters to an appropriate range,we achieve the final optimization results of the IEFG method without the checkerboard phenomenon.Comparative studies between the Element-Free Galerkin(EFG)and IEFG methods reveal that both methods yield consistent optimization results under identical parameter settings.However,the IEFG method significantly reduces computational time,highlighting its efficiency and suitability for orthotropic materials.
